R2.2.11 Rate constant (Higher Level Only)

The Rate Constant and Its Temperature Dependence

You’re trying to boil water for tea. If the stove is set to a low temperature, the water heats up slowly.
Turn the heat up, and the water boils much faster.

In the same way, temperature significantly influences the speed of chemical reactions.

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:

$k = A e^{- \frac{E_{a}}{R T}}$

Where:

$k$ : Rate constant.
$A$ : Arrhenius (frequency) factor, representing the likelihood of correctly oriented collisions.
$E_{a}$ : Activation energy ( ${J mol}^{- 1}$ ), the minimum energy needed for a reaction to occur.
$R$ : Gas constant ( $8.31 {J K}^{- 1} {mol}^{- 1}$ ).
$T$ : Absolute temperature (kelvin).

As temperature increases, the term $e^{- \frac{E_{a}}{R T}}$ 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:

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

Where:

$[A]$ and $[B]$ : Reactant concentrations, measured in ${mol dm}^{- 3}$ .
$n$ 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 ${mol dm}^{- 3} s^{- 1}$ , so the units of $k$ must balance the equation.

Examples of Units for $k$

Zero-order reaction ( $n + m = 0$ ):
- $rate = k$
- Units of $k$ : ${mol dm}^{- 3} s^{- 1}$ .
First-order reaction ( $n + m = 1$ ):
- $rate = k [A]$
- Units of $k$ : $s^{- 1}$ .
Second-order reaction ( $n + m = 2$ ):
- $rate = k [A]^{2}$ or $rate = k [A] [B]$
- Units of $k$ : ${dm}^{3} {mol}^{- 1} s^{- 1}$ .
Third-order reaction ( $n + m = 3$ ):
- $rate = k [A]^{2} [B]$
- Units of $k$ : ${dm}^{6} {mol}^{- 2} s^{- 1}$ .

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 $[RX] = 0.20 {mol dm}^{- 3}$ and $[{OH}^{-}] = 0.10 {mol dm}^{- 3}$ is $1.5 \times 10^{- 3} {mol dm}^{- 3} s^{- 1}$ . Calculate $k$ .

Solution

Rate equation: $rate = k [RX] [{OH}^{-}]$ .
Rearrange for $k$ : $k = \frac{rate}{[RX] [{OH}^{-}]} = \frac{1.5 \times 10^{- 3}}{(0.20) (0.10)} = 0.075 {dm}^{3} {mol}^{- 1} s^{- 1} .$

Reflection

Theory of Knowledge

How do reaction mechanisms demonstrate the relationship between theoretical models and experimental evidence in chemistry?

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Introduction to the Rate Constant

The rate constant ( $k$ ) is a fundamental parameter in chemical kinetics that quantifies the speed of a reaction under specific conditions.
Unlike reaction rate, which changes with concentration, the rate constant is fixed for a given reaction at a specific temperature.
Think of the rate constant as the "speed limit" of a reaction - it sets the maximum pace at which reactants can convert to products under optimal conditions.

Analogy

Imagine a factory assembly line. The rate constant is like the maximum production capacity per hour, while the reaction rate is the actual number of products made, which depends on how many workers (reactant molecules) are present.

Example

For the decomposition of hydrogen peroxide (

2H_2O_2 \rightarrow 2H_2O + O_2

), the rate constant might be

0.03 \text{s}^{-1}

at 25°C.