Chemistry · General chemistry II · Concept

Finding rate laws from experimental data

Find each reactant’s order and the rate constant k from initial-rate data, then use integrated rate law plots and half-life to find the order and k from concentration–time data.

What the rate law asks you to find

A rate law says how the rate depends on each reactant’s concentration. The exponents are the orders, and k is the rate constant. Neither can be read from the balanced equation: coefficients describe how much reacts, not how the rate responds. Orders come from experiments, usually small whole numbers such as 0, 1 or 2, though fractions and negative orders occur.

Once the orders are known, k follows from any single measurement. Its units depend on the overall order, so they are part of the answer.

rate = k {[A]}^{m} {[B]}^{n}, overall order = m + n

Method of initial rates: one reactant at a time

Run the reaction several times, changing one starting concentration while holding every other concentration and the temperature fixed, and measure the rate at the very start. Divide the two rate laws: k and every unchanged concentration cancel, leaving only the ratio for the reactant you changed.

If doubling [A] doubles the rate, m = 1. If it quadruples the rate, m = 2. If the rate does not change, m = 0. When the ratios are not tidy, take logarithms.

\frac{{rate}_{2}}{{rate}_{1}} = {(\frac{{[A]}_{2}}{{[A]}_{1}})}^{m} \Rightarrow m = \frac{ln ({rate}_{2} / {rate}_{1})}{ln ({[A]}_{2} / {[A]}_{1})}

Reading an order when one concentration doubles
Rate changes by	Order in that reactant	Why
× 1	0	2⁰ = 1
× 2	1	2¹ = 2
× 4	2	2² = 4
× 8	3	2³ = 8
× 2.83	1.5 (3/2)	2^1.5 ≈ 2.83

Finding k and its units

Substitute one trial’s rate and concentrations into the complete rate law and solve for k. Using a second trial should give the same k within measurement error, which is a useful check on the orders you chose. The units of k are whatever makes rate come out in M/s.

k = \frac{rate}{{[A]}^{m} {[B]}^{n}}

Units of k (time in seconds)
Overall order	Rate law form	Units of k
0	rate = k	M s⁻¹
1	rate = k[A]	s⁻¹
2	rate = k[A]² or k[A][B]	M⁻¹ s⁻¹
3	rate = k[A]²[B]	M⁻² s⁻¹

Concentration over time: which plot is straight?

A second kind of experiment follows one reactant as it is used up. Integrating the rate law gives an equation that is linear in time for exactly one choice of y-axis. Plot [A], ln[A] and 1/[A] against t: the plot that is straight tells you the order, and its slope gives k.

Compare the three plots, not one R² value. Over a short time range a curved plot can still have R² near 0.98, so look for the plot whose points have no systematic bend.

ln {[A]}_{t} = - k t + ln {[A]}_{0} \frac{1}{{[A]}_{t}} = k t + \frac{1}{{[A]}_{0}} {[A]}_{t} = - k t + {[A]}_{0}

Integrated rate laws
Order	Straight-line plot	Slope	Half-life
0	[A] vs t	−k	[A]₀ / 2k
1	ln[A] vs t	−k	0.693 / k
2	1/[A] vs t	+k	1 / (k[A]₀)

Half-life as a quick check

For a first-order reaction the half-life does not depend on concentration: every successive half takes the same time. If the time to fall from 0.50 M to 0.25 M equals the time to fall from 0.25 M to 0.125 M, the reaction is first order and k = 0.693/t½. For zero and second order, successive half-lives shrink or grow.

t_{1 / 2} = \frac{ln 2}{k} = \frac{0.693}{k}

Common mistakes

Taking orders from the coefficients of the balanced equation.
Comparing two trials in which more than one concentration changed.
Leaving k without units, or giving units that don’t match the overall order.
Reading the slope of a ln[A] plot as +k instead of −k.
Choosing the integrated rate law from one R² near 1 instead of comparing all three plots.
Using the first-order half-life formula for a zero- or second-order reaction.

Key terms

Rate law: An equation linking reaction rate to reactant concentrations, such as rate = k[A]²[B]. The units of the rate constant k depend on the overall order.
Reaction order: The exponent on a concentration in the rate law (the order in that reactant), or the sum of all the exponents (the overall order). Orders come from experiments, not from the balanced equation.
Overall order: The sum of the exponents on all the concentrations in a rate law.
Initial rate: The reaction rate measured right at the start, before concentrations change much. Comparing initial rates from runs with different starting concentrations reveals the reaction orders.
Integrated rate law: An equation for concentration versus time, derived from the rate law, such as ln[A] = −kt + ln[A]₀ for first order. Zero-, first- and second-order reactions each have their own form.
Half-life: The time it takes for a quantity to drop to half its starting value. For a first-order reaction it is ln 2/k and does not depend on the starting concentration.

Work through an example

For 2NO(g) + O₂(g) → 2NO₂(g) at a fixed temperature, three trials give these initial rates. Find the order in NO, the order in O₂, the rate law and k with units.

Find a rate law from initial rates →

Order and k from concentration–time data →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

Tro, Chemistry: A Molecular Approach, 4th ed., §14.3 The Rate Law: The Effect of Concentration on Reaction Rate, pp. 629–633 (method of initial rates, p. 630)
Tro, Chemistry: A Molecular Approach, 4th ed., §14.4 The Integrated Rate Law: The Dependence of Concentration on Time, pp. 634–641
OpenStax Chemistry 2e — Rate laws
OpenStax Chemistry 2e — Integrated rate laws

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