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Introductory Chemistry - 1st Canadian Edition: Rate Laws

Introductory Chemistry - 1st Canadian Edition
Rate Laws
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table of contents
  1. Cover
  2. Title Page
  3. Copyright
  4. Table Of Contents
  5. Acknowledgments
  6. Dedication
  7. About BCcampus Open Education
  8. Chapter 1. What is Chemistry
    1. Some Basic Definitions
    2. Chemistry as a Science
  9. Chapter 2. Measurements
    1. Expressing Numbers
    2. Significant Figures
    3. Converting Units
    4. Other Units: Temperature and Density
    5. Expressing Units
    6. End-of-Chapter Material
  10. Chapter 3. Atoms, Molecules, and Ions
    1. Acids
    2. Ions and Ionic Compounds
    3. Masses of Atoms and Molecules
    4. Molecules and Chemical Nomenclature
    5. Atomic Theory
    6. End-of-Chapter Material
  11. Chapter 4. Chemical Reactions and Equations
    1. The Chemical Equation
    2. Types of Chemical Reactions: Single- and Double-Displacement Reactions
    3. Ionic Equations: A Closer Look
    4. Composition, Decomposition, and Combustion Reactions
    5. Oxidation-Reduction Reactions
    6. Neutralization Reactions
    7. End-of-Chapter Material
  12. Chapter 5. Stoichiometry and the Mole
    1. Stoichiometry
    2. The Mole
    3. Mole-Mass and Mass-Mass Calculations
    4. Limiting Reagents
    5. The Mole in Chemical Reactions
    6. Yields
    7. End-of-Chapter Material
  13. Chapter 6. Gases
    1. Pressure
    2. Gas Laws
    3. Other Gas Laws
    4. The Ideal Gas Law and Some Applications
    5. Gas Mixtures
    6. Kinetic Molecular Theory of Gases
    7. Molecular Effusion and Diffusion
    8. Real Gases
    9. End-of-Chapter Material
  14. Chapter 7. Energy and Chemistry
    1. Formation Reactions
    2. Energy
    3. Stoichiometry Calculations Using Enthalpy
    4. Enthalpy and Chemical Reactions
    5. Work and Heat
    6. Hess’s Law
    7. End-of-Chapter Material
  15. Chapter 8. Electronic Structure
    1. Light
    2. Quantum Numbers for Electrons
    3. Organization of Electrons in Atoms
    4. Electronic Structure and the Periodic Table
    5. Periodic Trends
    6. End-of-Chapter Material
  16. Chapter 9. Chemical Bonds
    1. Lewis Electron Dot Diagrams
    2. Electron Transfer: Ionic Bonds
    3. Covalent Bonds
    4. Other Aspects of Covalent Bonds
    5. Violations of the Octet Rule
    6. Molecular Shapes and Polarity
    7. Valence Bond Theory and Hybrid Orbitals
    8. Molecular Orbitals
    9. End-of-Chapter Material
  17. Chapter 10. Solids and Liquids
    1. Properties of Liquids
    2. Solids
    3. Phase Transitions: Melting, Boiling, and Subliming
    4. Intermolecular Forces
    5. End-of-Chapter Material
  18. Chapter 11. Solutions
    1. Colligative Properties of Solutions
    2. Concentrations as Conversion Factors
    3. Quantitative Units of Concentration
    4. Colligative Properties of Ionic Solutes
    5. Some Definitions
    6. Dilutions and Concentrations
    7. End-of-Chapter Material
  19. Chapter 12. Acids and Bases
    1. Acid-Base Titrations
    2. Strong and Weak Acids and Bases and Their Salts
    3. Brønsted-Lowry Acids and Bases
    4. Arrhenius Acids and Bases
    5. Autoionization of Water
    6. Buffers
    7. The pH Scale
    8. End-of-Chapter Material
  20. Chapter 13. Chemical Equilibrium
    1. Chemical Equilibrium
    2. The Equilibrium Constant
    3. Shifting Equilibria: Le Chatelier’s Principle
    4. Calculating Equilibrium Constant Values
    5. Some Special Types of Equilibria
    6. End-of-Chapter Material
  21. Chapter 14. Oxidation and Reduction
    1. Oxidation-Reduction Reactions
    2. Balancing Redox Reactions
    3. Applications of Redox Reactions: Voltaic Cells
    4. Electrolysis
    5. End-of-Chapter Material
  22. Chapter 15. Nuclear Chemistry
    1. Units of Radioactivity
    2. Uses of Radioactive Isotopes
    3. Half-Life
    4. Radioactivity
    5. Nuclear Energy
    6. End-of-Chapter Material
  23. Chapter 16. Organic Chemistry
    1. Hydrocarbons
    2. Branched Hydrocarbons
    3. Alkyl Halides and Alcohols
    4. Other Oxygen-Containing Functional Groups
    5. Other Functional Groups
    6. Polymers
    7. End-of-Chapter Material
  24. Chapter 17. Kinetics
    1. Factors that Affect the Rate of Reactions
    2. Reaction Rates
    3. Rate Laws
    4. Concentration–Time Relationships: Integrated Rate Laws
    5. Activation Energy and the Arrhenius Equation
    6. Reaction Mechanisms
    7. Catalysis
    8. End-of-Chapter Material
  25. Chapter 18. Chemical Thermodynamics
    1. Spontaneous Change
    2. Entropy and the Second Law of Thermodynamics
    3. Measuring Entropy and Entropy Changes
    4. Gibbs Free Energy
    5. Spontaneity: Free Energy and Temperature
    6. Free Energy under Nonstandard Conditions
    7. End-of-Chapter Material
  26. Appendix A: Periodic Table of the Elements
  27. Appendix B: Selected Acid Dissociation Constants at 25°C
  28. Appendix C: Solubility Constants for Compounds at 25°C
  29. Appendix D: Standard Thermodynamic Quantities for Chemical Substances at 25°C
  30. Appendix E: Standard Reduction Potentials by Value
  31. Glossary
  32. About the Authors
  33. Versioning History

Rate Laws

Jessie A. Key

Learning Objectives

  • To gain an understanding of rate laws and determine rate laws from initial rates.
  • To gain an understanding of and the ability to determine reaction orders (including units).

The mathematical relationship of reaction rate with reactant concentrations is known as the rate law. This relationship may rely more heavily on the concentration of one particular reactant, and the resulting rate law may include some, all, or none of the reactant species involved in the reaction.

For the following hypothetical reaction

aA+bB\rightarrow cC

the rate law can be expressed as:

\text{rate}=k[A]^y[B]^z

The proportionality constant, k, is known as the rate constant and is specific for the reaction shown at a particular temperature. The rate constant changes with temperature, and its units depend on the sum of the concentration term exponents in the rate law. The exponents (y and z) must be experimentally determined and do not necessarily correspond to the coefficients in the balanced chemical equation.

Reaction Order

The sum of the concentration term exponents in a rate law equation is known as its reaction order. We can also refer to the relationship for each reactant in terms of its exponent as an order.

For the following reaction between nitrogen dioxide and carbon monoxide:

NO2(g) + CO(g) → NO(g) + CO2(g)

The rate law is experimentally determined to be: rate = k[NO2]2

Therefore, we would say that the overall reaction order for this reaction is second-order (the sum of all exponents in the rate law is 2), but zero-order for [CO] and second-order for [NO2].

The reaction order is most often a whole number such as 0, 1, or 2; however, there are instances where the reaction order may be a fraction or even a negative value.

Earlier, it was mentioned that the units of the rate constant depend on the order of the reaction. Let’s quickly examine why this occurs. A simplified rate law can be expressed generically in the following way:

    \begin{align*} \text{rate}&=k[\text{reactant}]^y \\ \\ \text{Units of rate}&=\text{(units of rate constant)}\text{(units of concentration)}^y \\ \\ \text{Units of rare constant}&=\dfrac{\text{Units of rate}}{(\text{Units of concentration})^y} \\ \\ &=\dfrac{\text{M/s}}{\text{M}^y} \end{align*}

Therefore, the units of the rate constant should be:

Reaction OrderUnits of rate constant
Zero-orderM s−1
First-orders−1
Second-orderL mol−1 s−1

Determining Rate Laws from Initial Rates

The rate law can be determined experimentally using the method of initial rates, where the instantaneous reaction rate is measured immediately on mixing the reactants. The process is repeated over several runs or trials, varying the concentration one reactant at a time. These runs can then be compared to elucidate how changing the concentration of each reactant affects the initial rate.

Example 17.3

The initial rate of reaction for the reaction E + F → G was measured at three different initial concentrations of reactants as shown in the table.

  1. Determine the rate law of the reaction.
  2. Determine the rate constant.
Trial Initial Rate
(mol L−1 s−1)
[E] (mol L−1) [F] (mol L−1)
12.73 × 10−50.1000.100
25.47 × 10−50.2000.100
32.71 × 10−50.1000.200

Solution

  1. Comparing trials 1 and 2, [E] is doubled, while [F] and the rate constant are held constant. This comparison will allow us to determine the order of reactant E:

        \begin{align*} \dfrac{\text{initial rate 2}}{\text{initial rate 1}}&=\left(\dfrac{[\text{E}] \ _2}{[\text{E}] \ _1}\right)^y \\ \\ \dfrac{5.47\times10^{-5}\text{M s}^{-1}}{2.73\times 10^{-5}\text{ M s}^{-1}}&=\left(\dfrac{0.200\text{ M}}{0.100\text{ M}}\right)^y \\ \\ 2.00&=2.00^y \\ \\ y&=1 \end{align*}

    Therefore, the reaction is first order with respect to [E].

    Comparing trials 1 and 3, [F] is doubled, while [E] and the rate constant are held constant. This comparison will allow us to determine the order of reactant F:

        \begin{align*} \dfrac{\text{initial rate 3}}{\text{initial rate 1}}&=\left(\dfrac{[\text{F}] \ _3}{[\text{F}] \ _1}\right)^z \\ \\ \dfrac{2.71\times 10^{-5}\text{ M s}^{-1}}{2.73\times 10^{-5}\text{ M s}^{-1}}&=\left(\dfrac{0.200\text{ M}}{0.100\text{ M}}\right)^z \\ \\ 0.993&=2.00^z \\ \\ z&=0 \end{align*}

    Therefore, the reaction is zero order with respect to [F].

    The rate law can now be written as:

    \text{rate}=k[\text{E}]^1

  2. Using the rate law we have just determined, substitute in the initial concentration values and initial rate for any trial and solve for the rate constant:

    \text{rate}=k[\text{E}]^1

    Using Trial 1:

        \begin{align*} 2.73\times 10^{-5}\text{ M s}^{-1}&=k(0.100\text{ M}) \\ \\ k&=\dfrac{2.73\times 10^{-5}\text{ M s}^{-1}}{(0.100\text{ M})} \\ \\ k&=2.73\times 10^{-4}\text{ s}^{-1} \end{align*}

Key Takeaways

  • The rate law is a mathematical relationship obtained by comparing reaction rates with reactant concentrations.
  • The reaction order is the sum of the concentration term exponents in a rate law equation.
  • A reaction’s rate law may be determined by the initial rates method.

Icon for the Creative Commons Attribution 4.0 International License

Rate Laws by Jessie A. Key is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

Annotate

Next Chapter
Concentration–Time Relationships: Integrated Rate Laws
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Chemistry

Copyright © 2014

                                by Jessie A. Key

            Introductory Chemistry - 1st Canadian Edition by Jessie A. Key is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
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