The rate equation

A rate equation shows how the rate of a reaction depends on the concentrations of the reactants: rate = k[A]ᵐ[B]ⁿ. The square brackets mean concentration in mol dm⁻³, m and n are the orders with respect to A and B, and k is the rate constant. The orders are found by experiment, not from the balanced equation. If rate = k[A][B]², doubling [A] doubles the rate, and doubling [B] makes it four times faster.

The rate equation

AQA's A-level specification gives the rate equation in this form, restricts the orders to 0, 1 and 2, and says that the rate equation is an experimentally determined relationship. Only reactants that affect the rate appear in it, and its orders do not have to match the numbers in the balanced equation.

Orders of reaction

A reactant that is zero order does not appear in the rate equation at all, because [A]⁰ = 1. More generally, multiplying a concentration by x multiplies the rate by xⁿ, where n is the order.

Here x is the concentration of the reactant in mol dm⁻³ and y is the rate in arbitrary units, for rate = 2[A]². From 0.5 to 1.0 mol dm⁻³ the rate rises from 0.5 to 2: four times as much.

Finding orders by the initial rates method

AQA expects students to deduce orders from rate-concentration data. Run the reaction several times, changing one concentration at a time, and measure the initial rate each time.

Example 1: deducing a rate equation

• From 1 to 2, [A] doubles with [B] fixed, and the rate doubles: first order in A.
• From 1 to 3, [B] doubles with [A] fixed, and the rate goes up 4 times: second order in B.

So the rate equation is rate = k[A][B]², and the overall order is 1 + 2 = 3.

Example 2: calculating k and its units

Rearrange the rate equation and use Experiment 1:

For the units, cancel: (mol dm⁻³ s⁻¹) ÷ (mol dm⁻³)³ = mol⁻² dm⁶ s⁻¹. So k = 0.015 mol⁻² dm⁶ s⁻¹. AQA's specification asks students to use given rate data to deduce a rate equation, then calculate the rate constant including its units.

Example 3: using the rate equation

For the same reaction at the same temperature, find the rate when [A] = 0.10 mol dm⁻³ and [B] = 0.30 mol dm⁻³.

Now try one in the course. Use the rate equation, then press Mark to check it there.

Units of the rate constant

The units of k depend on the overall order. Work them out by rearranging rate = k[ ]ⁿ and cancelling:

Concentration-time graphs

You can also find orders from how a concentration falls with time:

• Zero order: the concentration falls in a straight line. AQA mentions finding the rate constant of a zero-order reaction from the gradient of a concentration-time graph.
• First order: a curve with a constant half-life: the concentration halves in the same time, whatever it starts at.
• Any order: drawing a tangent to the curve gives the rate at that moment, as in our rate of reaction formula guide.

The rate equation and the rate-determining step

AQA's specification says the orders can provide information about the mechanism, including the rate-determining (slowest) step. A reaction often happens in several steps, and the slowest one controls the overall rate. The species in the rate equation, with their orders, are the ones involved in or before that slow step.

For rate = k[A][B]², a likely slow step involves one A and two B. A reactant that is zero order is only involved after the slow step, so changing its concentration does not change the rate.

Temperature and the rate constant

The rate equation holds at one temperature. Raising the temperature does not change the orders, but it makes k bigger, so the reaction goes faster at the same concentrations. The link between k and temperature is the Arrhenius equation, k = Ae^−Ea/RT. The Rates and Equilibrium Calculator finds orders from initial-rate data, works with the rate equation, and handles the Arrhenius equation too.

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Rates and Equilibrium Calculator

Work with rate equations, the Arrhenius equation and Kc

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Common mistakes

• Taking orders from the balanced equation. Orders come only from experiments.
• Changing two concentrations at once when comparing experiments. Compare two runs where only one concentration changes.
• Wrong units for k. Cancel the units properly; they depend on the overall order.
• Forgetting zero-order reactants. They are left out of the rate equation, but they still take part in the reaction.
• Thinking k changes with concentration. k only changes with temperature (or a catalyst).

Practise kinetics in a short timed paper

A-level Chemistry: Year 2 gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so rate equations turn up regularly but not in every paper. The Reaction Kinetics course has sections on rate equations and finding orders from data.

Read next

Chemistry · Rates and equilibriumRate of reaction formula: mean rate, units and tangentsThe rate of reaction formula: mean rate from reactant used or product formed, units g/s, cm³/s and mol/s, rate from a tangent, the practical and examples.