Rate Laws and Determining Reaction Orders in Chemical Kinetics

Rate Laws and Reaction Orders

Introduction to Rate Laws

Chemical kinetics studies the speed of chemical reactions and the factors that affect this speed. A rate law is a mathematical model that relates the rate of a chemical reaction to the concentration of its reactants.

• Rate Law: An equation that expresses the reaction rate as a function of the concentration of reactants, each raised to a specific power.

• General Format: For a reaction A + B → C + D, the rate law can be written as: $\text{Rate} = k [A]^x [B]^y$

• k: The rate constant, a proportionality constant specific to the reaction at a given temperature.

• x, y: The reaction orders with respect to A and B, respectively. These are determined experimentally.

• Overall Reaction Order: The sum of the exponents in the rate law ($x + y$).

Key Features of Rate Laws

• The rate law depends only on the concentrations of reactants, not products.

• The exponents in the rate law (reaction orders) are not generally equal to the stoichiometric coefficients in the balanced equation.

• Reaction orders must be determined experimentally, as they reflect the actual mechanism of the reaction.

• The rate constant k has units that depend on the overall reaction order and ensures proper unit cancellation.

Determining Reaction Orders

Reaction orders for overall reactions cannot be deduced from stoichiometry. Instead, they are found by analyzing experimental data, often using the method of initial rates.

• Experimental Determination: Measure the initial rate of reaction for different initial concentrations of reactants.

• Method of Initial Rates: Compare how changes in reactant concentrations affect the initial rate to deduce the order with respect to each reactant.

• Example: For the reaction $2\text{NO}(g) + 2\text{H}_2(g) \rightarrow \text{N}_2(g) + 2\text{H}_2\text{O}(g)$, initial rates are measured for various concentrations of NO and H2 to determine the rate law.

Worked Example: Method of Initial Rates

Suppose the following data are collected for a reaction:

Experiment	[NO] (M)	[H2] (M)	Initial Rate (M/s)
1	0.10	0.20	1.23 × 10-3
2	0.10	0.40	2.46 × 10-3
3	0.20	0.20	4.92 × 10-3

• To determine the order with respect to H2, compare experiments 1 and 2 (NO constant, H2 doubled): $\frac{2.46 \times 10^{-3}}{1.23 \times 10^{-3}} = 2$ Since the rate doubles when [H2] doubles, the reaction is first order in H2 ($y = 1$).

• To determine the order with respect to NO, compare experiments 1 and 3 (H2 constant, NO doubled): $\frac{4.92 \times 10^{-3}}{1.23 \times 10^{-3}} = 4$ Since the rate quadruples when [NO] doubles, the reaction is second order in NO ($x = 2$).

• Thus, the rate law is: $\text{Rate} = k [\text{NO}]^2 [\text{H}_2]^1$

Calculating the Rate Constant

• Substitute the concentrations and rate from any experiment into the rate law to solve for k:

$k = \frac{\text{Rate}}{[\text{NO}]^2 [\text{H}_2]}$

• For experiment 1: $k = \frac{1.23 \times 10^{-3}}{(0.10)^2 \times 0.20} = 0.615\ \text{M}^{-2}\text{s}^{-1}$

Summary Table: Key Concepts in Rate Laws

Concept	Description
Rate Law	Mathematical relationship between reaction rate and reactant concentrations
Order of Reaction	Exponent of each reactant in the rate law; must be determined experimentally
Overall Order	Sum of the exponents in the rate law
Rate Constant (k)	Proportionality constant specific to the reaction and temperature
Method of Initial Rates	Experimental technique to determine reaction orders by comparing initial rates

Additional info:

• Reaction mechanisms often involve multiple steps, and the rate law reflects the slowest (rate-determining) step, not the overall balanced equation.

• Units of the rate constant k vary depending on the overall order of the reaction.