The Rate Expression and Order of Reaction

This lesson covers: 

  1. What a rate equation is
  2. How orders of reaction relate to reactant concentrations
  3. What the rate constant (k) represents
  4. Calculations involving the rate equation

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.

The rate constant k relates reactant concentrations to rate at a given temperature

The rate constant, k, is a proportionality constant that relates the rate of a reaction to the concentrations of the reactants at a specific temperature.

  • A larger k value indicates a faster rate of reaction.
  • k remains constant for a given reaction at a fixed temperature.
  • Increasing the temperature expoentially increases the value of k, as collisions between reactant molecules are more frequent and more energetic.


The units of k depend on the overall order of the reaction.

Calculating reaction rate from the rate constant and orders

If the rate constant and orders of a reaction are known, the rate can be calculated.

Worked example 1 - Calculating the rate of an acid-catalysed reaction

Calculate the rate of the acid-catalysed reaction between propanone and iodine, given that the reaction is first order with respect to propanone, zero order with respect to iodine, and first order with respect to H+.

CH3COCH3 + I2 ➔ CH3COCH2I + HI

The rate constant (k) at a certain temperature is 630 mol-1 dm3 s-1 and the concentrations of propanone, iodine, and H+ are each 2.00×10−3 mol dm−3.


Step 1: Write the rate equation

Rate = k[CH3​COCH3​]1[I2​]0[H+]1= k[CH3​COCH3​][H+]


Step 2: Substitution and correct evaluation

Rate =630×(2.00×10−3)×(2.00×10−3)=2.52×10−3 mol dm−3 s−1


Thus, at this temperature, the rate of the acid-catalysed reaction between propanone and iodine is 2.52×10−3 mol dm−3  s−1.

Calculating the rate constant from experimental data

If the rate and orders of a reaction are known from experiments, the rate constant k can be calculated at a given temperature.

Worked example 2 - Calculating the rate constant for a gas-phase reaction

The following reaction is second order with respect to NO and zero order with respect to CO and O2:

NO(g) + CO(g) + O2(g) ➔ NO2(g) + CO2(g)

At a certain temperature, the reaction rate is 4.00×10−3 mol dm-3 s-1, with the concentrations of NO, CO, and O2 each at 4.50×10−3 mol dm-3.

Calculate the rate constant (k) for the reaction.


Step 1: Write the rate equation

Rate = k[NO]2[CO]0[O2​]0= k[NO]2


Step 2: Rearrange rate equation

k =[NO]2Rate​


Step 3: Substitution and correct evaluation

k =(4.50×10−3)24.00×10−3​=198


Step 3: Determine the units of k

Given that the rate is expressed in mol dm-3 s-1 and the concentration squared in (mol dm-3)2, solving for k in the equation results in units of mol-1 dm3 s-1.

Thus, at this temperature, the rate constant is 198 mol-1 dm3 s-1 for the given gas-phase reaction.