The method of initial rates determines the concentration exponents in a rate law by comparing experiments that begin with different reactant concentrations.

Suppose the unknown rate law is

$$r=k[A]^m[B]^n.$$

If two experiments differ only in $[A]$, then dividing their initial-rate equations cancels $k$ and the unchanged concentration of B:

$$\frac{r_2}{r_1} =\left(\frac{[A]_2}{[A]_1}\right)^m.$$

The exponent $m$ can then be inferred from how the rate changes. A second comparison that changes only B determines $n$.

Example

Consider these initial-rate data at one temperature:

Compare experiments 1 and 2. Only A changes, doubling from $0.10$ M to $0.20$ M, and the rate doubles:

$$\frac{4.0\times10^{-3}}{2.0\times10^{-3}} =2 =2^m,$$

so

$$m=1.$$

Now compare experiments 2 and 3. Only B doubles, while the rate increases by a factor of four:

$$\frac{1.6\times10^{-2}}{4.0\times10^{-3}} =4 =2^n,$$

so

$$n=2.$$

The rate law is therefore

$$\boxed{r=k[A][B]^2}.$$

Use any experiment to determine $k$. From experiment 1,

$$k=\frac{2.0\times10^{-3}}{(0.10)(0.10)^2} =2.0,\mathrm{M^{-2},s^{-1}}.$$

Initial-rate measurements are useful because they compare experiments before large composition changes or product accumulation complicate the interpretation.

The method determines an empirical rate law. Agreement between that law and a proposed molecular mechanism can support the mechanism, but the rate law by itself does not prove one unique sequence of elementary steps.