Reaction Kinetics and Rate Equations

The rate equation relates the rate of reaction to the concentrations of reactants:

$\text{rate} = k[A]^m[B]^n$

where:

• $k$ = rate constant (varies with temperature)
• $[A]$, $[B]$ = concentrations of reactants
• $m$, $n$ = orders of reaction with respect to A and B
• Overall order = $m + n$

Important: The rate equation can ONLY be determined experimentally — you CANNOT deduce it from the balanced equation.

Determining Order from Experimental Data

Compare pairs of experiments where only one concentration changes:

For A (compare exps 1 and 2): [A] doubles, [B] constant. Rate × 4. Order wrt A = 2

For B (compare exps 1 and 3): [B] doubles, [A] constant. Rate × 2. Order wrt B = 1

$\text{rate} = k[A]^2[B]$

Calculating k

Using experiment 1: $2.0 \times 10^{-3} = k \times (0.10)^2 \times (0.10)$ $k = \frac{2.0 \times 10^{-3}}{0.001} = 2.0 \text{ dm}^6 \text{ mol}^{-2} \text{ s}^{-1}$

Half-Life and Order

• Zero order: half-life decreases over time
• First order: half-life is constant (independent of concentration)
• Second order: half-life increases over time

For first order: $t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}$

The rate-determining step (RDS) is the slowest step in a multi-step reaction mechanism. It acts as a bottleneck.

Key Rule

The rate equation tells you what species are involved in the rate-determining step (and all steps up to and including it).

If the rate equation is $\text{rate} = k[A][B]$, then both A and B are involved in (or before) the RDS.

If the rate equation is $\text{rate} = k[A]^2$, then two molecules of A are involved in (or before) the RDS, and B is involved in a later, faster step.

Example

Reaction: $\text{A} + 2\text{B} \rightarrow \text{C} + \text{D}$

Rate equation: $\text{rate} = k[A][B]$

Possible mechanism:

• Step 1 (slow, RDS): $\text{A} + \text{B} \rightarrow \text{X}$ (intermediate)
• Step 2 (fast): $\text{X} + \text{B} \rightarrow \text{C} + \text{D}$

This is consistent because the RDS involves one A and one B, matching the rate equation.

The Arrhenius equation quantifies how the rate constant $k$ varies with temperature:

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

where:

• $A$ = pre-exponential factor (Arrhenius constant) — related to collision frequency and orientation
• $E_a$ = activation energy (J mol⁻¹)
• $R$ = gas constant ($8.314 \text{ J K}^{-1} \text{mol}^{-1}$)
• $T$ = temperature (K)

Logarithmic Form

$\ln k = \ln A - \frac{E_a}{RT}$

This is in the form $y = c + mx$ where:

• Plot $\ln k$ (y-axis) vs $\frac{1}{T}$ (x-axis)
• Gradient = $-\frac{E_a}{R}$
• y-intercept = $\ln A$

$E_a = -\text{gradient} \times R$

Example

Question: At 300 K, $k = 1.5 \times 10^{-3}$. At 350 K, $k = 8.7 \times 10^{-3}$. Calculate $E_a$.

$\ln k_1 = \ln A - \frac{E_a}{R \times 300}$ $\ln k_2 = \ln A - \frac{E_a}{R \times 350}$

Subtract: $\ln k_2 - \ln k_1 = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

$\ln\left(\frac{8.7 \times 10^{-3}}{1.5 \times 10^{-3}}\right) = \frac{E_a}{8.314}\left(\frac{1}{300} - \frac{1}{350}\right)$

$1.758 = \frac{E_a}{8.314} \times 4.76 \times 10^{-4}$

$E_a = \frac{1.758 \times 8.314}{4.76 \times 10^{-4}} = 30700 \text{ J mol}^{-1} = 30.7 \text{ kJ mol}^{-1}$

Catalysts lower $E_a$ by providing an alternative reaction pathway.

In the Arrhenius equation: lower $E_a$ → larger $k$ → faster rate.

A catalyst does not change:

• The enthalpy change ($\Delta H$)
• The equilibrium position
• The overall stoichiometry

• Rate equations are ALWAYS determined experimentally
• To find order: compare experiments where only one concentration changes
• Units of $k$ depend on the overall order
• The RDS is consistent with the rate equation — practice matching mechanisms to rate equations
• In Arrhenius calculations, ensure $E_a$ is in J (not kJ) when using $R = 8.314$ J K⁻¹ mol⁻¹

• Rate equation: $\text{rate} = k[A]^m[B]^n$ (determined experimentally)
• Orders: 0 (no effect), 1 (proportional), 2 (rate ∝ [A]²)
• First order: constant half-life, $t_{1/2} = 0.693/k$
• Rate-determining step: slowest step; rate equation reveals what's in/before the RDS
• Arrhenius equation: $k = Ae^{-E_a/RT}$; plot $\ln k$ vs $1/T$ to find $E_a$