Kinetics: Reaction Rates and Rate Laws in General Chemistry

Kinetics

Introduction to Kinetics

Kinetics is the branch of chemistry that studies the speed or rate at which chemical reactions occur and the factors that affect these rates. Understanding kinetics is essential for predicting how fast reactions proceed and for controlling chemical processes in both laboratory and industrial settings.

Reaction Rates

Definition and Measurement

• Reaction rate is defined as the change in concentration of a reactant or product per unit time.

• It is commonly expressed in units of molarity per second (M·s−1).

• The rate can be measured by monitoring the concentration of reactants or products over time.

General formula for reaction rate:

$\text{Rate} = -\frac{1}{a} \frac{\Delta [A]}{\Delta t} = -\frac{1}{b} \frac{\Delta [B]}{\Delta t} = \frac{1}{c} \frac{\Delta [C]}{\Delta t}$

• Where a, b, and c are the stoichiometric coefficients for reactants A, B and product C, respectively.

Example: For the reaction $2N_2O \rightarrow 2N_2 + O_2$:

$\text{Rate} = -\frac{1}{2} \frac{\Delta [N_2O]}{\Delta t} = \frac{1}{2} \frac{\Delta [N_2]}{\Delta t} = \frac{\Delta [O_2]}{\Delta t}$

The Rate Law and Reaction Orders

Rate Law Expression

• The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants.

• General form: $\text{Rate} = k[A]^m[B]^n$

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

Determining Reaction Order:

• If the rate does not change when the concentration of a reactant changes, the reaction is zero order with respect to that reactant.

• If the rate doubles when the concentration doubles, it is first order.

• If the rate quadruples when the concentration doubles, it is second order.

Example: If $\text{Rate} = k[A]^0$, then the reaction is zero order in A, and the rate does not depend on [A].

Integrated Rate Laws and Half-Life

Integrated Rate Laws

• Integrated rate laws relate the concentration of reactants to time.

• They differ for zero, first, and second order reactions.

Zero Order: $[A]_t = [A]_0 - kt$

First Order: $\ln [A]_t = \ln [A]_0 - kt$

Second Order: $\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$

• Half-life (t1/2) is the time required for the concentration of a reactant to decrease by half.

• For a first-order reaction: $t_{1/2} = \frac{0.693}{k}$

The Effect of Temperature and the Collision Model

Temperature Dependence of Rate

• Reaction rates generally increase with temperature.

• The Arrhenius equation describes this relationship:

$k = Ae^{-E_a/(RT)}$

• Where A is the frequency factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin.

Reaction Mechanisms

Elementary Steps and Rate-Determining Step

• A reaction mechanism is a sequence of elementary steps that describes the pathway from reactants to products.

• The rate-determining step is the slowest step in the mechanism and controls the overall reaction rate.

Catalysis

Role of Catalysts

• A catalyst increases the rate of a reaction by lowering the activation energy without being consumed in the reaction.

• Catalysts provide an alternative pathway for the reaction.

Summary Table: Reaction Order and Rate Laws

Order	Rate Law	Integrated Rate Law	Half-life Expression
Zero	Rate = k	$[A]_t = [A]_0 - kt$	$t_{1/2} = \frac{[A]_0}{2k}$
First	Rate = k[A]	$\ln [A]_t = \ln [A]_0 - kt$	$t_{1/2} = \frac{0.693}{k}$
Second	Rate = k[A]^2	$\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$	$t_{1/2} = \frac{1}{k[A]_0}$

Worked Examples

Example 1: Calculating Rate from Concentration Change

• Given: $\Delta [N_2O]/\Delta t = -0.0080$ M·s−1

• For $2N_2O \rightarrow 2N_2 + O_2$:

• $\text{Rate} = -\frac{1}{2} \frac{\Delta [N_2O]}{\Delta t} = 0.0040$ M·s−1

Example 2: Zero Order Reaction

• If the rate does not change with [A]: $\text{Rate} = k[A]^0 = k$

• Given $k = 0.011$ M·s−1, the rate is constant regardless of [A].

Example 3: Rate Law with Different Orders

• For $\text{Rate} = k[Y]^2$:

• If $[Y]$ is doubled, rate increases by a factor of 4.

• If $[Y]$ is tripled, rate increases by a factor of 9.

Additional info: These notes are based on standard General Chemistry kinetics topics and the provided problem set, with context and explanations expanded for clarity and completeness.