Rate equations relate reaction rate to reactant concentrations

A rate equation is a mathematical expression that shows how the rate of a chemical reaction depends on the concentrations of the reactants.

For a general reaction: A + B ➔ C + D

The rate equation takes the form:

Rate = k[A]^m[B]ⁿ

Where:

• Rate is the reaction rate (mol dm^-3 s^-1).
• k is the rate constant.
• [A] and [B] are the concentrations of reactants A and B (mol dm^-3).
• m and n are the orders of reaction with respect to A and B.

Reaction orders show how reactant concentrations affect the rate

The values of m and n in the rate equation are the reaction orders with respect to each reactant.

They indicate how changing the concentration of a reactant influences the reaction rate.

1. Zero order (m or n = 0) - The rate is independent of the reactant concentration. Doubling or tripling the concentration has no effect on the rate. [A]⁰ = 1, so zero order terms are often written without the concentration term.

1. First order (m or n = 1) - The rate is directly proportional to the reactant concentration. Doubling the concentration doubles the rate, tripling the concentration triples the rate.

1. Second order (m or n = 2) - The rate is proportional to the square of the reactant concentration. Doubling the concentration quadruples the rate (2² = 4), tripling the concentration increases the rate ninefold (3² = 9).

The overall order of the reaction is the sum of m and n.

Importantly, reaction orders can only be determined experimentally, not from balanced chemical equations.

Determining reaction orders from experimental data

To find the order of a reactant, we measure the initial reaction rate while varying the concentration of that reactant and keeping all other concentrations constant.

Worked example 1 - Determining the rate equation

Consider the reaction below, carried out at a constant temperature.

2NO_(g) + O_2(g) ➔ 2NO_2(g)

Determine the rate equation for this reaction.

Step 1: Analyse NO concentration change

Between experiments 1 and 2, the concentration of [NO] doubles while [O₂] remains constant, and the rate quadruples. This suggests the reaction is second order with respect to [NO].

Step 2: Analyse O₂ concentration change

Between experiments 1 and 3, doubling the concentration of [O₂] while keeping [NO] constant doubles the rate, indicating first-order dependence on [O₂].

Step 3: Write the rate equation

rate = k[NO]²[O₂]

Worked example 2 - Calculating the rate constant

Using the same reaction and data from the previous worked example 1, calculate the rate constant (k), including its units. Give your answer to 3 significant figures.

Step 1: Rearrange rate equation

First, rearrange the rate equation to solve for k.

$\text{k }=\frac{\text{rate}}{\text{[NO]}^2[\text{O}_2]}$

Step 2: Substitution and correct evaluation

Substitute values from one of the experiments into the equation to find k. For example, substituting values from experiment 1:

$\text{k }=\frac{\text{0.020}}{(0.05)^2 \times 0.05} =160$

Step 3: Determine the units

Given that the rate is expressed in mol dm^-3 s^-1 and the concentration cubed in (mol dm^-3)³, solving for k in the equation results in units of mol^-2 dm⁶ s^-1.

Therefore the rate constant for the reaction above is 160 mol^-2 dm⁶ s^-1.

Graphical representation of zero, first and second order reactions

The order of a reaction with respect to a reactant can be visualized using two types of plots:

1. Rate-concentration curves - These plots show the reaction rate ($\nu$) against reactant concentration [X] for a series of experiments. The shape of the curve indicates the order:

2. Concentration-time curves - These plots show reactant concentration against time for a single experiment. The shape of the curve depends on the order:

First and second order concentration-time plots can be difficult to distinguish visually. To identify the order, the data can be replotted to give a straight line graph whose gradient relates to k:

In the table above, [X]_0_ represents the initial concentration of reactant X at time t = 0.