The Rate Constant and Its Temperature Dependence
What is Rate Constant?
Definition
Rate constant
The rate constant, $ k $, serves as a measure of how fast a reaction proceeds, and it varies with temperature.
This relationship is captured by the Arrhenius equation:
$$
k = A e^{-\frac{E_a}{RT}}
$$
where:
- $ k $: Rate constant.
- $ A $: Arrhenius (frequency) factor, representing the likelihood of correctly oriented collisions.
- $ E_a $: Activation energy ($ \text{J mol}^{-1} $), the minimum energy needed for a reaction to occur.
- $ R $: Gas constant ($ 8.31 \, \text{J K}^{-1} \text{mol}^{-1} $).
- $ T $: Absolute temperature (kelvin).
Hint
- As temperature increases, the term $ e^{-\frac{E_a}{RT}} $ becomes larger, leading to an increase in $ k $.
- This explains why reactions generally occur more quickly at higher temperatures.
Tip
- Reactions with higher activation energies ($ E_a $) are more sensitive to temperature changes.
- Even a small increase in temperature can significantly increase the rate constant.
Units of the Rate Constant
The units of the rate constant, $ k $, depend on the overall order of the reaction. To determine these units, consider the rate equation:
$$
\text{rate} = k [A]^n [B]^m
$$
where:
- $ [A] $ and $ [B] $: Reactant concentrations, measured in $ \text{mol dm}^{-3} $.
- $ n $ and $ m $: Orders of reaction with respect to $ A $ and $ B $, respectively.
Note
- The overall order of the reaction is $ n + m $.
- The rate of reaction is typically measured in $ \text{mol dm}^{-3} \text{s}^{-1} $, so the units of $ k $ must balance the equation.
Examples of Units for $ k $
- Zero-order reaction ($ n + m = 0 $):
- $\text{rate} = k$
- Units of $ k $: $ \text{mol dm}^{-3} \text{s}^{-1} $.
- First-order reaction ($ n + m = 1 $):
- $\text{rate} = k [A]$
- Units of $ k $: $ \text{s}^{-1} $.
- Second-order reaction ($ n + m = 2 $):
- $\text{rate} = k [A]^2$ or $ \text{rate} = k [A][B] $
- Units of $ k $: $ \text{dm}^3 \text{mol}^{-1} \text{s}^{-1} $.
- Third-order reaction ($ n + m = 3 $):
- $\text{rate} = k [A]^2[B]$
- Units of $ k $: $ \text{dm}^6 \text{mol}^{-2} \text{s}^{-1} $.
Common Mistake
- Students sometimes confuse the units of $ k $ with the units of concentration or rate.
- Always verify units by balancing the rate equation.
Solving Problems Involving Rate Equations
To solve problems involving rate equations, follow these steps:
- Write the rate equation based on the reaction mechanism.
- Identify the reaction order and derive the units of $ k $.
- Substitute known values into the rate equation and solve for the unknown.
Example question
The reaction rate for a primary halogenoalkane with $[\text{RX}] = 0.20 \, \text{mol dm}^{-3} $ and $ [\text{OH}^-] = 0.10 \, \text{mol dm}^{-3} $ is $1.5 \times 10^{-3} \, \text{mol dm}^{-3} \text{s}^{-1} $. Calculate $ k $.
Solution
- Rate equation: $ \text{rate} = k [\text{RX}] [\text{OH}^-] $.
- Rearrange for $ k $:$$
k = \frac{\text{rate}}{[\text{RX}] [\text{OH}^-]} = \frac{1.5 \times 10^{-3}}{(0.20)(0.10)} = 0.075 \, \text{dm}^3 \text{mol}^{-1} \text{s}^{-1}.
$$
