How to Calculate Rate Order

Determining the order of a chemical reaction with respect to each reactant is a fundamental concept in chemical kinetics. The rate law for a reaction expresses how the rate of the reaction depends on the concentration of the reactants. For a general reaction:

aA + bB → products

The rate law is typically written as:

Rate = k[A]^m[B]^n

Where:

• Rate is the speed at which reactants are consumed or products are formed.
• k is the rate constant, which is specific to the reaction and temperature.
• [A] and [B] are the molar concentrations of reactants A and B.
• m and n are the reaction orders with respect to reactants A and B, respectively. These orders are typically small whole numbers (0, 1, 2) but can also be fractions or negative. They must be determined experimentally and cannot be predicted from the stoichiometry of the balanced equation.

The overall reaction order is the sum of the individual orders (m + n).

The Method of Initial Rates

The most common experimental method for determining reaction orders is the "method of initial rates." This involves conducting a series of experiments where the initial concentrations of reactants are systematically varied, and the initial reaction rate is measured for each experiment. By comparing how the rate changes when a specific reactant's concentration is changed (while keeping others constant), we can deduce the order with respect to that reactant.

Steps to Calculate Rate Order:

1. Set up Experiments: Design at least three experiments where you vary the initial concentrations of reactants and measure the initial rate for each.
2. Determine Order with Respect to Reactant A:
   • Find two experiments where the initial concentration of reactant B is the same, but the initial concentration of reactant A is different.
   • Let these be Experiment 1 (initial concentrations [A]₁, [B]₁, initial rate₁) and Experiment 2 (initial concentrations [A]₂, [B]₂, initial rate₂).
   • The ratio of the rates will be:

     Rate₂ / Rate₁ = (k[A]₂^m[B]₂^n) / (k[A]₁^m[B]₁^n)

   • Since [B]₁ = [B]₂, [B]₂^n / [B]₁^n = 1. Also, k cancels out.

     Rate₂ / Rate₁ = [A]₂^m / [A]₁^m = ([A]₂ / [A]₁)^m

   • Solve for m. This often involves logarithms or by inspection if the ratio of concentrations is simple (e.g., doubling).
3. Determine Order with Respect to Reactant B:
   • Find two experiments where the initial concentration of reactant A is the same, but the initial concentration of reactant B is different.
   • Let these be Experiment 1 (initial concentrations [A]₁, [B]₁, initial rate₁) and Experiment 3 (initial concentrations [A]₃, [B]₃, initial rate₃).
   • Similarly,

     Rate₃ / Rate₁ = ([B]₃ / [B]₁)^n

   • Solve for n.
4. Write the Rate Law: Once m and n are determined, substitute them into the general rate law equation: Rate = k[A]^m[B]^n.
5. Calculate the Rate Constant (k): Using the determined orders (m and n) and the data from any one of the experiments, you can solve for the rate constant k.

Example Calculation

Consider the reaction:

2NO(g) + O₂(g) → 2NO₂(g)

The following initial rate data was collected:

Experiment	[NO] (M)	[O₂] (M)	Initial Rate (M/s)
1	1.0	1.0	0.010
2	2.0	1.0	0.040
3	1.0	2.0	0.020

Determining the order with respect to NO (m):

Compare Experiment 1 and Experiment 2:

• [NO] doubles (from 1.0 M to 2.0 M)
• [O₂] remains constant (1.0 M)
• Rate quadruples (from 0.010 M/s to 0.040 M/s)

Rate₂ / Rate₁ = ([NO]₂ / [NO]₁)^m

0.040 / 0.010 = (2.0 / 1.0)^m

4 = 2^m

Therefore, m = 2 (The reaction is second order with respect to NO).

Determining the order with respect to O₂ (n):

Compare Experiment 1 and Experiment 3:

• [NO] remains constant (1.0 M)
• [O₂] doubles (from 1.0 M to 2.0 M)
• Rate doubles (from 0.010 M/s to 0.020 M/s)

Rate₃ / Rate₁ = ([O₂]₃ / [O₂]₁)^n

0.020 / 0.010 = (2.0 / 1.0)^n

2 = 2^n

Therefore, n = 1 (The reaction is first order with respect to O₂).

The Rate Law:

Rate = k[NO]²[O₂]¹

Calculating the Rate Constant (k):

Using data from Experiment 1:

0.010 M/s = k (1.0 M)² (1.0 M)¹

0.010 M/s = k (1.0 M³)

k = 0.010 M⁻²s⁻¹

This calculator helps you perform these calculations quickly by inputting experimental data.

${orderA}

${orderB}

Rate = k[A]^${orderA}[B]^${orderB}

${k.toExponential(4)}