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Chemistry Problems

Use chemistry problems as a tool for mastering chemistry concepts. Some of these examples show using formulas while others include lists of examples.

Acids, Bases, and pH Chemistry Problems

Learn about acids and bases. See how to calculate pH, pOH, K a , K b , pK a , and pK b .

  • Practice calculating pH.
  • Get example pH, pK a , pK b , K a , and K b calculations.
  • Get examples of amphoterism.

Atomic Structure Problems

Learn about atomic mass, the Bohr model, and the part of the atom.

  • Practice identifying atomic number, mass number, and atomic mass.
  • Get examples showing ways to find atomic mass.
  • Use Avogadro’s number and find the mass of a single atom .
  • Review the Bohr model of the atom.
  • Find the number of valence electrons of an element’s atom.

Chemical Bonds

Learn how to use electronegativity to determine whether atoms form ionic or covalent bonds. See chemistry problems drawing Lewis structures.

  • Identify ionic and covalent bonds.
  • Learn about ionic compounds and get examples.
  • Practice identifying ionic compounds.
  • Get examples of binary compounds.
  • Learn about covalent compounds and their properties.
  • See how to assign oxidation numbers.
  • Practice drawing Lewis structures.
  • Practice calculating bond energy.

Chemical Equations

Practice writing and balancing chemical equations.

  • Learn the steps of balancing equations.
  • Practice balancing chemical equations (practice quiz).
  • Get examples finding theoretical yield.
  • Practice calculating percent yield.
  • Learn to recognize decomposition reactions.
  • Practice recognizing synthesis reactions.
  • Practice recognizing single replacement reactions.
  • Recognize double replacement reactions.
  • Find the mole ratio between chemical species in an equation.

Concentration and Solutions

Learn how to calculate concentration and explore chemistry problems that affect chemical concentration, including freezing point depression, boiling point elevation, and vapor pressure elevation.

  • Get example concentration calculations in several units.
  • Practice calculating normality (N).
  • Practice calculating molality (m).
  • Explore example molarity (M) calculations.
  • Get examples of colligative properties of solutions.
  • See the definition and examples of saturated solutions.
  • See the definition and examples of unsaturated solutions.
  • Get examples of miscible and immiscible liquids.

Error Calculations

Learn about the types of error and see worked chemistry example problems.

  • See how to calculate percent.
  • Practice absolute and relative error calculations.
  • See how to calculate percent error.
  • See how to find standard deviation.
  • Calculate mean, median, and mode.
  • Review the difference between accuracy and precision.

Equilibrium Chemistry Problems

Learn about Le Chatelier’s principle, reaction rates, and equilibrium.

  • Solve activation energy chemistry problems.
  • Review factors that affect reaction rate.
  • Practice calculating the van’t Hoff factor.

Practice chemistry problems using the gas laws, including Raoult’s law, Graham’s law, Boyle’s law, Charles’ law, and Dalton’s law of partial pressures.

  • Calculate vapor pressure.
  • Solve Avogadro’s law problems.
  • Practice Boyle’s law problems.
  • See Charles’ law example problems.
  • Solve combined gas law problems.
  • Solve Gay-Lussac’s law problems.

Some chemistry problems ask you identify examples of states of matter and types of mixtures. While there are any chemical formulas to know, it’s still nice to have lists of examples.

  • Practice density calculations.
  • Identify intensive and extensive properties of matter.
  • See examples of intrinsic and extrinsic properties of matter.
  • Get the definition and examples of solids.
  • Get the definition and examples of gases.
  • See the definition and examples of liquids.
  • Learn what melting point is and get a list of values for different substances.
  • Get the azeotrope definition and see examples.
  • See how to calculate specific volume of a gas.
  • Get examples of physical properties of matter.
  • Get examples of chemical properties of matter.
  • Review the states of matter.

Molecular Structure Chemistry Problems

See chemistry problems writing chemical formulas. See examples of monatomic and diatomic elements.

  • Practice empirical and molecular formula problems.
  • Practice simplest formula problems.
  • See how to calculate molecular mass.
  • Get examples of the monatomic elements.
  • See examples of binary compounds.
  • Calculate the number of atoms and molecules in a drop of water.

Nomenclature

Practice chemistry problems naming ionic compounds, hydrocarbons, and covalent compounds.

  • Practice naming covalent compounds.
  • Learn hydrocarbon prefixes in organic chemistry.

Nuclear Chemistry

These chemistry problems involve isotopes, nuclear symbols, half-life, radioactive decay, fission, fusion.

  • Review the types of radioactive decay.

Periodic Table

Learn how to use a periodic table and explore periodic table trends.

  • Know the trends in the periodic table.
  • Review how to use a periodic table.
  • Explore the difference between atomic and ionic radius and see their trends on the periodic table.

Physical Chemistry

Explore thermochemistry and physical chemistry, including enthalpy, entropy, heat of fusion, and heat of vaporization.

  • Practice heat of vaporization chemistry problems.
  • Practice heat of fusion chemistry problems.
  • Calculate heat required to turn ice into steam.
  • Practice calculating specific heat.
  • Get examples of potential energy.
  • Get examples of kinetic energy.
  • See example activation energy calculations.

Spectroscopy and Quantum Chemistry Problems

See chemistry problems involving the interaction between light and matter.

  • Calculate wavelength from frequency or frequency from wavelength.

Stoichiometry Chemistry Problems

Practice chemistry problems balancing formulas for mass and charge. Learn about reactants and products.

  • Get example mole ratio problems.
  • Calculate percent yield.
  • Learn how to assign oxidation numbers.
  • Get the definition and examples of reactants in chemistry.
  • Get the definition and examples of products in chemical reactions.

Unit Conversions

There are some many examples of unit conversions that they have their own separate page!

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How to Solve a Chemistry Problem

Last Updated: February 15, 2024

This article was co-authored by Anne Schmidt . Anne Schmidt is a Chemistry Instructor in Wisconsin. Anne has been teaching high school chemistry for over 20 years and is passionate about providing accessible and educational chemistry content. She has over 9,000 subscribers to her educational chemistry YouTube channel. She has presented at the American Association of Chemistry Teachers (AATC) and was an Adjunct General Chemistry Instructor at Northeast Wisconsin Technical College. Anne was published in the Journal of Chemical Education as a Co-Author, has an article in ChemEdX, and has presented twice and was published with the AACT. Anne has a BS in Chemistry from the University of Wisconsin, Oshkosh, and an MA in Secondary Education and Teaching from Viterbo University. This article has been viewed 17,961 times.

Chemistry problems can vary in many different ways. Some questions are conceptual and others are quantitative. Each problem requires its own approach, and each has a different way to solve it correctly. What you can do is make a set of steps that can help us with any problems that you come across in the field of chemistry. Using these steps should help give you a guideline to working on any chemistry problem you encounter.

Starting the Problem

Step 1 Read the problem completely.

Finishing the Problem

Step 1 Check your units again.

Expert Q&A

Anne Schmidt

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Thanks for reading our article! If you’d like to learn more about chemistry, check out our in-depth interview with Anne Schmidt .

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Marking a Milestone: Four Years of Daily Study Groups

From data to discovery: studying computational biology with wolfram, navigating quantum computing: accelerating next-generation innovation, chemistry step-by-step solutions: chemical reactions.

Chemistry Step-by-Step Solutions: Chemical Reactions

If you’re studying chemistry or are in a discipline requiring chemistry prerequisite courses, then you know how expensive the required textbooks can be. To combat this, the chemical education community has developed open educational resources to provide free chemistry textbooks. However, although free textbooks keep cash in your wallet, they don’t include solution guides for all the homework problems.

Luckily, the Step-by-Step Solutions feature of Wolfram|Alpha has got your back! Whether you’re studying remotely or collaborating via video conferencing, Wolfram|Alpha helps you learn and apply the problem-solving frameworks for chemical word problems. The step-by-step solutions provide stepwise solution guides that can be viewed one step at a time or all at once. The guides not only hone efficient problem solving, but also facilitate digging deeper into concepts that might still be murky.

Over the next few weeks, we’ll be exploring some of the popular topics that middle-school, high-school and college students encounter in their chemistry courses and final exams: chemical reactions, structure and bonding , chemical solutions , and finally, quantum chemistry . Read on for example problems in chemical reactions and their step-by-step solutions!

Balancing Chemical Equations

A fundamental aspect of chemistry is balancing chemical equations . If chemical equations are the language in which chemical processes are expressed, then balancing chemical equations is the corresponding grammar. The step-by-step solution walks you through a robust algebraic approach to identifying the stoichiometric coefficients.

Example Problem

Write the balanced equation for the reaction of copper with nitric acid to produce copper nitrate, nitrogen oxide and water.

Step-by-Step Solution

For this class of problem, just enter “ balance copper + nitric acid -> copper nitrate + nitrogen dioxide + water ”.

"balance copper + nitric acid -> copper nitrate + nitrogen dioxide + water"

After balancing the related chemical equations, the next step in planning a laboratory experiment is computing how much of each reactant must be measured out. To do this, one needs the molar mass for each reactant. Step-by-step solutions are available for the molecular mass and relative molecular mass in addition to the molar mass . In all cases, a general framework for solving these types of problems is provided via the Plan step. Details of which formula to use and how to gather the necessary information are provided.

Calculate the molar mass of silver sulfate, Ag 2 SO 4 .

In this case, just enter “ molar mass silver sulfate ”.

"molar mass silver sulfate"

Mass Composition

One way to analyze individual chemicals is to compute and compare the mass and atom percentages. The step-by-step solution provides a general framework for solving this class of problem in the Plan step. Details of the relevant equations, as well as how to compute the necessary intermediate values, are provided. Ways in which you can check your work during the calculations are also available via the “Show intermediate steps” buttons.

Antihemophilic factor is a coagulant with the formula C 11794 H 18314 N 3220 O 355 S 83 . What is its percent composition?

For the answer, just enter “ antihemophilic factor elemental composition ”.

"antihemophilic factor elemental composition"

Chemical Conversions

Chemical conversions crop up in nearly every chemistry homework or research problem. As such, step-by-step solutions are available for converting among moles , mass , volume , molecules and atoms . Unit conversions and dimensional analysis details are provided.

How many atoms are in five milliliters of a 1.5 mM magnesium hydroxide solution?

To solve this, just enter “ convert 5 mL of 1.5 mM magnesium hydroxide to atoms ”.

"convert 5 mL of 1.5 mM magnesium hydroxide to atoms"

Stoichiometry

After running a chemical reaction, one often wants to know how the reaction went by computing the reaction yields . Step-by-step solutions are available for computing the amount of reactants needed and the theoretical yield in addition to the percent yield . The use of stoichiometric factors to generate the desired values is explained in detail.

Upon reaction of 1.274 grams of copper sulfate with excess zinc metal, 0.392 grams of copper metal was obtained according to the following equation: CuSO 4 (aq)+Zn(s)⟶Cu(s)+ZnSO 4 (aq). What is the percent yield?

To find the percent yield, just append the mass values to the corresponding chemical species and ask for the stoichiometry, “ 1.274 g CuSO4 + Zn -> 0.392 g Cu + ZnSO4 stoichiometry ”.

"1.274 g CuSO4 + Zn -> 0.392 g Cu + ZnSO4 stoichiometry"

Challenge Problems

Test your chemical reaction problem-solving skills by using the Wolfram|Alpha tools described to solve these word problems. Answers will be provided in the next blog post in this series.

  • Compute the molecular mass of acetaminophen. Is the element with the largest atom count also the element with the largest mass percent?
  • What is the limiting reactant and theoretical yield when 24.8 grams of white phosphorus and 0.200 moles of oxygen react to form 10.0 grams of phosphorus pentoxide?

And More Chemistry to Come

Whether you’re studying for upcoming final exams, puzzling out homework or just looking for a refresher, chemical reactions are one of many chemistry topics covered by the Wolfram|Alpha knowledgebase. Next week we’ll cover step-by-step solutions for chemical solutions , followed by structure and bonding , and then quantum chemistry . If you have suggestions for other step-by-step content (in chemistry or other subjects), please let us know! You can reach us by leaving a comment below or sending in feedback at the bottom of any Wolfram|Alpha query page.

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Problems and Problem Solving in Chemistry Education: Analysing Data, Looking for Patterns and Making Deductions

Problem solving is central to the teaching and learning of chemistry at secondary, tertiary and post-tertiary levels of education, opening to students and professional chemists alike a whole new world for analysing data, looking for patterns and making deductions. As an important higher-order thinking skill, problem solving also constitutes a major research field in science education. Relevant education research is an ongoing process, with recent developments occurring not only in the area of quantitative/computational problems, but also in qualitative problem solving.

The following situations are considered, some general, others with a focus on specific areas of chemistry: quantitative problems, qualitative reasoning, metacognition and resource activation, deconstructing the problem-solving process, an overview of the working memory hypothesis, reasoning with the electron-pushing formalism, scaffolding organic synthesis skills, spectroscopy for structural characterization in organic chemistry, enzyme kinetics, problem solving in the academic chemistry laboratory, chemistry problem-solving in context, team-based/active learning, technology for molecular representations, IR spectra simulation, and computational quantum chemistry tools. The book concludes with methodological and epistemological issues in problem solving research and other perspectives in problem solving in chemistry.

With a foreword by George Bodner.

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Problems and Problem Solving in Chemistry Education: Analysing Data, Looking for Patterns and Making Deductions, The Royal Society of Chemistry, 2021.

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  • Front Matter
  • Acknowledgments
  • Author Biographies
  • Chapter 1: Introduction − The Many Types and Kinds of Chemistry Problems p1-14 By Georgios Tsaparlis Georgios Tsaparlis University of Ioannina, Department of Chemistry Ioannina Greece [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 1: Introduction − The Many Types and Kinds of Chemistry Problems in another window
  • Chapter 2: Qualitative Reasoning in Problem-solving in Chemistry p15-37 By Vicente Talanquer Vicente Talanquer Department of Chemistry and Biochemistry, University of Arizona Tucson AZ 85721 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 2: Qualitative Reasoning in Problem-solving in Chemistry in another window
  • Chapter 3: Scaffolding Metacognition and Resource Activation During Problem Solving: A Continuum Perspective p38-67 By Nicole Graulich ; Nicole Graulich Justus-Liebig-Universität Gießen Germany Search for other works by this author on: This Site PubMed Google Scholar Axel Langner ; Axel Langner Justus-Liebig-Universität Gießen Germany Search for other works by this author on: This Site PubMed Google Scholar Kimberly Vo ; Kimberly Vo Monash University Australia [email protected] Search for other works by this author on: This Site PubMed Google Scholar Elizabeth Yuriev Elizabeth Yuriev Monash University Australia [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 3: Scaffolding Metacognition and Resource Activation During Problem Solving: A Continuum Perspective in another window
  • Chapter 4: Deconstructing the Problem-solving Process: Beneath Assigned Points and Beyond Traditional Assessment p68-92 By Ozcan Gulacar ; Ozcan Gulacar University of California, Davis, Department of Chemistry One Shields Avenue Davis CA 95616 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Charlie Cox ; Charlie Cox Duke University, Department of Chemistry Box 90346, 128 Science Drive Durham NC 27708-0346 USA Search for other works by this author on: This Site PubMed Google Scholar Herb Fynewever Herb Fynewever Calvin University, Department of Chemistry 3201 Burton SE Grand Rapids MI 49546 USA Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 4: Deconstructing the Problem-solving Process: Beneath Assigned Points and Beyond Traditional Assessment in another window
  • Chapter 5: It Depends on the Problem and on the Solver: An Overview of the Working Memory Overload Hypothesis, Its Applicability and Its Limitations p93-126 By Georgios Tsaparlis Georgios Tsaparlis University of Ioannina, Department of Chemistry Ioannina Greece [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 5: It Depends on the Problem and on the Solver: An Overview of the Working Memory Overload Hypothesis, Its Applicability and Its Limitations in another window
  • Chapter 6: Mechanistic Reasoning Using the Electron-pushing Formalism p127-144 By Gautam Bhattacharyya Gautam Bhattacharyya Missouri State University, Department of Chemistry 901 South National Avenue Springfield MO 65897 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 6: Mechanistic Reasoning Using the Electron-pushing Formalism in another window
  • Chapter 7: Scaffolding Synthesis Skills in Organic Chemistry p145-165 By Alison B. Flynn Alison B. Flynn Department of Chemistry and Biomolecular Sciences, University of Ottawa 10 Marie Curie Ottawa Ontario K1N 6N5 Canada [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 7: Scaffolding Synthesis Skills in Organic Chemistry in another window
  • Chapter 8: Problem Solving Using NMR and IR Spectroscopy for Structural Characterization in Organic Chemistry p166-198 By Megan C. Connor ; Megan C. Connor Department of Chemistry, University of Michigan Ann Arbor Michigan USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Ginger V. Shultz Ginger V. Shultz Department of Chemistry, University of Michigan Ann Arbor Michigan USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 8: Problem Solving Using NMR and IR Spectroscopy for Structural Characterization in Organic Chemistry in another window
  • Chapter 9: Assessing System Ontology in Biochemistry: Analysis of Students’ Problem Solving in Enzyme Kinetics p199-216 By Jon-Marc G. Rodriguez ; Jon-Marc G. Rodriguez University of Iowa, Department of Chemistry E355 Chemistry Building Iowa City Iowa 52242-1294 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Sven J. Philips ; Sven J. Philips Purdue University, Department of Chemistry 560 Oval Drive West Lafayette IN 47907 USA Search for other works by this author on: This Site PubMed Google Scholar Nicholas P. Hux ; Nicholas P. Hux Purdue University, Department of Chemistry 560 Oval Drive West Lafayette IN 47907 USA Search for other works by this author on: This Site PubMed Google Scholar Marcy H. Towns Marcy H. Towns Purdue University, Department of Chemistry 560 Oval Drive West Lafayette IN 47907 USA Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 9: Assessing System Ontology in Biochemistry: Analysis of Students’ Problem Solving in Enzyme Kinetics in another window
  • Chapter 10: Problem Solving in the Chemistry Teaching Laboratory: Is This Something That Happens? p217-252 By Ian Hawkins ; Ian Hawkins Welch College Gallatin TN 37066 USA Search for other works by this author on: This Site PubMed Google Scholar Vichuda K. Hunter ; Vichuda K. Hunter Middle Tennessee State University, Department of Chemistry PO Box 68 Murfreesboro TN 37132 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Michael J. Sanger ; Michael J. Sanger Middle Tennessee State University, Department of Chemistry PO Box 68 Murfreesboro TN 37132 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Amy J. Phelps Amy J. Phelps Middle Tennessee State University, Department of Chemistry PO Box 68 Murfreesboro TN 37132 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 10: Problem Solving in the Chemistry Teaching Laboratory: Is This Something That Happens? in another window
  • Chapter 11: Problems and Problem Solving in the Light of Context-based Chemistry p253-278 By Karolina Broman Karolina Broman Umeå University Sweden [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 11: Problems and Problem Solving in the Light of Context-based Chemistry in another window
  • Chapter 12: Using Team Based Learning to Promote Problem Solving Through Active Learning p279-319 By Natalie J. Capel ; Natalie J. Capel Keele University UK [email protected] Search for other works by this author on: This Site PubMed Google Scholar Laura M. Hancock ; Laura M. Hancock Keele University UK [email protected] Search for other works by this author on: This Site PubMed Google Scholar Chloe Howe ; Chloe Howe Keele University UK [email protected] Search for other works by this author on: This Site PubMed Google Scholar Graeme R. Jones ; Graeme R. Jones Keele University UK [email protected] Search for other works by this author on: This Site PubMed Google Scholar Tess R. Phillips ; Tess R. Phillips Keele University UK [email protected] Search for other works by this author on: This Site PubMed Google Scholar Daniela Plana Daniela Plana Keele University UK [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 12: Using Team Based Learning to Promote Problem Solving Through Active Learning in another window
  • Chapter 13: Technology, Molecular Representations, and Student Understanding in Chemistry p321-339 By Jack D. Polifka ; Jack D. Polifka Department of Chemistry, Human Computer Interaction Program, Iowa State University Ames IA 50011 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar John Y. Baluyut ; John Y. Baluyut Math and Science Division, University of Providence Great Falls MT, 59405 USA Search for other works by this author on: This Site PubMed Google Scholar Thomas A. Holme Thomas A. Holme Department of Chemistry, Human Computer Interaction Program, Iowa State University Ames IA 50011 USA [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 13: Technology, Molecular Representations, and Student Understanding in Chemistry in another window
  • Chapter 14: An Educational Software for Supporting Students’ Learning of IR Spectral Interpretation p340-360 By Maria Limniou ; Maria Limniou School of Psychology, University of Liverpool UK [email protected] Search for other works by this author on: This Site PubMed Google Scholar Nikos Papadopoulos ; Nikos Papadopoulos Department of Chemistry, Aristotle University of Thessaloniki Greece Search for other works by this author on: This Site PubMed Google Scholar Dimitris Gavril ; Dimitris Gavril Department of Chemistry, Aristotle University of Thessaloniki Greece Search for other works by this author on: This Site PubMed Google Scholar Aikaterini Touni ; Aikaterini Touni Department of Chemistry, Aristotle University of Thessaloniki Greece Search for other works by this author on: This Site PubMed Google Scholar Markella Chatziapostolidou Markella Chatziapostolidou Department of Chemistry, Aristotle University of Thessaloniki Greece Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 14: An Educational Software for Supporting Students’ Learning of IR Spectral Interpretation in another window
  • Chapter 15: Exploring Chemistry Problems with Computational Quantum Chemistry Tools in the Undergraduate Chemistry Curriculum p361-384 By Michael P. Sigalas Michael P. Sigalas Aristotle University of Thessaloniki, Laboratory of Quantum and Computational Chemistry, Department of Chemistry Thessaloniki 54124 Greece [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 15: Exploring Chemistry Problems with Computational Quantum Chemistry Tools in the Undergraduate Chemistry Curriculum in another window
  • Chapter 16: Methodological and Epistemological Issues in Science Education Problem-solving Research: Linear and Nonlinear Paradigms p385-413 By Dimitrios Stamovlasis ; Dimitrios Stamovlasis Aristotle University of Thessaloniki Thessaloniki Greece [email protected] Search for other works by this author on: This Site PubMed Google Scholar Julie Vaiopoulou Julie Vaiopoulou Democritus University of Thrace Alexandroupolis Greece [email protected] University of Nicosia Nicosia Cyprus Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 16: Methodological and Epistemological Issues in Science Education Problem-solving Research: Linear and Nonlinear Paradigms in another window
  • Chapter 17: Issues, Problems and Solutions: Summing It All Up p414-444 By Georgios Tsaparlis Georgios Tsaparlis University of Ioannina, Department of Chemistry Ioannina Greece [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 17: Issues, Problems and Solutions: Summing It All Up in another window
  • Chapter 18: Postscript – Two Issues for Provocative Thought: (a) The Potential Synergy Between HOTS and LOTS (b) When Problem Solving Might Descend to Chaos Dynamics p445-456 By Georgios Tsaparlis Georgios Tsaparlis University of Ioannina, Department of Chemistry Ioannina Greece [email protected] Search for other works by this author on: This Site PubMed Google Scholar Abstract Open the PDF Link PDF for Chapter 18: Postscript – Two Issues for Provocative Thought: (a) The Potential Synergy Between HOTS and LOTS (b) When Problem Solving Might Descend to Chaos Dynamics in another window
  • Subject Index p457-467 Open the PDF Link PDF for Subject Index in another window

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Explore a range of topics through open-ended experiments, where learners can devise their own testing plans

Four solutions cover image

Identifying four unknown solutions

Allow learner’s the opportunity to devise their own testing protocols to identify chloride ions in four solutions.

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Amino acids and hair growth

Discover how quickly human hair grows, and develop learner’s understanding of amino acids too. 

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Carbon and copper oxide

Create tests to identify carbon and copper oxide, using knowledge of acids, redox reactions, and metal oxides.

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Catalase enzyme reactions

Using four distinct questions, with open-ended methods, to explore catalysts and enzymes. 

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Chemical and physical properties of pheromones

Help learner’s to understand what pheromones are, and how they are used by animals such as moths.

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Cobalt complexes

Devise structures for cobalt complexes given, decide how to distinguish between the four complexes, and identify some properties that are the same for all four complexes.

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Defining a ‘weak’ acid

Starting from its K a (or pK a ) value, learners calculate as much information as you need to show what is meant by ‘weak’ in weak acid.

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Detecting copper in black solids

Devise experiments to identify a black solid sample by using chemicals and apparatus in the laboratory.

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Driving force: what makes reactions go

Carry out certain experiments in a given order

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Empty space in gaseous argon

How much ‘empty space’ is in a sample of gaseous argon? Students use their knowledge of Avogadro’s number and the concept of atomic size to find out. 

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Find the chloride

Devise experiments to determine which of five solids is the chloride by using chemicals and apparatus in the laboratory.

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Find the pattern - alcohols

Consider data on the densities of six primary alcohols and solve four problems based on those data.

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Find the pattern - metals

Consider data on the specific heat capacities for some metals and solve three problems related to those data.

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Finding the right ionic compound

Devise a procedure to identify four solids and then use this to carry out the identification.

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H⁺ ions in water

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Helping an industrial chemist

Devise a method for making a copper catalyst dispersed on an aluminium oxide support.

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Identifying five unknown white solids

Learners are provided with five mystery white solids, and need to devise their own experiments to identify each.

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Identifying six unknown solutions

Devise experiments to label six numbered solutions correctly using chemicals and apparatus in the laboratory.

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Identifying three unknown white solids

Devise experiments to label three white solids correctly by using chemicals and apparatus in the laboratory.

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Jets of liquids: the effect of charged plastic on water

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Making copper

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Measuring CO2 from reactions

Use chemicals and apparatus, devise experiments and determine the amount of carbon dioxide that can be obtained from a mixture of two solids

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The thickness of titanium atoms in paint

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Thermodynamic quantities in a reaction

Calculate the thermodynamic quantities for a reaction using the standard enthalpy of neutralisation (strong acid and strong base) and any other information.

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Using the problems

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What compound? Identify and analyse

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Explain two observations related to the production of a jelly-like solid when two detergents were used in combination.

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  • Research Matters — to the Science Teacher

Problem Solving in Chemistry

One of the major difficulties in teaching introductory chemistry courses is helping students become efficient problem solvers. Most beginning chemistry students find this one of the most difficulty aspects of the introductory chemistry course. What does research tell us about problem solving in chemistry? Just why do students have such difficulty in solving chemistry problems? Are some ways of teaching students to solve problems more effective than others? Problem solving in any area is a very complex process. It involves an understanding of the language in which the problem is stated, the interpretation of what is given in the problem and what is sought, an understanding of the science concepts involved in the solution, and the ability to perform mathematical operations if these are involved in the problem. The first requirement for successful problem solving is that the problem solver understand the meaning of the problem. In order to do so there must be an understanding of the vocabulary and its usage in the problem. There are two types of words that occur in problems, ordinary words that science teachers generally assume that students know and more technical terms that require understanding of concepts specific to the discipline. Researchers have found that many students do not know the meaning of common words such as contrast, displace, diversity, factor, fundamental, incident, negligible, relevant, relative, spontaneous and valid. Slight changes in the way a problem is worded may make a difference in whether a students is able to solve it correctly. For example, when "least" is changed to "most" in a problem, the percentage getting the question correct may increase by 25%. Similar improvements occur for changing negative to positive forms, for rewording long and complex questions, and for changing from the passive to the active voice. Although teachers would like students to solve problems in whatever way they are framed they must be cognizant of the fact that these subtle changes will make a difference in students' success in solving problems. From several research studies on problem solving in chemistry, it is clear that the major reason why students are unable to solve problems is that they do not understand the concepts on which the problems are based. Studies that compare the procedures used by students who are inexperienced in solving problems with experts show that experts were able to retrieve relevant concepts more readily from their long term memory. Studies have also shown that experts concepts are linked to one another in a network. Experts spend a considerable period of time planning the strategy that will be used to solve the problem whereas novices jump right in using a formula or trying to apply an algorithm. In the past few years, science educators have been trying to determine which science concepts students understand and which they do not. Because chemistry is concerned with the nature of matter, and matter is defined as anything that has mass and volume, students must understand these concepts to be successful problem solvers in chemistry. Research studies have shown that a surprising number of high school students do not understand the meaning of mass, volume, heat, temperature and changes of state. One reason why students do not understand these concepts is because when they have been taught in the classroom, they have not been presented in a variety of contexts. Often the instruction has been verbal and formal. This will be minimally effective if students have not had the concrete experiences. Hence, misconceptions arise. Although the very word "misconception" has a negative connotation, this information is important for chemistry teachers. They are frameworks by which the students view the world around them. If a teacher understands these frameworks, then instruction can be formulated that builds on student's existing knowledge. It appears that students build conceptual frameworks as they try to make sense out of their surroundings. In addition to the fundamental properties of matter mentioned above, there are other concepts that are critical to chemical calculations. One of these is the mole concept and another is the particulate nature of matter. There is mounting evidence that many students do not understand either of these concepts sufficiently well to use them in problem solving. It appears that if chemistry problem solving skills of students are to improve, chemistry teachers will need to spend a much greater period of time on concept acquisition. One way to do this will be to present concepts in a variety of contexts, using hands-on activities.

What does this research imply about procedures that are useful for helping students become more successful at problem solving?

Chemistry problems can be solved using a variety of techniques. Many chemistry teachers and most introductory chemistry texts illustrate problem solutions using the factor-label method. It has been shown that this is not the best technique for high school students of high mathematics anxiety and low proportional reasoning ability. The use of analogies and schematic diagrams results in higher achievement on problems involving moles, stoichiometry, and molarity. The use of analogs is not profitable for certain types of problems. When problems became complex (such as in dilution problems) students are unable to solve even the analog problems. For these types of problems, using analogs in instruction would be useless unless teachers are willing to spend additional time teaching students how to solve problems using the analog. Many students are unable to match analogs with the chemistry problems even after practice in using analogs. Students need considerable practice if analogs are used in instruction. When teaching chemistry by the lecture method, concept development needed for problem solving may be enhanced by pausing for a two minute interval at about 8 to 12 minute intervals during the lecture. This provides students time to review what has been presented, fill in the gaps, and interpret the information for others, and thus learn it themselves. The use of concept maps may also help students understand concepts and to relate them to one another. Requiring students to use a worksheet with each problem may help them solve them in a more effective way. The worksheet might include a place for them to plan a problem, that is list what is given and what is sought; to describe the problem situation by writing down other concepts they retrieve from memory (the use of a picture may integrate these); to find the mathematical solution; and to appraise their results. Although the research findings are not definitive, the above approaches offer some promise that students' problem solving skills can be improved and that they can learn to solve problems in a meaningful way.

For further information about this research area, please contact:

Dr. Dorothy Gabel Education Building 3rd and Jordan Bloomington, Indiana 47405

Problem-Solving in Chemistry

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what is the problem solving of chemistry

  • George M. Bodner 19 &
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George M. Bodner

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John K. Gilbert

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Curtin University of Technology, Australia

David F. Treagust

Leiden University, The Netherlands

Jan H. Van Driel

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Bodner, G.M., Herron, J.D. (2002). Problem-Solving in Chemistry. In: Gilbert, J.K., De Jong, O., Justi, R., Treagust, D.F., Van Driel, J.H. (eds) Chemical Education: Towards Research-based Practice. Science & Technology Education Library, vol 17. Springer, Dordrecht. https://doi.org/10.1007/0-306-47977-X_11

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what is the problem solving of chemistry

Chemistry Assistant

Ai-powered chemistry problem solver.

  • Homework Help: Students can use the Chemistry Assistant to help understand and work through chemistry problems in their homework.
  • Teaching Aid: Teachers can use this tool to generate solutions to chemistry problems, aiding in lesson planning and student instruction.
  • Exam Preparation: Use the Chemistry Assistant to prepare for chemistry exams by solving practice problems and getting explanations of chemistry terms and principles.
  • Research Assistance: Researchers can use this tool to help work through chemistry problems in their work.

Yes, the Chemistry Assistant is designed to handle a wide range of chemistry problems, from basic to advanced. However, it's always important to cross-verify the solutions provided by the AI with trusted resources or professionals in the field to ensure accuracy and understanding, especially with more complex problems and principles.

While the Chemistry Assistant is specifically designed for chemistry problems, HyperWrite offers other AI tools for different subjects and needs. You can explore more tools at app.hyperwriteai.com/tools .

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11 Best AI Chemistry Solvers

Yakov Itai Samelson

The evolution of AI technology has led to the development of tools that can interpret and solve intricate chemistry questions, ranging from basic compound identification to advanced reaction mechanisms. These solvers are designed to cater to a variety of needs, whether it’s a high school student preparing for an exam or a researcher analyzing molecular interactions. The integration of AI into chemistry has opened new possibilities for exploration and discovery, making it an indispensable asset for anyone looking to excel in this scientific discipline.

The Need for AI Chemistry Solver

The need for an AI chemistry solver stem from the inherent challenges of the subject. Chemistry, with its vast array of elements, compounds, and reactions, can be daunting even for the most dedicated learners. The complexity of chemical equations and the precision required for laboratory work demand a level of accuracy and understanding that can be overwhelming. This is where an AI chemistry solver becomes invaluable, offering step-by-step guidance and explanations that demystify the subject.

Moreover, the pace of modern education and research does not always allow the luxury of time. Students are often required to balance multiple subjects, while researchers are under pressure to deliver results swiftly. An AI chemistry solver addresses these time constraints by providing instant solutions and insights, enabling users to learn more efficiently and effectively. It also serves as a personal tutor, available at any hour, to assist with homework, exam preparation, or research queries, ensuring that no one falls behind due to a lack of resources or support.

AI Chemistry Solvers

How does AI Chemistry Solver work?

An AI chemistry solver operates by utilizing advanced algorithms and machine learning techniques to process and interpret chemical data. When a user inputs a chemistry problem, the AI analyzes the information, identifies the underlying principles, and computes the solution. This process often involves pattern recognition, natural language processing, and predictive analytics, which allow the AI to handle a wide range of queries, from simple molecular weight calculations to complex organic synthesis pathways.

The solver’s capability to provide detailed explanations for each step is a testament to the sophistication of the AI. It not only delivers the final answer but also educates the user on the methodology, reinforcing learning and enabling the application of these methods to future problems. This interactive approach to problem-solving ensures that users are not just passively receiving information but are actively engaging with the material, leading to a more profound comprehension of chemistry.

  • Chemistry AI Homework Solver
  • Chemistry Answers – AP Chemistry
  • Chem Calculator
  • Chemistry Assistant
  • StudyMonkey
  • Chemistry Problem Solver
  • Chemistry X10
  • Chemistry Answers

How to choose the Best AI Chemistry Solver?

Choosing the best AI chemistry solver requires careful consideration of several factors. The first aspect to consider is the accuracy of the solutions provided. The solver must be reliable, with a proven track record of delivering correct answers consistently. This is crucial as incorrect information can lead to misunderstandings and setbacks in learning or research. Another important feature is the breadth of topics covered. The ideal solver should be versatile, capable of addressing a wide spectrum of chemistry disciplines, from inorganic to physical chemistry.

The user interface is also a key element. A user-friendly platform that is easy to navigate ensures that users can focus on learning without being hindered by complicated software. Additionally, the solver should be responsive, with a quick turnaround time for solutions, catering to the fast-paced nature of academic and professional environments. Lastly, the support for various educational levels is essential. A top-tier AI chemistry solver should be adaptable, providing appropriate assistance for different expertise levels, making it a valuable resource for everyone from novices to seasoned chemists.

AI Chemistry Solver

Mathway

Mathway is a AI chemistry solver for students and professionals alike, navigating the complex waters of chemistry with ease and precision. This digital platform extends beyond the traditional boundaries of education, offering a bridge between intricate chemical equations and those seeking to understand or solve them. Its design caters to a wide audience, from high school students grappling with the basics to university scholars and professionals delving into more advanced topics. Mathway’s approachable interface and robust computational power simplify the process of learning and applying chemistry, making it an indispensable tool in the academic and professional toolkit.

What does Mathway do?

Mathway serves as a versatile assistant, adept at tackling a broad spectrum of chemistry problems. Whether it’s balancing chemical equations, calculating molar masses, or understanding stoichiometry, Mathway provides step-by-step solutions that demystify complex concepts. Its functionality extends to assisting with organic chemistry, thermodynamics, and even quantum chemistry problems, offering explanations that bridge the gap between question and understanding. By inputting a problem, users receive not just an answer but a pathway to comprehension, making Mathway an essential companion for homework, exam preparation, and beyond.

Mathway Key Features

Comprehensive Chemistry Solver : Mathway excels in offering solutions across a wide range of chemistry topics, from basic chemical reactions to intricate organic chemistry problems. This feature ensures that users of all levels find value in the platform.

Step-by-Step Explanations : One of Mathway’s standout features is its ability to break down solutions into understandable steps. This educational approach aids in the learning process, allowing users to grasp the methodology behind the answers.

User-Friendly Interface : The platform is designed with simplicity in mind, making it accessible to users with varying degrees of familiarity with technology. Its intuitive design ensures that finding solutions to complex chemistry problems is straightforward and hassle-free.

Instant Problem Solver : Mathway provides immediate answers to the chemistry problems inputted by users. This rapid response capability is invaluable for students and professionals who are pressed for time.

Mobile Accessibility : With a mobile app available, Mathway offers the convenience of solving chemistry problems on the go. This feature is particularly beneficial for students and educators who need access to a powerful chemistry solver outside of traditional settings.

Educational Tool : Beyond solving problems, Mathway serves as an educational resource. It supports the learning process by offering insights into problem-solving strategies and fostering a deeper understanding of chemistry concepts.

2. Chemistry AI Homework Solver

Chemistry AI Homework Solver

Chemistry AI Homework Solver is a cutting-edge AI chemistry solver designed to alleviate these challenges. This AI-powered solution stands out as a beacon of support for both high school and college students grappling with the intricacies of chemistry. By leveraging advanced algorithms and machine learning techniques, it offers a seamless and efficient way to navigate through a wide array of chemistry problems. From balancing chemical equations to understanding organic chemistry concepts, the solver provides step-by-step solutions that not only aid in completing assignments but also enhance the user’s comprehension of the subject matter. Its user-friendly interface and quick response time make it an indispensable resource for students aiming to improve their academic performance and grasp complex chemistry topics with ease.

What does Chemistry AI Homework Solver do?

The Chemistry AI Homework Solver is a revolutionary tool that transforms the way students approach their chemistry homework. By inputting their assignment questions or problems into the system, users are met with accurate, step-by-step solutions within seconds. This AI chemistry solver platform is adept at tackling a broad spectrum of chemistry challenges, ranging from basic chemical reactions to more advanced topics such as stoichiometry, thermodynamics, and organic synthesis. Beyond merely providing answers, it educates users by breaking down complex processes into understandable steps, thereby improving their conceptual understanding and problem-solving skills. This feature is particularly beneficial for students seeking to deepen their knowledge and apply what they learn to future assignments. The solver’s ability to deliver quick and reliable solutions not only saves valuable study time but also empowers students to tackle their chemistry coursework with confidence and efficiency.

Chemistry AI Homework Solver Key Features

Advanced Algorithms and Machine Learning : The backbone of the Chemistry AI Homework Solver is its sophisticated algorithms and machine learning capabilities. These technologies enable the tool to process and solve a wide range of chemistry problems with remarkable accuracy and speed.

Step-by-Step Solutions : One of the solver’s most valuable features is its ability to generate detailed, step-by-step solutions. This approach not only helps students complete their assignments but also enhances their understanding of the underlying concepts and methodologies.

Wide Range of Chemistry Topics : Whether it’s basic chemistry principles or more complex topics like organic chemistry and thermodynamics, the solver is equipped to handle questions across the chemistry spectrum. This versatility makes it a comprehensive resource for students at various levels of study.

User-Friendly Interface : The platform’s intuitive interface ensures that students can easily navigate and use the solver without any hassle. This accessibility is crucial for facilitating a smooth and productive learning experience.

Quick Response Time : In the fast-paced academic world, time is of the essence. The Chemistry AI Homework Solver stands out for its ability to deliver solutions swiftly, enabling students to work more efficiently and effectively.

Enhanced Learning and Comprehension : Beyond solving problems, the tool plays a pivotal role in improving students’ comprehension of chemistry concepts. By providing clear explanations and breaking down complex processes, it supports a deeper level of learning and retention.

3. Chemistry Answers – AP Chemistry

Chemistry Answers

Chemistry Answers – AP Chemistry is an innovative AI chemistry solver designed to assist students in mastering the complexities of chemistry, particularly those preparing for AP Chemistry exams. This app stands out as a comprehensive tool, offering a wide array of features aimed at enhancing the learning experience. From detailed glossaries to graded quizzes, Chemistry Answers – AP Chemistry provides an interactive platform that caters to various learning levels, making it an indispensable resource for students aiming to excel in their chemistry studies. Its user-friendly interface and extensive content coverage make it a go-to solution for both secondary school and university students seeking to deepen their understanding of chemistry concepts.

What does Chemistry Answers – AP Chemistry do?

Chemistry Answers – AP Chemistry serves as a virtual tutor, offering an extensive range of services designed to support students through their chemistry education journey. It provides thousands of glossary terms with definitions and illustrations, making complex concepts easier to grasp. The AI chemistry solver also features learning material graded over three levels, allowing students to progress at their own pace and challenge themselves as they master each level. Additionally, Chemistry Answers – AP Chemistry offers the unique ability to repeat quizzes, enabling students to improve their results through practice. This focus on repetition and progressive learning helps solidify understanding and retention of chemistry concepts, making it an invaluable tool for students aiming for excellence in their studies.

Chemistry Answers – AP Chemistry Key Features

Comprehensive Glossary : The app boasts thousands of glossary terms, each defined and illustrated to aid in the understanding of complex chemistry concepts. This feature serves as an invaluable reference tool, allowing students to quickly look up and learn terms anytime, anywhere.

Graded Learning Material : Chemistry Answers offers learning materials across three levels, catering to students at different stages of their chemistry education. This approach ensures that learners can start with the basics and gradually tackle more challenging content as they build their knowledge and confidence.

Quiz Repetition : Students have the opportunity to retake quizzes, providing a means to practice and improve their scores. This feature emphasizes the importance of learning from mistakes and reinforces the material through repetition.

User-Friendly Interface : The intuitive design of the app makes navigating through its vast content straightforward, ensuring a seamless learning experience. This ease of use encourages students to engage with the material regularly, enhancing their study habits.

Tailored for AP Chemistry : Specifically designed with AP Chemistry students in mind, the app includes quizzes and content that align with the curriculum and exam requirements. This targeted approach helps students focus their study efforts on what’s most important for exam success.

Accessibility : Available for download on various devices, Chemistry Answers provides flexible access to its resources, making it easy for students to study on the go or from the comfort of their home. This accessibility ensures that valuable study time is maximized, regardless of the student’s location.

4. Chem Calculator

Chem Calculator

Chem Calculator stands out as an indispensable tool for organic chemists, offering a robust solution for calculating molecular weights with unparalleled ease and accuracy. Designed with the needs of the chemistry community in mind, this app has quickly become a must-have in the App Store for professionals and students alike. This AI chemistry solver user-friendly interface allows for mixed input of elements, numbers, and parentheses, delivering real-time results that streamline the calculation process. With features like a full-sized periodic table and the ability to save, edit, or delete results, Chem Calculator not only simplifies complex calculations but also enhances productivity and learning in the field of organic chemistry.

What does Chem Calculator do?

Chem Calculator is designed to simplify the life of organic chemists by providing a comprehensive solution for calculating molecular weights directly from their smartphones or tablets. This a AI chemistry solver supports mixed inputs, including elements, numbers, and parentheses, offering real-time results that make it easier to focus on research and study without getting bogged down in manual calculations. Beyond molecular weight calculation, Chem Calculator features a full-sized periodic table, allowing users to check element details with a long press on the element button. It also offers functionalities such as saving, editing, or deleting results, making it a versatile tool for managing chemical calculations efficiently.

Chem Calculator Key Features

Mixed Input Support : This feature allows users to input a combination of elements, numbers, and parentheses, providing real-time results for molecular weight calculations. It streamlines the calculation process, making it faster and more intuitive for chemists.

Full-Sized Periodic Table : Chem Calculator includes a comprehensive periodic table, where periods 1, 2, and 3 elements are combined in one line for easy access. This feature aids in quick reference and element detail checking, enhancing the app’s utility.

Save, Delete, or Edit Results : Users can save their calculation results for future reference, and have the flexibility to delete or edit them as needed. This functionality is particularly useful for managing multiple calculations and keeping track of important data.

Element Detail by Long Pressing : By long pressing the element button, users can access detailed information about each element. This feature provides valuable insights and data that can assist in making informed decisions during chemical analysis.

Real-Time Result Display : The app displays calculation results in real time, offering immediate feedback as users input their data. This feature ensures accuracy and efficiency in molecular weight calculations, saving time and reducing the potential for errors.

Periodic Table Customization : The first three periods of elements are combined in one line within the periodic table, offering a unique and streamlined view that enhances usability. This customization makes it easier for users to navigate the table and find the information they need quickly.

Chem Calculator is a powerful tool that brings a new level of efficiency and accuracy to the field of organic chemistry. Its key features are thoughtfully designed to meet the specific needs of chemists, making it an essential app for professionals and students alike.

5. NoProblem

NoProblem

NoProblem is a versatile AI Chemistry Solver designed to assist students in conquering the complexities of chemistry homework. This innovative app leverages the power of GPT technology to provide not just answers, but also detailed explanations of the processes behind each solution. It’s a tool that aims to enhance understanding and retention of chemistry concepts, making it a valuable companion for students at various levels of education.

What does NoProblem do?

NoProblem serves as a personal AI-powered homework assistant that simplifies academic challenges. By allowing users to scan homework problems with their device’s camera, the app delivers comprehensive, step-by-step solutions in real-time. It’s particularly adept at handling a multitude of academic disciplines, including the intricate world of chemistry, making it an indispensable resource for students seeking to deepen their comprehension and excel in their studies.

NoProblem Key Features

Scan and Solve : This feature transforms your device into a powerful educational tool. By simply taking a picture of a homework problem, you receive instant solutions, making it easier to tackle complex tasks.

Step-by-Step Guidance : Unlike conventional solvers that provide the final answer, NoProblem elucidates the solution process, fostering a deeper understanding of the underlying chemistry concepts.

Broad Subject Coverage : Whether it’s basic algebra or advanced chemistry, NoProblem is equipped to assist with a wide spectrum of subjects, ensuring comprehensive academic support.

Interactive Learning : The app promotes an engaging learning experience with interactive solutions and detailed explanations, which are instrumental in enhancing students’ grasp of the material.

User-Friendly Interface : NoProblem boasts an intuitive design, making it accessible for students of all ages and facilitating a stress-free educational journey.

Accessibility and Convenience : With the ability to use the app anytime and anywhere, students can study effectively and efficiently, fitting learning into their busy schedules.

6. Chemistry Assistant

Chemistry Assistant

Chemistry Assistant is an AI chemistry solver for those seeking to navigate the complexities of chemical problems. This AI-powered chemistry problem solver from HyperWrite stands out for its ability to process and provide solutions to a wide array of chemistry questions. By harnessing the capabilities of advanced AI models, the Chemistry Assistant demystifies intricate chemistry challenges, offering clear and comprehensive solutions. This tool is not only a boon for students grappling with homework but also serves as a valuable resource for teachers crafting lesson plans and for researchers who require assistance in their scientific inquiries.

What does Chemistry Assistant do?

Chemistry Assistant is a versatile tool designed to assist users across various levels of chemistry proficiency. It functions by accepting chemistry problems or questions input by the user and then generating detailed answers. The process involves the AI’s analysis of the problem, followed by the creation of a solution that elucidates the steps and principles involved. This tool is adept at tackling a spectrum of chemistry problems, from elementary to advanced, making it an indispensable asset for homework help, teaching aid, exam preparation, and research assistance. Its user-friendly interface simplifies the interaction, allowing for a seamless experience from problem submission to solution review.

Chemistry Assistant Key Features

Homework Help : The Chemistry Assistant is particularly beneficial for students who are seeking to deepen their understanding of chemistry concepts and work through problems encountered in their homework.

Teaching Aid : Educators can leverage this tool to generate solutions for chemistry problems, which can be integrated into lesson planning and enhance the instructional process.

Exam Preparation : For those preparing for chemistry exams, the Chemistry Assistant offers a means to practice with various problems and gain a better grasp of chemistry terms and principles.

Research Assistance : Researchers find the Chemistry Assistant to be a helpful ally when dealing with complex chemistry problems within their work, streamlining the problem-solving aspect of their projects.

Advanced AI Models : Utilizing cutting-edge AI models like GPT-4 and ChatGPT, the Chemistry Assistant can handle a broad range of chemistry problems, providing users with confidence in the tool’s capabilities.

Accessibility and Ease of Use : The tool’s straightforward interface allows users to input their chemistry problems with ease and receive solutions without the need for extensive technical knowledge, making it accessible to a wide audience.

7. StudyMonkey

StudyMonkey

StudyMonkey is dynamic AI-powered chemistry tutor designed to assist students in navigating the complexities of chemistry homework and assignments. This innovative platform offers a blend of step-by-step tutoring, essay enhancement, and work assessment capabilities, all delivered with remarkable speed. StudyMonkey aims to alleviate the stress and anxiety associated with challenging chemistry questions and last-minute essay requirements by providing solutions in a matter of seconds. Its 24/7 availability ensures that students can receive immediate assistance whenever needed, effectively eliminating the need for late-night study sessions and the reliance on peers for help.

What does StudyMonkey do?

StudyMonkey serves as a versatile educational companion, offering comprehensive support across various facets of chemistry learning. The platform is adept at tackling a wide array of chemistry homework questions, from multiple-choice queries to short answer questions and even essay writing. StudyMonkey’s AI is designed to cater to every educational level, from elementary to master’s degree students, ensuring that no student’s query goes unanswered. The ability to review past questions and answers further enriches the learning experience, allowing students to revisit material and solidify their understanding in preparation for exams.

StudyMonkey Key Features

24/7 Availability : StudyMonkey stands out for its round-the-clock service, providing students with the flexibility to seek help at any hour, which is particularly beneficial for those with unpredictable schedules.

Step-by-Step Guidance : The platform offers detailed explanations for complex chemistry problems, ensuring that students not only receive answers but also understand the underlying concepts.

Essay Enhancement : Beyond numerical problems, StudyMonkey assists with essay writing, helping students to improve the quality of their written work in chemistry.

Assessment Capabilities : The AI tutor evaluates students’ work, offering insights that can guide learning and help improve academic performance.

Support for All Educational Levels : Catering to a broad spectrum of students, StudyMonkey is equipped to handle queries from the most basic to advanced levels of education.

History Review : An invaluable feature for exam preparation, students can access their question history to review and learn from their past inquiries.

8. Chemistry Problem Solver

Chemistry Problem Solver

Chemistry Problem Solver is a pivotal tool designed to demystify the complexities of chemistry for students and professionals alike. This innovative AI chemistry solver leverages artificial intelligence to provide comprehensive solutions to a wide array of chemistry problems, ranging from basic chemical equations to intricate molecular structure analysis. Its intuitive interface and sophisticated algorithms enable users to not only find answers but also to understand the underlying principles of chemistry, making it an indispensable resource for anyone looking to enhance their grasp of the subject.

What does Chemistry Problem Solver do?

The Chemistry Problem Solver is a versatile tool that serves as a virtual tutor for individuals navigating the challenging waters of chemistry. It excels in breaking down complex chemical reactions, calculating molar masses, and offering step-by-step solutions to a variety of chemistry-related queries. Whether it’s balancing chemical equations, understanding stoichiometry, or exploring the properties of different substances, this AI-driven assistant provides clear, accurate, and immediate answers. Its ability to simplify complicated concepts into understandable chunks of information makes learning chemistry more accessible and less intimidating for students at all levels.

Chemistry Problem Solver Key Features

Comprehensive Chemical Equation Solver : This feature stands out by offering users the ability to input any chemical equation and receive a detailed, step-by-step solution. It simplifies the process of balancing equations, making it easier for users to understand the reaction mechanisms and stoichiometric relationships involved.

Molar Mass Calculator : A crucial tool for chemistry students and professionals, this calculator allows for the quick computation of the molar mass of any given compound. By inputting the formula of the compound, users can instantly obtain its molar mass, facilitating calculations related to chemical reactions and solutions.

Interactive Learning Interface : The platform’s user-friendly interface encourages interactive learning, enabling users to actively engage with the material. This feature promotes a deeper understanding of chemistry concepts through hands-on practice and exploration.

AI-Driven Problem Analysis : Leveraging advanced artificial intelligence algorithms, the Chemistry Problem Solver can analyze and interpret a wide range of chemistry problems, providing accurate solutions and explanations. This AI-driven approach ensures that users receive reliable and up-to-date information.

Customizable Study Aids : Tailored to meet the individual needs of its users, the platform offers customizable study aids, including practice problems and quizzes. These resources are designed to reinforce learning and help users master chemistry concepts at their own pace.

Real-Time Feedback and Support : Offering real-time feedback on user inputs and solutions, this feature enhances the learning experience by providing immediate clarification and support. It ensures that users can learn from their mistakes and gain confidence in their chemistry skills.

Chem AI

Chem AI is an AI chemistry solver designed to address the wide spectrum of challenges encountered in the field of chemistry. This innovative AI chemistry solver stands out by offering instant, step-by-step solutions to a diverse array of chemistry problems, ranging from multiple choice questions to complex mathematical equations. Its ability to recognize and interpret problems through uploaded or photographed questions simplifies the learning and problem-solving process for students and professionals alike. By covering topics across all levels of chemistry, from high school fundamentals to intricate doctoral research, Chem AI positions itself as a versatile and indispensable resource. The app’s commitment to enhancing the educational experience is further demonstrated through its provision of detailed mathematical solutions and a user-friendly trial option, making advanced chemistry more accessible and comprehensible.

What does Chem AI do?

Chem AI is designed to simplify the often complex and daunting task of solving chemistry problems. By leveraging advanced artificial intelligence, it provides users with immediate, detailed explanations for a wide range of chemistry questions. Whether you’re grappling with a tricky multiple choice question, a word problem, a diagram-based query, or a mathematical chemistry equation, Chem AI is equipped to offer step-by-step guidance. Users can easily upload or take photos of their chemistry problems, and the app swiftly recognizes and processes the query to deliver comprehensive solutions. This functionality not only aids in academic learning and exam preparation but also supports research and professional work, making Chem AI a valuable tool for anyone looking to deepen their understanding of chemistry.

Chem AI Key Features

Photo Recognition : Chem AI’s photo recognition feature allows users to upload or snap a picture of their chemistry problem, whether it’s handwritten or digital. This innovative capability ensures that users can quickly and effortlessly get help with their specific questions, enhancing the learning experience by making it more interactive and accessible.

Detailed Solutions : The app provides detailed, step-by-step solutions to chemistry problems. This feature is particularly beneficial for users who are not just looking for answers but also wish to understand the underlying principles and methodologies involved in solving chemistry-related questions.

Wide Range of Topics : Chem AI covers an extensive array of topics within the chemistry domain. From high school chemistry basics to complex questions at the doctoral research level, the app serves as a comprehensive resource for users at various stages of their educational or professional journey.

Mathematical Problem Solving : Beyond chemical concepts, Chem AI is adept at solving mathematical problems related to chemistry. It offers detailed steps for mathematical solutions, making it easier for users to grasp the calculations and formulae involved in chemistry.

Free Trial : Understanding the importance of user satisfaction, Chem AI provides a free trial, allowing potential users to explore the app’s features and capabilities before committing to it. This approach demonstrates the app’s confidence in its value and effectiveness.

User-Friendly Interface : The app boasts a user-friendly interface that simplifies navigation and problem-solving. This design consideration ensures that users of all ages and proficiency levels can effectively utilize the app, thereby democratizing access to advanced chemistry assistance.

Chem AI stands out as a versatile and user-friendly tool that significantly aids in the understanding and solving of chemistry problems, making it an asset for students, educators, and professionals in the field of chemistry.

10. Chemistry X10

Chemistry X10

Chemistry X10 is a versatile AI chemistry solver and educational tool designed to assist students from middle to high school levels in mastering chemistry. With its ability to solve a wide array of chemistry problems, the app serves as a reliable companion for homework and test preparation. It operates seamlessly without the need for an internet connection, which is a significant advantage for users with limited access to the web. The app’s popularity is reflected in its impressive download figures, indicating a strong trust in its capabilities among learners and educators alike.

What does Chemistry X10 do?

Chemistry X10 is essentially a digital tutor that simplifies the process of learning chemistry. By inputting the ‘Given’ and ‘Find’ parameters of a problem, users receive a fully worked out solution with a clear explanation and answer. The AI chemistry solver covers various types of chemistry tasks, including those involving formulas, solutions, mixtures, reaction equations, and calculations of excess and deficiency. It also boasts a database of chemical reactions and the ability to solve transformation chains and balance equations using different methods. For students looking to deepen their understanding of chemistry concepts, Chemistry X10 is a comprehensive resource that enhances learning outside the classroom.

Chemistry X10 Key Features

Task Solutions : Chemistry X10 excels in providing detailed solutions to chemistry problems. Users can enter the known variables and the app will generate a step-by-step solution, complete with explanations, making it an invaluable study aid.

Chemical Reactions : The app includes a vast repository of chemical reactions, allowing users to find reaction equations even when offline. This feature is particularly useful for students who need to study on the go.

Transformation Chains : For more complex chemistry problems, Chemistry X10 can solve open transformation chains, providing users with a sequence of reactions that lead from one compound to another.

Coefficient Arrangement : Balancing chemical equations is made easier with the app’s ability to arrange coefficients either by fitting or by the method of electronic balance, ensuring that students learn the correct stoichiometry.

Molar Masses : Calculating molar masses of substances is a fundamental skill in chemistry, and Chemistry X10 simplifies this process by providing accurate molar mass calculations for complex substances.

Periodic Table : The app features a mobile-oriented periodic table, offering quick access to essential information about chemical elements, which is crucial for solving chemistry problems and understanding the properties of elements.

11. Chemistry Answers

Chemistry Answers

Chemistry Answers is an AI chemistry solver app designed to support both students and enthusiasts in navigating the complexities of chemistry. With its clear interface and straightforward functions, it stands out as a reliable companion for learning and reference, requiring no internet connection for use. This AI chemistry solver app caters to a global audience with its bilingual support, offering both Chinese and English versions. From solving chemical equations to exploring the periodic table, calculating molar masses, and beyond, Chemistry Answers provides a comprehensive suite of tools to enhance the learning experience. Its commitment to simplicity and accessibility makes it an invaluable resource for anyone looking to deepen their understanding of chemistry.

What does Chemistry Answers do?

Chemistry Answers serves as a multifunctional tool designed to simplify the study of chemistry. It aids users in solving chemical equations, understanding the periodic table, and calculating molar masses with ease. Additionally, the app offers functionalities such as common name queries, substance color identification, and a metal activity table, making it a comprehensive resource for students and chemistry enthusiasts alike. Its ability to function without an internet connection adds to its convenience, allowing users to access information and solve problems anytime, anywhere. Whether for academic purposes or personal interest, Chemistry Answers equips users with the knowledge and tools needed to explore the fascinating world of chemistry.

Chemistry Answers Key Features

Chemical Equation Solver : This feature allows users to input chemical equations and receive solutions, simplifying the process of balancing equations and understanding chemical reactions.

Periodic Table Access : Users can explore an interactive periodic table, gaining insights into elements’ properties, atomic structures, and more, fostering a deeper comprehension of the building blocks of chemistry.

Molar Mass Calculator : This tool enables the calculation of molar masses for various substances, aiding in stoichiometry and chemical calculations, essential for academic and research purposes.

Common Name Query : Users can search for substances by their common names, bridging the gap between everyday language and scientific terminology, enhancing learning and communication.

Substance Color Query : This unique feature allows users to identify substances based on color, adding a visual dimension to chemical understanding and making experiments more accessible.

Metal Activity Table : By providing a metal activity table, the app helps users predict reaction outcomes, understand reactivity series, and make informed decisions in experimental setups.

Chemistry Answers stands out as a comprehensive tool for anyone interested in chemistry, offering a blend of educational resources, practical tools, and interactive features to support learning and exploration in the field.

FAQs on AI Chemistry Solver

What is an ai chemistry solver.

An AI Chemistry Solver is a digital tool that uses artificial intelligence to assist in solving chemical equations, understanding chemical reactions, and providing educational support in the field of chemistry. It’s designed to interpret complex chemical data and provide accurate solutions, explanations, and predictions.

Who can benefit from using an AI Chemistry Solver?

Students, educators, researchers, and professionals in the field of chemistry can all benefit from using an AI Chemistry Solver. It serves as an educational aid for students, a teaching tool for educators, a research assistant for scientists, and a problem-solving resource for industry professionals.

Can AI Chemistry Solvers handle organic chemistry problems?

Yes, many AI Chemistry Solvers are equipped to handle organic chemistry problems, including reaction mechanisms, synthesis pathways, and molecular structure elucidation.

Are AI Chemistry Solvers accurate?

The accuracy of an AI Chemistry Solver depends on the sophistication of its algorithms and the quality of its data sources. The best solvers are highly accurate, with a strong track record of providing correct solutions.

Do AI Chemistry Solvers provide explanations for their solutions?

Yes, the top AI Chemistry Solvers provide step-by-step explanations for their solutions, helping users understand the process and learn from the problem-solving experience.

How do AI Chemistry Solvers differ from traditional problem-solving methods?

AI Chemistry Solvers differ from traditional methods in their ability to process large amounts of data quickly, provide instant feedback, and offer interactive learning experiences that adapt to the user’s needs.

Can AI Chemistry Solvers predict chemical reactions?

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Computational Problem Solving in the Chemical Sciences

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  • Acceleration of aircraft carrier take-off
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  • Deriving displacement as a function of time, acceleration, and initial velocity
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The kinematic equations

  • Setting up problems with constant acceleration
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What are the kinematic equations?

What is a free falling object (a projectile), how do you select and use a kinematic equation, more problem solving tips.

  • Don't forget that the kinematic equations are only true assuming the acceleration is constant during the time interval considered.
  • Sometimes a known variable will not be explicitly given in a problem, but rather implied. For instance, "starts from rest" means v 0 = 0 ‍   , "dropped" often means v 0 = 0 ‍   , and "comes to a stop" means v = 0 ‍   . Also, the magnitude of the acceleration due to gravity on all free falling projectiles is assumed to be g = 9.81 m s 2 ‍   , so this acceleration will usually not be given explicitly in a problem but will just be implied for a free falling object.
  • Don't forget that all the kinematic variables, except for t , ‍   can be negative. A missing negative sign is a very common source of error. If upward is assumed to be positive, then the acceleration due to gravity for a free falling object must be negative: a g = − 9.81 m s 2 ‍   .
  • Solving a problem involving the third kinematic equation, Δ x = v 0 t + 1 2 a t 2 ‍   , might require the use of the quadratic formula.
  • Don't forget that even though you can choose any time interval during the constant acceleration, the kinematic variables you plug into a kinematic equation must be consistent with that time interval. In other words, the initial velocity v 0 ‍   has to be the velocity of the object at the initial position and start of the time interval t ‍   . Similarly, the final velocity v ‍   must be the velocity at the final position and end of the time interval t ‍   being analyzed.

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Chapter 15.3: Solving Equilibrium Problems

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Learning Objective

  • To solve quantitative problems involving chemical equilibriums.

There are two fundamental kinds of equilibrium problems: (1) those in which we are given the concentrations of the reactants and the products at equilibrium (or, more often, information that allows us to calculate these concentrations), and we are asked to calculate the equilibrium constant for the reaction; and (2) those in which we are given the equilibrium constant and the initial concentrations of reactants, and we are asked to calculate the concentration of one or more substances at equilibrium. In this section, we describe methods for solving both kinds of problems.

Calculating an Equilibrium Constant from Equilibrium Concentrations

We saw in the exercise in Example 6 in Section 15.2 that the equilibrium constant for the decomposition of CaCO 3 (s) to CaO(s) and CO 2 (g) is K = [CO 2 ]. At 800°C, the concentration of CO 2 in equilibrium with solid CaCO 3 and CaO is 2.5 × 10 −3 M. Thus K at 800°C is 2.5 × 10 −3 . (Remember that equilibrium constants are unitless.)

Ball and stick models of n-butane and isobutane (2-methylpropane).

A more complex example of this type of problem is the conversion of n -butane, an additive used to increase the volatility of gasoline, to isobutane (2-methylpropane). This reaction can be written as follows:

\( n-butane \left ( g \right ) \rightleftharpoons isobutane \left ( g \right ) \tag{15.3.1} \)

and the equilibrium constant K = [isobutane]/[ n -butane]. At equilibrium, a mixture of n -butane and isobutane at room temperature was found to contain 0.041 M isobutane and 0.016 M n -butane. Substituting these concentrations into the equilibrium constant expression,

\( K= \dfrac{isobutane}{n-butane}=\dfrac{0.041\;\cancel{M}}{0.016 \; \cancel{M}} = 2.6 \tag{15.3.2} \)

Thus the equilibrium constant for the reaction as written is 2.6.

Example 15.3.1

The reaction between gaseous sulfur dioxide and oxygen is a key step in the industrial synthesis of sulfuric acid:

\( 2SO_{2}\left ( g \right ) + O_{2}\left ( g \right ) \rightleftharpoons 2SO_{3}\left ( g \right ) \)

A mixture of SO 2 and O 2 was maintained at 800 K until the system reached equilibrium. The equilibrium mixture contained 5.0 × 10 −2 M SO 3 , 3.5 × 10 −3 M O 2 , and 3.0 × 10 −3 M SO 2 . Calculate K and K p at this temperature.

Given: balanced equilibrium equation and composition of equilibrium mixture

Asked for: equilibrium constant

Write the equilibrium constant expression for the reaction. Then substitute the appropriate equilibrium concentrations into this equation to obtain K .

Substituting the appropriate equilibrium concentrations into the equilibrium constant expression,

\( K=\dfrac{\left [ SO_{3} \right ]^{2}}{\left [ SO_{2} \right ]^{2}\left [ O_{2} \right ]}=\dfrac{\left ( 5.0\times 10^{-2} \right )^{2}}{\left ( 3.0\times 10^{-3} \right )^{2}\left ( 3.5\times 10^{-3} \right )}=7.9\times 10^{4} \)

To solve for K p , we use Equation 15.2.17 , where Δ n = 2 − 3 = −1:

\( K_{p}= K\left ( RT \right )^{\Delta n} \) \( =7.9\times 10^{4}\left [ \left (0.082606\; L\cdot atm/mol\cdot \cancel{K} \right ) \left ( 800 \; \cancel{K} \right )\right ] \) \( =1.2\times 10^{3}\)

Hydrogen gas and iodine react to form hydrogen iodide via the reaction

\( H_{2}\left ( g \right ) + I_{2}\left ( g \right ) \rightleftharpoons 2HI\left ( g \right ) \)

A mixture of H 2 and I 2 was maintained at 740 K until the system reached equilibrium. The equilibrium mixture contained 1.37 × 10 −2 M HI, 6.47 × 10 −3 M H 2 , and 5.94 × 10 −4 M I 2 . Calculate K and K p for this reaction.

Answer: K = 48.8; K p = 48.8

Chemists are not often given the concentrations of all the substances, and they are not likely to measure the equilibrium concentrations of all the relevant substances for a particular system. In such cases, we can obtain the equilibrium concentrations from the initial concentrations of the reactants and the balanced chemical equation for the reaction, as long as the equilibrium concentration of one of the substances is known. Example 9 shows one way to do this.

Example 15.3.2

A 1.00 mol sample of NOCl was placed in a 2.00 L reactor and heated to 227°C until the system reached equilibrium. The contents of the reactor were then analyzed and found to contain 0.056 mol of Cl 2 . Calculate K at this temperature. The equation for the decomposition of NOCl to NO and Cl 2 is as follows:

\( 2NOCl \left ( g \right ) \rightleftharpoons 2NO\left ( g \right ) + Cl_{2}\left ( g \right ) \)

Given: balanced equilibrium equation, amount of reactant, volume, and amount of one product at equilibrium

Asked for: K

A Write the equilibrium constant expression for the reaction. Construct a table showing the initial concentrations, the changes in concentrations, and the final concentrations (as initial concentrations plus changes in concentrations).

B Calculate all possible initial concentrations from the data given and insert them in the table.

C Use the coefficients in the balanced chemical equation to obtain the changes in concentration of all other substances in the reaction. Insert those concentration changes in the table.

D Obtain the final concentrations by summing the columns. Calculate the equilibrium constant for the reaction.

A The first step in any such problem is to balance the chemical equation for the reaction (if it is not already balanced) and use it to derive the equilibrium constant expression. In this case, the equation is already balanced, and the equilibrium constant expression is as follows:

\( K=\dfrac{\left [ NO_{2} \right ]^{2}\left [ Cl_{2} \right ]}{\left [ NOCl \right ]^{2}} \)

To obtain the concentrations of NOCl, NO, and Cl 2 at equilibrium, we construct a table showing what is known and what needs to be calculated. We begin by writing the balanced chemical equation at the top of the table, followed by three lines corresponding to the initial concentrations, the changes in concentrations required to get from the initial to the final state, and the final concentrations.

B Initially, the system contains 1.00 mol of NOCl in a 2.00 L container. Thus [NOCl] i = 1.00 mol/2.00 L = 0.500 M. The initial concentrations of NO and Cl 2 are 0 M because initially no products are present. Moreover, we are told that at equilibrium the system contains 0.056 mol of Cl 2 in a 2.00 L container, so [Cl 2 ] f = 0.056 mol/2.00 L = 0.028 M. We insert these values into the following table:

C We use the stoichiometric relationships given in the balanced chemical equation to find the change in the concentration of Cl 2 , the substance for which initial and final concentrations are known:

Δ[Cl 2 ] = [0.028 M (final) − 0.00 M (initial)] = +0.028 M

According to the coefficients in the balanced chemical equation, 2 mol of NO are produced for every 1 mol of Cl 2 , so the change in the NO concentration is as follows:

\( \Delta \left [ NO \right ] = \left ( \dfrac{0.028 \; \cancel{mol\;Cl_{2}}}{L} \right )\left ( \dfrac{2\; mol\; NO}{1\;\cancel{mol\;Cl_{2}}} \right )=0.056\; M \)

Similarly, 2 mol of NOCl are consumed for every 1 mol of Cl 2 produced, so the change in the NOCl concentration is as follows:

\( \Delta \left [ NOCl \right ] = \left ( \dfrac{0.028 \; \cancel{mol\;Cl_{2}}}{L} \right )\left ( \dfrac{-2\; mol\; NO}{1\;\cancel{mol\;Cl_{2}}} \right )=-0.056\; M \)

We insert these values into our table:

D We sum the numbers in the [NOCl] and [NO] columns to obtain the final concentrations of NO and NOCl:

[NO] f = 0.000 M + 0.056 M = 0.056 M [NOCl] f = 0.500 M + (−0.056 M) = 0.444 M

We can now complete the table:

We can now calculate the equilibrium constant for the reaction:

\( K=\dfrac{\left [ NO_{2} \right ]^{2}\left [ Cl_{2} \right ]}{\left [ NOCl \right ]^{2}}=\dfrac{\left ( 0.056 \right )^{2}\left ( 0.028 \right )}{0.444}^{2}=4.5\times 10^{-4} \)

The German chemist Fritz Haber (1868–1934; Nobel Prize in Chemistry 1918) was able to synthesize ammonia (NH 3 ) by reacting 0.1248 M H 2 and 0.0416 M N 2 at about 500°C. At equilibrium, the mixture contained 0.00272 M NH 3 . What is K for the reaction N 2 + 3 H 2 ⇌ 2NH 3 at this temperature? What is K p ?

Answer: K = 0.105; K p = 2.61 × 10 −5

The original laboratory apparatus designed by Fritz Haber and Robert Le Rossignol in 1908 for synthesizing ammonia from its elements. A metal catalyst bed, where ammonia was produced, is in the large cylinder at the left. The Haber-Bosch process used for the industrial production of ammonia uses essentially the same process and components but on a much larger scale. Unfortunately, Haber’s process enabled Germany to prolong World War I when German supplies of nitrogen compounds, which were used for explosives, had been exhausted in 1914.

Calculating Equilibrium Concentrations from the Equilibrium Constant

To describe how to calculate equilibrium concentrations from an equilibrium constant, we first consider a system that contains only a single product and a single reactant, the conversion of n -butane to isobutane (Equation 15.26), for which K = 2.6 at 25°C. If we begin with a 1.00 M sample of n -butane, we can determine the concentration of n -butane and isobutane at equilibrium by constructing a table showing what is known and what needs to be calculated, just as we did in Example 9.

The initial concentrations of the reactant and product are both known: [ n -butane] i = 1.00 M and [isobutane] i = 0 M. We need to calculate the equilibrium concentrations of both n -butane and isobutane. Because it is generally difficult to calculate final concentrations directly, we focus on the change in the concentrations of the substances between the initial and the final (equilibrium) conditions. If, for example, we define the change in the concentration of isobutane (Δ[isobutane]) as + x , then the change in the concentration of n -butane is Δ[ n -butane] = − x . This is because the balanced chemical equation for the reaction tells us that 1 mol of n -butane is consumed for every 1 mol of isobutane produced. We can then express the final concentrations in terms of the initial concentrations and the changes they have undergone.

Substituting the expressions for the final concentrations of n -butane and isobutane from the table into the equilibrium equation,

\( K=\dfrac{\left [ isobutane \right]}{\left [ n-butane \right ]}=\dfrac{x}{1.00-x}=2.6 \)

Rearranging and solving for x ,

\( x = 2.6\left ( 1.00-x \right )=2.6-2.6x \) \( x + 2.6x =2.6 \) \( x = 0.72 \)

We obtain the final concentrations by substituting this x value into the expressions for the final concentrations of n -butane and isobutane listed in the table:

[ n -butane] f = (1.00 − x ) M = (1.00 − 0.72) M = 0.28 M [isobutane] f = (0.00 + x ) M = (0.00 + 0.72) M = 0.72 M

We can check the results by substituting them back into the equilibrium constant expression to see whether they give the same K that we used in the calculation:

\( K=\dfrac{\left [ isobutane \right]}{\left [ n-butane \right ]}=\dfrac{0.72 \; \cancel{M}}{0.28 \; \cancel{M}}=2.6 \)

This is the same K we were given, so we can be confident of our results.

Example 10 illustrates a common type of equilibrium problem that you are likely to encounter.

Example 15.3.3

The water–gas shift reaction is important in several chemical processes, such as the production of H 2 for fuel cells. This reaction can be written as follows:

\( H_{2}\left ( g \right ) + CO_{2}\left ( g \right ) \rightleftharpoons H_{2}O\left ( g \right ) + CO\left ( g \right )\)

K = 0.106 at 700 K. If a mixture of gases that initially contains 0.0150 M H 2 and 0.0150 M CO 2 is allowed to equilibrate at 700 K, what are the final concentrations of all substances present?

Given: balanced equilibrium equation, K , and initial concentrations

Asked for: final concentrations

A Construct a table showing what is known and what needs to be calculated. Define x as the change in the concentration of one substance. Then use the reaction stoichiometry to express the changes in the concentrations of the other substances in terms of x . From the values in the table, calculate the final concentrations.

B Write the equilibrium equation for the reaction. Substitute appropriate values from the table to obtain x .

C Calculate the final concentrations of all species present. Check your answers by substituting these values into the equilibrium constant expression to obtain K .

A The initial concentrations of the reactants are [H 2 ] i = [CO 2 ] i = 0.0150 M. Just as before, we will focus on the change in the concentrations of the various substances between the initial and final states. If we define the change in the concentration of H 2 O as x , then Δ[H 2 O] = + x . We can use the stoichiometry of the reaction to express the changes in the concentrations of the other substances in terms of x . For example, 1 mol of CO is produced for every 1 mol of H 2 O, so the change in the CO concentration can be expressed as Δ[CO] = + x . Similarly, for every 1 mol of H 2 O produced, 1 mol each of H 2 and CO 2 are consumed, so the change in the concentration of the reactants is Δ[H 2 ] = Δ[CO 2 ] = − x . We enter the values in the following table and calculate the final concentrations.

B We can now use the equilibrium equation and the given K to solve for x :

\( K=\dfrac{\left [ H_{2}O] \right ] \left [ CO \right ]}{\left [ H_{2} \right ]\left [ CO_{2} \right ]}=\dfrac{\left (x \right )\left ( x \right ) }{\left ( 0.0150-x \right )\left ( 0.0150-x \right )}=\dfrac{x^{2}}{\left ( 0.0150-x \right )^{2}}=0.160 \notag \)

We could solve this equation with the quadratic formula, but it is far easier to solve for x by recognizing that the left side of the equation is a perfect square; that is,

\[\dfrac{x^2}{(0.0150−x)^2}=\left(\dfrac{x}{0.0150−x}\right)^2=0.106 \notag \]

(The quadratic formula is presented in Essential Skills 7 in Section 15.7 .) Taking the square root of the middle and right terms,

\[\dfrac{x^2}{(0.0150−x)^2} =(0.106)^{1/2}=0.326 \notag \]

\[x =(0.326)(0.0150)−0.326x \notag \]

\[1.326x=0.00489 \notag \]

\[x =0.00369=3.69 \times 10^{−3} \notag \]

C The final concentrations of all species in the reaction mixture are as follows:

  • \([H_2]_f=[H_2]_i+Δ[H_2]=(0.0150−0.00369) \;M=0.0113\; M\)
  • \([CO_2]_f =[CO_2]_i+Δ[CO_2]=(0.0150−0.00369)\; M=0.0113\; M\)
  • \([H_2O]_f=[H_2O]_i+Δ[H_2O]=(0+0.00369) \;M=0.00369\; M\)
  • \([CO]_f=[CO]_i+Δ[CO]=(0+0.00369)\; M=0.00369 \;M\)

We can check our work by inserting the calculated values back into the equilibrium constant expression:

\[K=\dfrac{[H_2O][CO]}{[H_2][CO_2]}=\dfrac{(0.00369)^2}{(0.0113)^2}=0.107 \notag \]

To two significant figures, this K is the same as the value given in the problem, so our answer is confirmed.

Hydrogen gas reacts with iodine vapor to give hydrogen iodide according to the following chemical equation:

\[H_{2(g)}+I_{2(g)} \rightleftharpoons 2HI_{(g)} \notag \]

K = 54 at 425°C. If 0.172 M H 2 and I 2 are injected into a reactor and maintained at 425°C until the system equilibrates, what is the final concentration of each substance in the reaction mixture?

Answer: [HI] f = 0.270 M; [H 2 ] f = [I 2 ] f = 0.037 M

In Example 10, the initial concentrations of the reactants were the same, which gave us an equation that was a perfect square and simplified our calculations. Often, however, the initial concentrations of the reactants are not the same, and/or one or more of the products may be present when the reaction starts. Under these conditions, there is usually no way to simplify the problem, and we must determine the equilibrium concentrations with other means. Such a case is described in Example 11.

Example 15.3.4

In the water–gas shift reaction shown in Example 10, a sample containing 0.632 M CO 2 and 0.570 M H 2 is allowed to equilibrate at 700 K. At this temperature, K = 0.106. What is the composition of the reaction mixture at equilibrium?

Given: balanced equilibrium equation, concentrations of reactants, and K

Asked for: composition of reaction mixture at equilibrium

A Write the equilibrium equation. Construct a table showing the initial concentrations of all substances in the mixture. Complete the table showing the changes in the concentrations ( x ) and the final concentrations.

B Write the equilibrium constant expression for the reaction. Substitute the known K value and the final concentrations to solve for x .

C Calculate the final concentration of each substance in the reaction mixture. Check your answers by substituting these values into the equilibrium constant expression to obtain K .

A [CO 2 ] i = 0.632 M and [H 2 ] i = 0.570 M. Again, x is defined as the change in the concentration of H 2 O: Δ[H 2 O] = + x . Because 1 mol of CO is produced for every 1 mol of H 2 O, the change in the concentration of CO is the same as the change in the concentration of H 2 O, so Δ[CO] = + x . Similarly, because 1 mol each of H 2 and CO 2 are consumed for every 1 mol of H 2 O produced, Δ[H 2 ] = Δ[CO 2 ] = − x . The final concentrations are the sums of the initial concentrations and the changes in concentrations at equilibrium.

B We can now use the equilibrium equation and the known K value to solve for x :

\[K=\dfrac{[H_2O][CO]}{[H_2][CO_2]}=\dfrac{x^2}{(0.570−x)(0.632−x)}=0.106 \notag \]

In contrast to Example 10, however, there is no obvious way to simplify this expression. Thus we must expand the expression and multiply both sides by the denominator:

\[x^2 = 0.106(0.360 − 1.20x + x^2) \notag \]

Collecting terms on one side of the equation,

\[0.894x^2 + 0.127x − 0.0382 = 0 \notag \]

This equation can be solved using the quadratic formula:

\[ x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} = \dfrac{−0.127 \pm \sqrt{(0.127)^2−4(0.894)(−0.0382)}}{2(0.894)} \notag \]

\[x =0.148 \text{ and } −0.290 \notag \]

Only the answer with the positive value has any physical significance, so Δ[H 2 O] = Δ[CO] = +0.148 M, and Δ[H 2 ] = Δ[CO 2 ] = −0.148 M.

  • \([H_2]_f[ = [H_2]_i+Δ[H_2]=0.570 \;M −0.148\; M=0.422 M\)
  • \([CO_2]_f =[CO_2]_i+Δ[CO_2]=0.632 \;M−0.148 \;M=0.484 M\)
  • \([H_2O]_f =[H_2O]_i+Δ[H_2O]=0\; M+0.148\; M =0.148\; M\)
  • \([CO]_f=[CO]_i+Δ[CO]=0 M+0.148\;M=0.148 M\)

We can check our work by substituting these values into the equilibrium constant expression:

\[K=\dfrac{[H_2O][CO]}{[H_2][CO_2]}=\dfrac{(0.148)^2}{(0.422)(0.484)}=0.107 \notag \]

Because K is essentially the same as the value given in the problem, our calculations are confirmed.

The exercise in Example 8 showed the reaction of hydrogen and iodine vapor to form hydrogen iodide, for which K = 54 at 425°C. If a sample containing 0.200 M H 2 and 0.0450 M I 2 is allowed to equilibrate at 425°C, what is the final concentration of each substance in the reaction mixture?

Answer: [HI] f = 0.0882 M; [H 2 ] f = 0.156 M; [I 2 ] f = 9.2 × 10 −4 M

In many situations it is not necessary to solve a quadratic (or higher-order) equation. Most of these cases involve reactions for which the equilibrium constant is either very small ( K ≤ 10 −3 ) or very large ( K ≥ 10 3 ), which means that the change in the concentration (defined as x ) is essentially negligible compared with the initial concentration of a substance. Knowing this simplifies the calculations dramatically, as illustrated in Example 12.

Example 15.3.5

Atmospheric nitrogen and oxygen react to form nitric oxide:

\[N_{2(g)}+O_{2(g)} \rightleftharpoons 2NO_{(g)} \notag \]

K p = 2.0 × 10 −31 at 25°C. What is the partial pressure of NO in equilibrium with N 2 and O 2 in the atmosphere (at 1 atm, P{N 2 } = 0.78 atm and P{O 2 } = 0.21 atm

Given: balanced equilibrium equation and values of K p , P{O 2 } and P{N 2 }

Asked for: partial pressure of NO

A Construct a table and enter the initial partial pressures, the changes in the partial pressures that occur during the course of the reaction, and the final partial pressures of all substances.

B Write the equilibrium equation for the reaction. Then substitute values from the table to solve for the change in concentration ( x ).

C Calculate the partial pressure of NO. Check your answer by substituting values into the equilibrium equation and solving for K .

A Because we are given K p and partial pressures are reported in atmospheres, we will use partial pressures. The initial partial pressure of O 2 is 0.21 atm and that of N 2 is 0.78 atm. If we define the change in the partial pressure of NO as 2 x , then the change in the partial pressure of O 2 and of N 2 is − x because 1 mol each of N 2 and of O 2 is consumed for every 2 mol of NO produced. Each substance has a final partial pressure equal to the sum of the initial pressure and the change in that pressure at equilibrium.

B Substituting these values into the equation for the equilibrium constant,

\[K_p=\dfrac{(P_{NO})^2}{(P_{N_2})(P_{O_2})}=\dfrac{(2x)^2}{(0.78−x)(0.21−x)}=2.0 \times 10^{−31} \notag \]

In principle, we could multiply out the terms in the denominator, rearrange, and solve the resulting quadratic equation. In practice, it is far easier to recognize that an equilibrium constant of this magnitude means that the extent of the reaction will be very small; therefore, the x value will be negligible compared with the initial concentrations. If this assumption is correct, then to two significant figures, (0.78 − x ) = 0.78 and (0.21 − x ) = 0.21. Substituting these expressions into our original equation,

\[\dfrac{(2x)^2}{(0.78)(0.21)} = 2.0 \times 10^{−31} \notag \]

\[\dfrac{4x^2}{0.16} =2.0 \times10^{−31} \notag \]

\[x^2=\dfrac{0.33 \times 10^{−31}}{4} \notag \]

\[x^=9.1 \times 10^{−17} \notag \]

C Substituting this value of x into our expressions for the final partial pressures of the substances,

  • \(P_{NO}=2x \; atm=1.8 \times 10^{−16} \;atm \)
  • \(P_{N_2}=(0.78−x) \;atm=0.78 \;atm \)
  • \(P_{O_2}=(0.21−x) \;atm=0.21\; atm\)

From these calculations, we see that our initial assumption regarding x was correct: given two significant figures, 2.0 × 10 −16 is certainly negligible compared with 0.78 and 0.21. When can we make such an assumption? As a general rule, if x is less than about 5% of the total, or 10 −3 > K > 10 3 , then the assumption is justified. Otherwise, we must use the quadratic formula or some other approach. The results we have obtained agree with the general observation that toxic NO, an ingredient of smog, does not form from atmospheric concentrations of N 2 and O 2 to a substantial degree at 25°C. We can verify our results by substituting them into the original equilibrium equation:

\[K_p=\dfrac{(P_{NO})^2}{(P_{N_2})(P_{O_2})}=\dfrac{(1.8 \times 10^{−16})^2}{(0.78)(0.21)}=2.0 times 10^{−31} \notag \]

The final K p agrees with the value given at the beginning of this example.

Under certain conditions, oxygen will react to form ozone, as shown in the following equation:

\[H_{2(g)}+C_2H_{4(g)} \overset{Ni}{\rightleftharpoons} C_2H_{6(g)} \notag \]

K p = 2.5 × 10 −59 at 25°C. What ozone partial pressure is in equilibrium with oxygen in the atmosphere P(O 2 ) =0.21 atm ?

Answer: 4.8 × 10 −31 atm

Another type of problem that can be simplified by assuming that changes in concentration are negligible is one in which the equilibrium constant is very large ( K ≥ 10 3 ). A large equilibrium constant implies that the reactants are converted almost entirely to products, so we can assume that the reaction proceeds 100% to completion. When we solve this type of problem, we view the system as equilibrating from the products side of the reaction rather than the reactants side. This approach is illustrated in Example 13.

Example 15.3.6

The chemical equation for the reaction of hydrogen with ethylene (C 2 H 4 ) to give ethane (C 2 H 6 ) is as follows:

K = 9.6 × 10 18 at 25°C. If a mixture of 0.200 M H 2 and 0.155 M C 2 H 4 is maintained at 25°C in the presence of a powdered nickel catalyst, what is the equilibrium concentration of each substance in the mixture?

Given: balanced chemical equation, K , and initial concentrations of reactants

Asked for: equilibrium concentrations

A Construct a table showing initial concentrations, concentrations that would be present if the reaction were to go to completion, changes in concentrations, and final concentrations.

B Write the equilibrium constant expression for the reaction. Then substitute values from the table into the expression to solve for x (the change in concentration).

C Calculate the equilibrium concentrations. Check your answers by substituting these values into the equilibrium equation.

A From the magnitude of the equilibrium constant, we see that the reaction goes essentially to completion. Because the initial concentration of ethylene (0.155 M) is less than the concentration of hydrogen (0.200 M), ethylene is the limiting reactant; that is, no more than 0.155 M ethane can be formed from 0.155 M ethylene. If the reaction were to go to completion, the concentration of ethane would be 0.155 M and the concentration of ethylene would be 0 M. Because the concentration of hydrogen is greater than what is needed for complete reaction, the concentration of unreacted hydrogen in the reaction mixture would be 0.200 M − 0.155 M = 0.045 M. The equilibrium constant for the forward reaction is very large, so the equilibrium constant for the reverse reaction must be very small. The problem then is identical to that in Example 12. If we define − x as the change in the ethane concentration for the reverse reaction, then the change in the ethylene and hydrogen concentrations is + x . The final equilibrium concentrations are the sums of the concentrations for the forward and reverse reactions.

B Substituting values into the equilibrium constant expression,

\[K=\dfrac{[C_2H_6]}{[H_2][C_2H_4]}=\dfrac{0.155−x}{(0.045+x)x}=9.6 \times 10^{18} \notag \]

Once again, the magnitude of the equilibrium constant tells us that the equilibrium will lie far to the right as written, so the reverse reaction is negligible. Thus x is likely to be very small compared with either 0.155 M or 0.045 M, and the equation can be simplified [(0.045 + x ) = 0.045 and (0.155 − x ) = 0.155] as follows:

\[K=\dfrac{0.155}{0.045x} = 9.6 \times 10^{18} \notag \]

\[x=3.6 \times 10^{−19} \notag \]

C The small x value indicates that our assumption concerning the reverse reaction is correct, and we can therefore calculate the final concentrations by evaluating the expressions from the last line of the table:

  • \([C_2H_6]_f = (0.155 − x)\; M = 0.155 \; M\)
  • \([C_2H_4]_f = x\; M = 3.6 \times 10^{−19} M \)
  • \([H_2]_f = (0.045 + x) \;M = 0.045 \;M\)

We can verify our calculations by substituting the final concentrations into the equilibrium constant expression:

\[K=\dfrac{[C_2H_6]}{[H_2][C_2H_4]}=\dfrac{0.155}{(0.045)(3.6 \times 10^{−19})}=9.6 \times 10^{18} \notag \]

This K value agrees with our initial value at the beginning of the example.

Hydrogen reacts with chlorine gas to form hydrogen chloride:

\[H_{2(g)}+Cl_{2(g)} \rightleftharpoons 2HCl_{(g)} \notag \]

K p = 4.0 × 10 31 at 47°C. If a mixture of 0.257 M H 2 and 0.392 M Cl 2 is allowed to equilibrate at 47°C, what is the equilibrium composition of the mixture?

  • \([H_2]_f = 4.8 \times 10^{−32}\; M\)
  • \([Cl_2]_f = 0.135\; M\)
  • \([HCl]_f = 0.514\; M\)

When an equilibrium constant is calculated from equilibrium concentrations, molar concentrations or partial pressures are substituted into the equilibrium constant expression for the reaction. Equilibrium constants can be used to calculate the equilibrium concentrations of reactants and products by using the quantities or concentrations of the reactants, the stoichiometry of the balanced chemical equation for the reaction, and a tabular format to obtain the final concentrations of all species at equilibrium.

Key Takeaway

  • Various methods can be used to solve the two fundamental types of equilibrium problems: (1) those in which we calculate the concentrations of reactants and products at equilibrium and (2) those in which we use the equilibrium constant and the initial concentrations of reactants to determine the composition of the equilibrium mixture.

Conceptual Problems

Describe how to determine the magnitude of the equilibrium constant for a reaction when not all concentrations of the substances are known.

Calculations involving systems with very small or very large equilibrium constants can be dramatically simplified by making certain assumptions about the concentrations of products and reactants. What are these assumptions when K is (a) very large and (b) very small? Illustrate this technique using the system A + 2B ⇌ C for which you are to calculate the concentration of the product at equilibrium starting with only A and B. Under what circumstances should simplifying assumptions not be used?

Numerical Problems

Please be sure you are familiar with the topics discussed in Essential Skills 7 ( Section 15.7 ) before proceeding to the Numerical Problems.

In the equilibrium reaction A + B ⇌C, what happens to K if the concentrations of the reactants are doubled? tripled? Can the same be said about the equilibrium reaction A ⇌B + C?

The following table shows the reported values of the equilibrium P{O 2 } at three temperatures for the reaction Ag 2 O(s) ⇌ 2 Ag(s) + 1/2 O 2 (g) for which Δ H ° = 31 kJ/mol. Are these data consistent with what you would expect to occur? Why or why not?

Given the equilibrium system N 2 O 4 (g) ⇌ 2 NO 2 (g), what happens to K p if the initial pressure of N 2 O 4 is doubled? If K p is 1.7 × 10 −1 at 2300°C, and the system initially contains 100% N 2 O 4 at a pressure of 2.6 × 10 2 atm, what is the equilibrium pressure of each component?

At 430°C, 4.20 mol of HI in a 9.60 L reaction vessel reaches equilibrium according to the following equation: H 2 (g) + I 2 (g) ⇌2HI(g) At equilibrium, [H 2 ] = 0.047 M and [HI] = 0.345 M. What are K and K p for this reaction?

Methanol, a liquid used as an automobile fuel additive, is commercially produced from carbon monoxide and hydrogen at 300°C according to the following reaction: CO(g) + 2H 2 (g) ⇌ CH 3 OH(g) and K p = 1.3 × 10 −4 . If 56.0 g of CO is mixed with excess hydrogen in a 250 mL flask at this temperature, and the hydrogen pressure is continuously maintained at 100 atm, what would be the maximum percent yield of methanol? What pressure of hydrogen would be required to obtain a minimum yield of methanol of 95% under these conditions?

Starting with pure A, if the total equilibrium pressure is 0.969 atm for the reaction A (s ⇌ 2 B(g) + C(g), what is K p ?

The decomposition of ammonium carbamate to NH 3 and CO 2 at 40°C is written as NH 4 CO 2 NH 2 (s) ⇌ 2NH 3 (g) + CO 2 If the partial pressure of NH 3 at equilibrium is 0.242 atm, what is the equilibrium partial pressure of CO 2 ? What is the total gas pressure of the system? What is K p ?

At 375 K, K p for the reaction SO 2 Cl 2 (g) ⇌ SO 2 (g) + Cl 2 g) is 2.4, with pressures expressed in atmospheres. At 303 K, K p is 2.9 × 10 −2 .

  • What is K for the reaction at each temperature?
  • If a sample at 375 K has 0.100 M Cl 2 and 0.200 M SO 2 at equilibrium, what is the concentration of SO 2 Cl 2 ?
  • If the sample given in part b is cooled to 303 K, what is the pressure inside the bulb?

For the gas-phase reaction aA ⇌ bB, show that K p = K ( RT ) Δ n assuming ideal gas behavior.

For the gas-phase reaction I 2 ⇌2I, show that the total pressure is related to the equilibrium pressure by the following equation:

\[P_T=\sqrt{K_pP_{I_2}} + P_{I_2} \notag \]

Experimental data on the system Br2(l) ⇌ Br 2 (aq) are given in the following table. Graph [Br 2 ] versus moles of Br 2 (l) present; then write the equilibrium constant expression and determine K .

Data accumulated for the reaction n- butane(g) ⇌ isobutane(g) at equilibrium are shown in the following table. What is the equilibrium constant for this conversion? If 1 mol of n -butane is allowed to equilibrate under the same reaction conditions, what is the final number of moles of n -butane and isobutane?

Solid ammonium carbamate (NH 4 CO 2 NH 2 ) dissociates completely to ammonia and carbon dioxide when it vaporizes:

\[ NH_4CO_2NH_{2(s)} \rightleftharpoons 2NH_{3(g)}+CO_{2(g)} \notag \]

At 25°C, the total pressure of the gases in equilibrium with the solid is 0.116 atm. What is the equilibrium partial pressure of each gas? What is K p ? If the concentration of CO 2 is doubled and then equilibrates to its initial equilibrium partial pressure + x atm, what change in the NH 3 concentration is necessary for the system to restore equilibrium?

The equilibrium constant for the reaction COCl 2 (g) ⇌ CO(g) + Cl 2 (g) is K p = 2.2 × 10 −10 at 100°C. If the initial concentration of COCl 2 is 3.05 × 10 −3 M, what is the partial pressure of each gas at equilibrium at 100°C? What assumption can be made to simplify your calculations?

Aqueous dilution of IO 4 − results in the following reaction:

\[IO^−_{4(aq)}+2H_2O_{(l)} \rightleftharpoons H_4IO^−_{6(aq)} \notag \]

and K = 3.5 × 10 −2 . If you begin with 50 mL of a 0.896 M solution of IO 4 − that is diluted to 250 mL with water, how many moles of H 4 IO 6 − are formed at equilibrium?

Iodine and bromine react to form IBr, which then sublimes. At 184.4°C, the overall reaction proceeds according to the following equation:

\[I_{2(g)}+Br_{2(g)} \rightleftharpoons 2IBr_{(g)} \notag \]

K p = 1.2 × 10 2 . If you begin the reaction with 7.4 g of I 2 vapor and 6.3 g of Br 2 vapor in a 1.00 L container, what is the concentration of IBr(g) at equilibrium? What is the partial pressure of each gas at equilibrium? What is the total pressure of the system?

For the reaction

\[C_{(s)} + 12N_{2(g)}+\frac{5}{2}H_{2(g)} \rightleftharpoons CH3NH2(g) \notag \]

K = 1.8 × 10 −6 . If you begin the reaction with 1.0 mol of N 2 , 2.0 mol of H 2 , and sufficient C(s) in a 2.00 L container, what are the concentrations of N 2 and CH 3 NH 2 at equilibrium? What happens to K if the concentration of H 2 is doubled?

Contributors

Modified by Joshua B. Halpern

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May 1, 2024

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Researchers create new chemical compound to solve 120-year-old problem

by University of Minnesota College of Science and Engineering

Researchers create new chemical compound to solve 120-year-old problem

For the first time, chemists in the University of Minnesota Twin Cities College of Science and Engineering have created a highly reactive chemical compound that has eluded scientists for more than 120 years. The discovery could lead to new drug treatments, safer agricultural products, and better electronics. The study is published in Science .

For decades, researchers have been investigating molecules called N-heteroarenes, which are ring-shaped chemical compounds that contain one or more nitrogen atoms . Bio-active molecules having a N-heteroarene core are widely used for numerous medicinal applications, lifesaving pharmaceuticals, pesticides and herbicides, and even electronics.

"While the average person does not think about heterocycles on a daily basis, these unique nitrogen-containing molecules are widely applied across all facets of human life," said Courtney Roberts, the senior author of the study and a University of Minnesota Department of Chemistry assistant professor who holds the 3M Alumni Professorship.

These molecules are highly sought out by many industries, but are extremely challenging for chemists to make. Previous strategies have been able to target these specific molecules, but scientists have not been able to create a series of these molecules.

One reason for this is that these molecules are extremely reactive. They are so active that chemists have used computational modeling to predict that they should be impossible to make. This has created challenges for more than a century and prevented a solution to create this chemical substance.

"What we were able to do was to run these chemical reactions with specialized equipment while getting rid of elements commonly found in our atmosphere," said Jenna Humke, a University of Minnesota chemistry graduate student and lead author on the paper. "Luckily, we have the tools to do that at the University of Minnesota. We ran experiments under nitrogen in a closed-chamber glovebox, which creates a chemically inactive environment to test and move samples."

These experiments were accomplished by using organometallic catalysis—the interaction between metals and organic molecules . The research required collaboration between both organic and inorganic chemists. This is something that is common at the University of Minnesota.

"We were able to solve this long-standing challenge because the University of Minnesota Department of Chemistry is unique in that we don't have formal divisions," Roberts added. "This allows us to put together a team of experts in all fields of chemistry, which was a vital component in completing this project."

After introducing the chemical compound in this paper, the next steps will be to make it widely available to chemists across multiple fields to streamline the creation process. This could help solve important problems like preventing food scarcity and treating illnesses to save lives.

Journal information: Science

Provided by University of Minnesota College of Science and Engineering

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Julie Radico Psy.D. ABPP

Self-Esteem

It’s ok you can’t solve every problem, trying to “fix" everything can leave you feeling like a failure..

Updated May 10, 2024 | Reviewed by Ray Parker

  • What Is Self-Esteem?
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  • Your intrinsic value is more than what you can do for other people.

You are still worthwhile and can be successful, even if you don’t have all the solutions.

  • Consider which decision will make you feel you’ve stayed true to your values.

In coaching others, I often discuss problem-solving strategies to help individuals think creatively and consider many options when they are faced with challenging situations.

Problem solving 1-2 includes the following:

  • Define the problem, identify obstacles, and set realistic goals .
  • Generate a variety of alternative solutions to overcome obstacles identified.
  • Choose which idea has the highest likelihood to achieve the goal.
  • Try out the solution in real-life and see if it worked or not.

Problem-solving strategies can be helpful in many situations. Thinking creatively and testing out different potential solutions can help you come up with alternative ways of solving your problems.

While many problems can be solved, there are also situations in which there is no “perfect” solution or in which what seems to be the best solution still leaves you feeling unsatisfied or like you’re not doing enough.

I encourage you to increase your comfort around the following three truths:

1. You can’t always solve everyone else’s problems.

2. You can’t always solve all of your own problems.

3. You are not a failure if you can’t solve every problem.

Source: Hans-Peter Gauster / Unsplash

You can’t always solve everyone else’s problems.

When someone around you needs help, do you feel compelled to find solutions to their problem?

Are you seen as the problem solver at your job or in your close relationships?

Does it feel uncomfortable for you to listen to someone tell you about a problem and not offer solutions?

There are times when others come to you because they know you can help them solve a problem. There are also times when the other person is coming to you not for a solution to their problem, but for support, empathy, and a listening ear.

Your relationships may be negatively impacted if others feel that you don’t fully listen and only try to “fix” everything for them. While this may feel like a noble act, it may lead the other person to feel like they have failed or that you think they are unable to solve their own problems.

Consider approaching such situations with curiosity by saying to the other person:

  • As you share this information with me, tell me how I can best support you.
  • What would be most helpful right now? Are you looking for an empathetic ear or want to brainstorm potential next steps?
  • I want to be sure I am as helpful as I can be right now; what are you hoping to get out of our conversation?

You can’t always solve all of your own problems.

We are taught from a young age that problems have a solution. For example, while solving word problems in math class may not have been your favorite thing to do, you knew there was ultimately a “right” answer. Many times, the real world is much more complex, and many of the problems that you face do not have clear or “right” answers.

You may often be faced with finding solutions that do the most good for the most amount of people, but you know that others may still be left out or feel unsatisfied with the result.

Your beliefs about yourself, other people, and the world can sometimes help you make decisions in such circumstances. You may ask for help from others. Some may consider their faith or spirituality for guidance. While others may consider philosophical theories.

Knowing that there often isn’t a “perfect” solution, you may consider asking yourself some of the following questions:

  • What’s the healthiest decision I can make? The healthiest decision for yourself and for those who will be impacted.
  • Imagine yourself 10 years in the future, looking back on the situation: What do you think the future-you would encourage you to do?
  • What would a wise person do?
  • What decision will allow you to feel like you’ve stayed true to your values?

You are not a failure if you can’t solve all of the problems.

If you have internalized feeling like you need to be able to solve every problem that comes across your path, you may feel like a failure each time you don’t.

It’s impossible to solve every problem.

what is the problem solving of chemistry

Your intrinsic value is more than what you can do for other people. You have value because you are you.

Consider creating more realistic and adaptive thoughts around your ability to help others and solve problems.

Some examples include:

  • I am capable, even without solving all of the problems.
  • I am worthwhile, even if I’m not perfect.
  • What I do for others does not define my worth.
  • In living my values, I know I’ve done my best.

I hope you utilize the information above to consider how you can coach yourself the next time you:

  • Start to solve someone else’s problem without being asked.
  • Feel stuck in deciding the best next steps.
  • Judge yourself negatively.

1. D'zurilla, T. J., & Goldfried, M. R. (1971). Problem solving and behavior modification. Journal of abnormal psychology, 78(1), 107.

2. D’Zurilla, T. J., & Nezu, A. M. (2010). Problem-solving therapy. Handbook of cognitive-behavioral therapies, 3(1), 197-225.

Julie Radico Psy.D. ABPP

Julie Radico, Psy.D. ABPP, is a board-certified clinical psychologist and coauthor of You Will Get Through This: A Mental Health First-Aid Kit.

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A psychologist explains love’s ‘chemistry-or-compatibility’ problem.

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Here’s why the effervescent "initial spark" cannot be used as a yardstick to predict the longevity ... [+] of a romantic bond.

Many people come to therapy carrying the weight of disappointment and disillusionment from their failed relationships. They may say things like:

  • “I’m still reeling from the shock. Everything felt perfect, like it was meant to be. Now, I question it all. How could something so right go so wrong?”
  • “I thought I found my soulmate. Now, I see cracks in our foundation. Arguments, misunderstandings, feeling distant—it’s become a daily occurrence. How do I move on from this?”

In the initial stages of romance, it’s common for people to be enchanted by the sparks of chemistry. The intense attraction, the palpable tension, the intimate moments—all contribute to a feeling of electricity that can cloud judgment. However, beyond this captivating charm lies the foundation of compatibility, the complex mix of shared beliefs, aspirations and communication methods that sustain a relationship over time.

Recognizing the vital difference between chemistry and compatibility is essential for building healthy and satisfying relationships. While chemistry ignites passion, compatibility nurtures the core of a lasting bond. Failing to distinguish between the two can lead to various consequences, from turbulent relationships filled with disagreement to the painful realization that the passion was only temporary. These experiences cut deep, leaving scars that linger long after the immediate wounds have healed.

Here are three reasons why we’re often drawn to initial chemistry, leading us to conveniently overlook long-term compatibility.

1. The Pull Of Attraction Bias

We’re naturally inclined to prioritize physical attraction and chemistry when selecting partners. This instinct, rooted in our evolutionary past, once ensured the survival of our species—guiding our ancestors towards mates who appeared healthy and fertile. Today, while survival concerns are less immediate, the allure of physical attraction remains strong. That initial spark between two people can be irresistible, capturing our attention and kindling our passions.

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Recent research supports the idea that both physical attraction and compatibility contribute to the trajectory of a relationship, starting from the moment potential partners’ first meet. This aligns with evolutionary theories of human pair bonding, suggesting that individuals seek partners with qualities advantageous for relationship success. Thus, initial impressions extend beyond physical attraction and can significantly influence how relationships evolve over time.

Fixating solely on immediate physical attraction may cause us to overlook critical compatibility factors like shared values, communication styles and life goals. In the heat of the moment, it’s easy to prioritize the thrill of chemistry over the deeper considerations of long-term compatibility.

This preference for chemistry over compatibility can lead to significant repercussions for relationships. Establishing connections solely based on surface attraction can result in an unstable foundation. While passion may be present, genuine compatibility might be absent. Without the solid groundwork of shared values and understanding, relationships may struggle when confronted with the realities of everyday life.

2. Cognitive Dissonance And Rationalization

When people experience intense chemistry with someone, the flood of emotions can overpower any doubts about long-term compatibility. This infatuation might blind them to warning signs or inconsistencies in the relationship, believing that their intense connection will overcome any challenge.

This gives rise to cognitive dissonance. While there's an undeniable attraction drawing people together, there may also be subtle indications of incompatibility—such as divergent values or communication styles, hinting at future challenges.

To alleviate this discomfort, people resort to rationalization and denial. Studies indicate that when making decisions, our brains rapidly spin justifications for our choices, often without extended thought. This occurs in the moment, with our brains possibly adjusting our emotions to match our decisions or vice versa. The findings also clarify that people might downplay the importance of compatibility, assuming their chemistry to be sufficient to sustain the relationship. Rationalizations such as “Our bond is so strong, we’ll overcome any obstacle” or “Our differences enhance our connection” act as psychological defenses against doubt and uncertainty.

Despite these efforts to rationalize their feelings, the tension between chemistry and compatibility persists. As the relationship progresses, challenges may arise, highlighting the differences that were overlooked in the beginning. Problems with communication, conflicts over values and differing life goals can weaken the foundation of the relationship.

3. Emotional Needs And Unconscious Desires

Our emotional needs and desires play a significant role in compatibility in relationships. Seeking partners who fulfill immediate emotional or psychological needs, such as validation, excitement or a sense of belonging drives this inclination.

At its core, individuals are drawn to partners who offer validation through compliments, attention or physical affection. Feeling understood and cherished can obscure potential incompatibilities, as they trigger the euphoria of feeling valued. Similarly, the thrill of the chase often overshadows practical considerations. In the heat of passion, people may become swept away by chemistry, overlooking long-term implications.

Furthermore, the desire for belonging and companionship can lead people to seek out partners who provide comfort from loneliness. Numerous studies suggest that individuals who fear being single tend to settle for less in their romantic relationships. Consequently, despite differences in values or communication styles, they may mistake mere chemistry for genuine compatibility.

Not understanding one’s emotional needs drives individuals to repeatedly choose partners who provide short-term satisfaction. The allure of chemistry may lead them to prioritize immediate gratification over long-term compatibility. However, through self-awareness and reflection, individuals can navigate their desires and cultivate relationships that are both thrilling and fulfilling.

Are you concerned about the compatibility piece of your relationship? Take the evidence-based Relationship Satisfaction Scale to learn more.

Mark Travers

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what is the problem solving of chemistry

New Journal of Chemistry

Study on the removal of so 4 2− and ca 2+ from potassium chloride brine via a method combining calcium chloride and carbon dioxide.

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* Corresponding authors

a National Fundamental Research Laboratory of New Hazardous Chemicals Assessment and Accident Analysis, Institute of Applied Electrochemistry, Beijing University of Chemical Technology, Beijing 100029, China E-mail: [email protected] Fax: +8610-64435452 Tel: +8610-64435452

b State Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology, Beijing 100029, China E-mail: [email protected]

Purification treatment is needed to control the concentration of SO 4 2− and Ca 2+ in brine to solve the problem of low electrolysis efficiency for the electrolysis of KCl solution in the chlor-alkali industry. However, the traditional barium method to remove SO 4 2− faces the problem of high cost and environmental pollution. Herein, inexpensive and environmentally friendly anhydrous calcium chloride was used as the precipitant to replace barium chloride. The conditions for SO 4 2− removal at room temperature were optimized by studying the precipitant doping ratio, reaction time, and stirring speed. The results indicated that the concentration of SO 4 2− in brine could be reduced to less than 5 g L −1 at a raw material ratio of n Ca 2+ to n SO 4 2− of 1.1 : 1, a stirring speed of 300 rpm, and a reaction time of 120 min. Ca 2+ in the solution could be removed by introducing carbon dioxide, and the removal rate of Ca 2+ could reach 99.89% by adjusting the pH of the solution to an appropriate value. This work paves a new way for efficient brine refining to boost sustainable development in the chlor-alkali industry.

Graphical abstract: Study on the removal of SO42− and Ca2+ from potassium chloride brine via a method combining calcium chloride and carbon dioxide

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what is the problem solving of chemistry

H. Ren, Q. Wang, Y. Sun, Y. Chen, P. Wan and J. Pan, New J. Chem. , 2024, Advance Article , DOI: 10.1039/D4NJ01044B

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What is climate change mitigation and why is it urgent?

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What is climate change mitigation and why is it urgent?

  • Climate change mitigation involves actions to reduce or prevent greenhouse gas emissions from human activities.
  • Mitigation efforts include transitioning to renewable energy sources, enhancing energy efficiency, adopting regenerative agricultural practices and protecting and restoring forests and critical ecosystems.
  • Effective mitigation requires a whole-of-society approach and structural transformations to reduce emissions and limit global warming to 1.5°C above pre-industrial levels.
  • International cooperation, for example through the Paris Agreement, is crucial in guiding and achieving global and national mitigation goals.
  • Mitigation efforts face challenges such as the world's deep-rooted dependency on fossil fuels, the increased demand for new mineral resources and the difficulties in revamping our food systems.
  • These challenges also offer opportunities to improve resilience and contribute to sustainable development.

What is climate change mitigation?

Climate change mitigation refers to any action taken by governments, businesses or people to reduce or prevent greenhouse gases, or to enhance carbon sinks that remove them from the atmosphere. These gases trap heat from the sun in our planet’s atmosphere, keeping it warm. 

Since the industrial era began, human activities have led to the release of dangerous levels of greenhouse gases, causing global warming and climate change. However, despite unequivocal research about the impact of our activities on the planet’s climate and growing awareness of the severe danger climate change poses to our societies, greenhouse gas emissions keep rising. If we can slow down the rise in greenhouse gases, we can slow down the pace of climate change and avoid its worst consequences.

Reducing greenhouse gases can be achieved by:

  • Shifting away from fossil fuels : Fossil fuels are the biggest source of greenhouse gases, so transitioning to modern renewable energy sources like solar, wind and geothermal power, and advancing sustainable modes of transportation, is crucial.
  • Improving energy efficiency : Using less energy overall – in buildings, industries, public and private spaces, energy generation and transmission, and transportation – helps reduce emissions. This can be achieved by using thermal comfort standards, better insulation and energy efficient appliances, and by improving building design, energy transmission systems and vehicles.
  • Changing agricultural practices : Certain farming methods release high amounts of methane and nitrous oxide, which are potent greenhouse gases. Regenerative agricultural practices – including enhancing soil health, reducing livestock-related emissions, direct seeding techniques and using cover crops – support mitigation, improve resilience and decrease the cost burden on farmers.
  • The sustainable management and conservation of forests : Forests act as carbon sinks , absorbing carbon dioxide and reducing the overall concentration of greenhouse gases in the atmosphere. Measures to reduce deforestation and forest degradation are key for climate mitigation and generate multiple additional benefits such as biodiversity conservation and improved water cycles.
  • Restoring and conserving critical ecosystems : In addition to forests, ecosystems such as wetlands, peatlands, and grasslands, as well as coastal biomes such as mangrove forests, also contribute significantly to carbon sequestration, while supporting biodiversity and enhancing climate resilience.
  • Creating a supportive environment : Investments, policies and regulations that encourage emission reductions, such as incentives, carbon pricing and limits on emissions from key sectors are crucial to driving climate change mitigation.

Photo: Stephane Bellerose/UNDP Mauritius

Photo: Stephane Bellerose/UNDP Mauritius

Photo: La Incre and Lizeth Jurado/PROAmazonia

Photo: La Incre and Lizeth Jurado/PROAmazonia

What is the 1.5°C goal and why do we need to stick to it?

In 2015, 196 Parties to the UN Climate Convention in Paris adopted the Paris Agreement , a landmark international treaty, aimed at curbing global warming and addressing the effects of climate change. Its core ambition is to cap the rise in global average temperatures to well below 2°C above levels observed prior to the industrial era, while pursuing efforts to limit the increase to 1.5°C.

The 1.5°C goal is extremely important, especially for vulnerable communities already experiencing severe climate change impacts. Limiting warming below 1.5°C will translate into less extreme weather events and sea level rise, less stress on food production and water access, less biodiversity and ecosystem loss, and a lower chance of irreversible climate consequences.

To limit global warming to the critical threshold of 1.5°C, it is imperative for the world to undertake significant mitigation action. This requires a reduction in greenhouse gas emissions by 45 percent before 2030 and achieving net-zero emissions by mid-century.

What are the policy instruments that countries can use to drive mitigation?

Everyone has a role to play in climate change mitigation, from individuals adopting sustainable habits and advocating for change to governments implementing regulations, providing incentives and facilitating investments. The private sector, particularly those businesses and companies responsible for causing high emissions, should take a leading role in innovating, funding and driving climate change mitigation solutions. 

International collaboration and technology transfer is also crucial given the global nature and size of the challenge. As the main platform for international cooperation on climate action, the Paris Agreement has set forth a series of responsibilities and policy tools for its signatories. One of the primary instruments for achieving the goals of the treaty is Nationally Determined Contributions (NDCs) . These are the national climate pledges that each Party is required to develop and update every five years. NDCs articulate how each country will contribute to reducing greenhouse gas emissions and enhance climate resilience.   While NDCs include short- to medium-term targets, long-term low emission development strategies (LT-LEDS) are policy tools under the Paris Agreement through which countries must show how they plan to achieve carbon neutrality by mid-century. These strategies define a long-term vision that gives coherence and direction to shorter-term national climate targets.

Photo: Mucyo Serge/UNDP Rwanda

Photo: Mucyo Serge/UNDP Rwanda

Photo: William Seal/UNDP Sudan

Photo: William Seal/UNDP Sudan

At the same time, the call for climate change mitigation has evolved into a call for reparative action, where high-income countries are urged to rectify past and ongoing contributions to the climate crisis. This approach reflects the UN Framework Convention on Climate Change (UNFCCC) which advocates for climate justice, recognizing the unequal historical responsibility for the climate crisis, emphasizing that wealthier countries, having profited from high-emission activities, bear a greater obligation to lead in mitigating these impacts. This includes not only reducing their own emissions, but also supporting vulnerable countries in their transition to low-emission development pathways.

Another critical aspect is ensuring a just transition for workers and communities that depend on the fossil fuel industry and its many connected industries. This process must prioritize social equity and create alternative employment opportunities as part of the shift towards renewable energy and more sustainable practices.

For emerging economies, innovation and advancements in technology have now demonstrated that robust economic growth can be achieved with clean, sustainable energy sources. By integrating renewable energy technologies such as solar, wind and geothermal power into their growth strategies, these economies can reduce their emissions, enhance energy security and create new economic opportunities and jobs. This shift not only contributes to global mitigation efforts but also sets a precedent for sustainable development.

What are some of the challenges slowing down climate change mitigation efforts?

Mitigating climate change is fraught with complexities, including the global economy's deep-rooted dependency on fossil fuels and the accompanying challenge of eliminating fossil fuel subsidies. This reliance – and the vested interests that have a stake in maintaining it – presents a significant barrier to transitioning to sustainable energy sources.

The shift towards decarbonization and renewable energy is driving increased demand for critical minerals such as copper, lithium, nickel, cobalt, and rare earth metals. Since new mining projects can take up to 15 years to yield output, mineral supply chains could become a bottleneck for decarbonization efforts. In addition, these minerals are predominantly found in a few, mostly low-income countries, which could heighten supply chain vulnerabilities and geopolitical tensions.

Furthermore, due to the significant demand for these minerals and the urgency of the energy transition, the scaled-up investment in the sector has the potential to exacerbate environmental degradation, economic and governance risks, and social inequalities, affecting the rights of Indigenous Peoples, local communities, and workers. Addressing these concerns necessitates implementing social and environmental safeguards, embracing circular economy principles, and establishing and enforcing responsible policies and regulations .

Agriculture is currently the largest driver of deforestation worldwide. A transformation in our food systems to reverse the impact that agriculture has on forests and biodiversity is undoubtedly a complex challenge. But it is also an important opportunity. The latest IPCC report highlights that adaptation and mitigation options related to land, water and food offer the greatest potential in responding to the climate crisis. Shifting to regenerative agricultural practices will not only ensure a healthy, fair and stable food supply for the world’s population, but also help to significantly reduce greenhouse gas emissions.  

Photo: UNDP India

Photo: UNDP India

Photo: Nino Zedginidze/UNDP Georgia

Photo: Nino Zedginidze/UNDP Georgia

What are some examples of climate change mitigation?

In Mauritius , UNDP, with funding from the Green Climate Fund, has supported the government to install battery energy storage capacity that has enabled 50 MW of intermittent renewable energy to be connected to the grid, helping to avoid 81,000 tonnes of carbon dioxide annually. 

In Indonesia , UNDP has been working with the government for over a decade to support sustainable palm oil production. In 2019, the country adopted a National Action Plan on Sustainable Palm Oil, which was collaboratively developed by government, industry and civil society representatives. The plan increased the adoption of practices to minimize the adverse social and environmental effects of palm oil production and to protect forests. Since 2015, 37 million tonnes of direct greenhouse gas emissions have been avoided and 824,000 hectares of land with high conservation value have been protected.

In Moldova and Paraguay , UNDP has helped set up Green City Labs that are helping build more sustainable cities. This is achieved by implementing urban land use and mobility planning, prioritizing energy efficiency in residential buildings, introducing low-carbon public transport, implementing resource-efficient waste management, and switching to renewable energy sources. 

UNDP has supported the governments of Brazil, Costa Rica, Ecuador and Indonesia to implement results-based payments through the REDD+ (Reducing emissions from deforestation and forest degradation in developing countries) framework. These include payments for environmental services and community forest management programmes that channel international climate finance resources to local actors on the ground, specifically forest communities and Indigenous Peoples. 

UNDP is also supporting small island developing states like the Comoros to invest in renewable energy and sustainable infrastructure. Through the Africa Minigrids Program , solar minigrids will be installed in two priority communities, Grand Comore and Moheli, providing energy access through distributed renewable energy solutions to those hardest to reach.

And in South Africa , a UNDP initative to boost energy efficiency awareness among the general population and improve labelling standards has taken over commercial shopping malls.

What is climate change mitigation and why is it urgent?

What is UNDP’s role in supporting climate change mitigation?

UNDP aims to assist countries with their climate change mitigation efforts, guiding them towards sustainable, low-carbon and climate-resilient development. This support is in line with achieving the Sustainable Development Goals (SDGs), particularly those related to affordable and clean energy (SDG7), sustainable cities and communities (SDG11), and climate action (SDG13). Specifically, UNDP’s offer of support includes developing and improving legislation and policy, standards and regulations, capacity building, knowledge dissemination, and financial mobilization for countries to pilot and scale-up mitigation solutions such as renewable energy projects, energy efficiency initiatives and sustainable land-use practices. 

With financial support from the Global Environment Facility and the Green Climate Fund, UNDP has an active portfolio of 94 climate change mitigation projects in 69 countries. These initiatives are not only aimed at reducing greenhouse gas emissions, but also at contributing to sustainable and resilient development pathways.

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson and Calcea Johnson

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Michelle Blouin Williams

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Bill Whitaker with Calcea Johnson and Ne'Kiya Jackson

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

St. Mary's Academy president and interim principal Pamela Rogers

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Bill Whitaker and Gloria Ladson-Billings

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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COMMENTS

  1. Introduction − The Many Types and Kinds of Chemistry Problems

    Problem solving is a complex set of activities, processes, and behaviors for which various models have been used at various times. Specifically, "problem solving is a process by which the learner discovers a combination of previously learned rules that they can apply to achieve a solution to a new situation (that is, the problem)". 2 Zoller identifies problem solving, along with critical ...

  2. 1.12: Scientific Problem Solving

    The scientific method, as developed by Bacon and others, involves several steps: Ask a question - identify the problem to be considered. Make observations - gather data that pertains to the question. Propose an explanation (a hypothesis) for the observations. Make new observations to test the hypothesis further.

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    2. Attempt a different method. If you have time, solve the problem again but this time try to use a different method. Choose the answer you are more comfortable with or the answer that makes more sense in the context of the problem. For example, if you are working on a reaction try a different method to reach your final product, doing this can ...

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    1.8: Solving Chemical Problems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. Measurements are made using a variety of units. It is often useful or necessary to convert a measured quantity from one unit into another.

  6. PDF PROBLEM SOLVING IN CHEMISTRY

    on problem solving for several reasons. First, problem solving is what chemists do, regardless of whether they work in the area of synthesis, spectroscopy, theory, analysis, or the characterization of compounds. Second, it was clear that individuals who were successful in chemistry courses either developed good problem solving skills — more or

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    Henry Agnew (UC Davis) 2.7: Solving Problems Involving Equations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. Many problems in chemistry involve manipulating equations and require the use of multiple conversion steps. Such problems easy to solve as numerical problems once you understand how to ...

  8. Practice Chemistry

    Chemistry Courses. Take a guided, problem-solving based approach to learning Chemistry. These compilations provide unique perspectives and applications you won't find anywhere else. The Chemical Reaction. What's inside. Introduction; Chemical Kinetics ...

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    After running a chemical reaction, one often wants to know how the reaction went by computing the reaction yields. Step-by-step solutions are available for computing the amount of reactants needed and the theoretical yield in addition to the percent yield. The use of stoichiometric factors to generate the desired values is explained in detail.

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    Problem solving is central to the teaching and learning of chemistry at secondary, tertiary and post-tertiary levels of education, opening to students and professional chemists alike a whole new world for analysing data, looking for patterns and making deductions.

  11. Visualization and Problem Solving for General Chemistry

    Visualization and Problem Solving for General Chemistry. Table of Contents: States of Matter: Elements, Compounds & Mixtures: Liquids: Solutions

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    Creative problem-solving in chemistry. Explore a range of topics through open-ended experiments, where learners can devise their own testing plans. Identifying four unknown solutions. Allow learner's the opportunity to devise their own testing protocols to identify chloride ions in four solutions.

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    Problem solving in any area is a very complex process. It involves an understanding of the language in which the problem is stated, the interpretation of what is given in the problem and what is sought, an understanding of the science concepts involved in the solution, and the ability to perform mathematical operations if these are involved in ...

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    Mental models: The role of representations in problem-solving in chemistry. University Chemistry Education, 4, 24-30. Google Scholar. Bowen, C.W. (1990). Representational systems used by graduate students while problem-solving in organic synthesis. Journal of Research in Science Teaching, 27, 351-370.

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    Preview. 71 terms. Preview. Study with Quizlet and memorize flashcards containing terms like What is a general approach to solving a problem?, What are the steps for solving numeric problems?, What are the steps of solving nonnumeric problems? and more.

  16. Chemistry Calculator

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  17. Mathway

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  18. Chemistry Assistant

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  20. Computational Problem Solving in the Chemical Sciences

    This course is designed to bridge this gap. It provides a comprehensive introduction to the mathematical and computational skills necessary to model chemical phenomena at the atomic level. We start by building a strong foundation in mathematical representations of chemical problems, utilizing open-source software tools for problem-solving, data ...

  21. AI Chemistry Solver

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  22. The kinematic equations (article)

    The kinematic equations are listed below. 1. v = v 0 + a t. 2. Δ x = ( v + v 0 2) t. 3. Δ x = v 0 t + 1 2 a t 2. 4. v 2 = v 0 2 + 2 a Δ x. Since the kinematic equations are only accurate if the acceleration is constant during the time interval considered, we have to be careful to not use them when the acceleration is changing.

  23. Chapter 15.3: Solving Equilibrium Problems

    Solution: A The first step in any such problem is to balance the chemical equation for the reaction (if it is not already balanced) and use it to derive the equilibrium constant expression. In this case, the equation is already balanced, and the equilibrium constant expression is as follows: K = [NO2]2[Cl2] [NOCl]2.

  24. Researchers create new chemical compound to solve 120-year-old problem

    DOI: 10.1126/science.adi1606. For the first time, chemists in the University of Minnesota Twin Cities College of Science and Engineering have created a highly reactive chemical compound that has ...

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    The project team will use the results of this research to develop and evaluate instructional modules to better support organic chemistry students? learning and problem solving. This IUSE: EHR project intends to systematically investigate the effects of different molecular representations on students? success in solving organic chemistry problems.

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    Without the solid groundwork of shared values and understanding, relationships may struggle when confronted with the realities of everyday life. 2. Cognitive Dissonance And Rationalization. When ...

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