Chemical Kinetics: The Mathematics of Reaction Rates

Introduction

Chemical kinetics is the study of reaction rates and the factors that influence them. Understanding the mathematical framework of reaction kinetics allows chemists to predict reaction behavior and optimize conditions for industrial processes.

Reaction Rates

The rate of a chemical reaction measures how quickly reactants are converted into products. It is typically expressed as:

\text{Rate} = -\frac{1}{a} \frac{d[A]}{dt} = \frac{1}{b} \frac{d[B]}{dt},

where:

$[A]$ and $[B]$ are the concentrations of reactants and products,
$a$ and $b$ are the stoichiometric coefficients in the balanced reaction,
$t$ is time.

Example

Consider the reaction:

2H_2 + O_2 \rightarrow 2H_2O.

The rate can be written as:

\text{Rate} = -\frac{1}{2} \frac{d[H_2]}{dt} = -\frac{d[O_2]}{dt} = \frac{1}{2} \frac{d[H_2O]}{dt}.

Rate Laws

The relationship between reaction rate and reactant concentrations is given by the rate law:

\text{Rate} = k [A]^m [B]^n,

where:

$k$ is the rate constant,
$m$ and $n$ are the reaction orders with respect to $A$ and $B$ .

The overall reaction order is $m + n$ .

Determining the Rate Law

The rate law is determined experimentally by measuring reaction rates under different initial concentrations of reactants.

Example

For a reaction $A + B \rightarrow C$ , if doubling $[A]$ doubles the rate and doubling $[B]$ quadruples the rate, the rate law is:

\text{Rate} = k [A]^1 [B]^2.

Integrated Rate Laws

Integrated rate laws describe how reactant concentrations change over time.

First-Order Reactions

For a first-order reaction $A \rightarrow B$ , the rate law is:

\text{Rate} = -\frac{d[A]}{dt} = k [A].

Integrating gives:

[A](t) = [A]_0 e^{-kt},

where $[A]_0$ is the initial concentration of $A$ .

Second-Order Reactions

For a second-order reaction $A \rightarrow B$ , the rate law is:

\text{Rate} = -\frac{d[A]}{dt} = k [A]^2.

Integrating gives:

\frac{1}{[A](t)} = \frac{1}{[A]_0} + kt.

Arrhenius Equation

The rate constant $k$ depends on temperature, as described by the Arrhenius equation:

k = A e^{-\frac{E_a}{RT}},

where:

$A$ is the pre-exponential factor,
$E_a$ is the activation energy,
$R$ is the gas constant,
$T$ is the temperature in kelvins.

Example

For a reaction with $E_a = 50 \text{kJ/mol}$ and $A = 10^7 \text{s}^{-1}$ , at $T = 298 \text{K}$ :

k = 10^7 e^{-\frac{50000}{8.314 \cdot 298}}.

Conclusion

Chemical kinetics provides a quantitative framework for understanding and controlling chemical reactions. By leveraging rate laws, integrated rate laws, and the Arrhenius equation, chemists can predict reaction behavior and design efficient processes.