How do you find rate law for a reaction?

You do a series of experiments to determine how the rate depends on the concentration of each component in the reaction.

See What is the rate law?

For a reaction aA +bB → products, the rate law is

rate = `k["A"]^m["B"]^n`

You must determine the values of `k`, `m`, and `n` experimentally for a given reaction at a given temperature. `m` and `n` are usually integers.

One way is to use initial concentrations. It is called the method of initial rates.

Example

Find the rate law for the reaction aA +bB → products at 27 °C, given the data below.

Trial; [A]₀; [B]₀; rate
1; 1.0 mol·L⁻¹; 1.0 mol·L⁻¹; 2.0 mol·L⁻¹s⁻¹
2: 1.0 mol·L⁻¹; 2.0 mol·L⁻¹; 8.1 mol·L⁻¹s⁻¹
3; 2.0 mol·L⁻¹; 2.0 mol·L⁻¹; 15.9 mol·L⁻¹s⁻¹

Solution:

Look for two trials in which all but one concentration stays constant.

Step 1

In Trials 1 and 2, [A] remains constant, but [B] changes.

`("rate"_2)/("rate"_1) = (k["A"]_2^m["B"]_2^n)/(k["A"]_1^m["B"]_1^n) = ((1.0"mol·L⁻¹")/(1.0"mol·L⁻¹"))^m × ((2.0"mol·L⁻¹")/(1.0"mol·L⁻¹"))^n = (8.1"mol·L⁻¹s⁻¹")/(2.0"mol·L⁻¹s⁻¹")`

`(2.0)^n` = 4.05

`nlog2.0` = log 4.05

`n = (log 4.05)/(log 2.0) = 0.607/0.30` = 2.0 ≈ 2

Step 2

In Trials 2 and 4, [B] remains constant, but [A] changes.

`("rate"_3)/("rate"_2) = (k["A"]_3^m["B"]_3^n)/(k["A"]_3^m["B"]_3^n) = ((2.0"mol·L⁻¹")/(1.0"mol·L⁻¹"))^m × ((2.0"mol·L⁻¹")/(2.0"mol·L⁻¹"))^n = (15.9"mol·L⁻¹s⁻¹")/(8.1"mol·L⁻¹s⁻¹")`

`(2.0)^m` = 1.96

`mlog2.0` = log 1.96

`m = (log 1.96)/(log 2.0) = 0.292/0.30` = 0.97 ≈ 1

So the rate law is

rate = `k["A"]["B"]^2`