Chemical Kinetics and Reaction Mechanisms

Integral Forms of Rate Laws

First-Order Reactions

• Rate Law: Rate = $\frac{-\Delta[A]}{\Delta t} = k[A]^1$
• Integrated Rate Law:
  • Integrating the rate law: $\int<em>{0}^{t} \frac{-d[A]}{dt} = \int</em>{0}^{t} k[A]^1$
  • Result: $-ln[A]<em>t - (-ln[A]</em>0) = k(t) - k(0)$
• Straight Line Form:
  • Rearranging the integrated rate law: $ln[A]<em>t = -k(t) + ln[A]</em>0$
  • Form: $y = m(x) + c$
  • Where:
    • $[A]_t$ is the concentration of A after time t.
    • k is the rate constant.
    • t is the specified time.
    • $[A]_0$ is the initial concentration of A.
• Graphical Representation:
  • Plot of $ln[A]_t$ vs. t yields a straight line with a slope of -k.
• Application: Pesticide Decomposition
  • Pesticides decompose into harmless products; half-life is important.
  • Example: A pesticide has a half-life of 10.2 years.
  • Current concentration in a lake is $3.1 × 10^{-5}$ g/mL.
  • What was its concentration 4 years ago?

Second-Order Reactions

• Rate Law: Initial rate = $-\frac{d[A]}{dt} = k[A]^2$
• Integrated Rate Law:
  • Integrating the rate law: $\int<em>{0}^{t} \frac{-d[A]}{dt} = \int</em>{0}^{t} k[A]^2$
  • Result: $\frac{1}{[A]<em>t} = k(t) + \frac{1}{[A]</em>0}$
• Straight Line Form:
  • $\frac{1}{[A]<em>t} = k(t) + \frac{1}{[A]</em>0}$
  • Form: $y = m(x) + c$
  • A plot of $\frac{1}{[A]_t}$ vs. t yields a straight line with a slope of k.

Half-Life

• Definition: The time taken for the concentration of a reactant to decrease to half its original concentration.
• First-Order Reaction:
  • $t_{1/2} = \frac{0.693}{k}$
• Second-Order Reaction:
  • $t<em>{1/2} = \frac{1}{k[A]</em>0}$

Reaction Mechanisms

Definition

• A step-by-step sequence of elementary reactions (elementary steps) that explains how the overall reaction proceeds.
• Net chemical change is directly observable.
• A mechanism describes what takes place at each stage of a chemical transformation.
• Information on which bonds are broken and formed and the relative rates of each step.

Understanding Reaction Mechanisms

• Mechanisms must:
  • Account for all species in a reaction.
  • Explain a rate law.
  • Account for intermediates (when observed).
• Intermediate:
  • A substance formed in one step and used up in another.
  • Appear in the mechanism but not the overall balanced equation.
• Rate Determining Step:
  • In many reaction mechanisms, there is usually one step that is slower than all other steps.
  • This step is known as the rate-determining step.

Rate Determining Step

• Analogy: The neck of a funnel; the rate at which water flows through the funnel is determined by the width of the neck.
• The rate of reaction depends on the rate of the slowest step.
• It is the slowest step in a chemical reaction.

Rate Determining Step and Rate Law

• Reaction mechanisms can be used as a replacement for the isolation method to determine the rate law of a reaction.
• Example:
  • Reaction: $NO<em>2(g) + CO(g) \rightarrow NO(g) + CO</em>2(g)$
  • Experimentally, this reaction was found to be second order w.r.t. $NO_2$ and zero order w.r.t. CO.
• Rate Law:
  • $Rate = k[NO_2]^2$
• The reaction rate depends on one molecule.

Reaction Mechanism Example

• Step One: $NO<em>2(g) + NO</em>2(g) \rightleftharpoons N<em>2O</em>4(g)$
• Step Two: $N<em>2O</em>4(g) + CO(g) \rightarrow NO(g) + CO<em>2(g) + NO</em>2(g)$
• Overall Reaction: $NO<em>2(g) + CO(g) \rightarrow NO(g) + CO</em>2(g)$

Determining Reaction Rates

• Step One is the slow step (the rate-determining step).
• The rate-determining step involves the collision of two $NO_2$ molecules.
• This is consistent with the rate law: $Rate = k[NO_2]^2$
• The reaction order for any single elementary step is equal to the coefficients for that step.
• If we have: $initial rate = k[A]^x[B]^y$
• x and y can be determined from the coefficients of the molecules in the rate-determining step.

Formulating a Reaction Mechanism

• Steps:
  1. Measure the rate of reaction.
  2. Formulate the rate law. This step will allow you to determine which molecules will form the reactants of the rate-determining step.
  3. Determine any possible intermediates that may be formed during the conversion of reactants to products.