Rate Laws, Rate Constants, and Reaction Order
A rate law is an equation that relates the rate of a reaction to the concentrations of the reactants, rate = k[A]m[B]n, where k is the rate constant and the exponents m and n are the reaction orders, which must be found by experiment. In brief: if doubling [A] quadruples the rate, the reaction is second order in A; if doubling [B] doubles the rate, it is first order in B, and the overall order is 3.
A rate law is the central result of a kinetic study, because it summarizes how the rate depends on the composition of the mixture and provides clues about the mechanism. The orders cannot be read from the balanced equation, except in the special case of elementary reactions. This article defines the terms, shows the method of initial rates with a worked example, gives the units of k and explains how to use the rate law for predictions.
What is a rate law?
For a reaction aA + bB → products, the experimental rate law has the form:
rate = k [A]m [B]n
The rate constant k is a proportionality constant. It is independent of the concentrations but depends on the temperature, and a catalyst changes it. A large k means a fast reaction. The exponents m and n are the orders with respect to A and B, and they are usually small integers (0, 1, 2), though fractional and negative orders are known for complex reactions. The sum m + n is the overall order.
What does reaction order mean?
- Zero order in A: the rate does not depend on [A].
- First order in A: the rate is proportional to [A].
- Second order in A: the rate is proportional to [A]2.
The exponents are not equal to the stoichiometric coefficients. The reaction 2NO + 2H2 → N2 + 2H2O has the coefficients 2 and 2, yet the experimental rate law is rate = k[NO]2[H2]. The orders are determined by the mechanism of the reaction.
How does the order affect the rate?
| Order in A | If [A] doubles | If [A] triples | If [A] is halved |
|---|---|---|---|
| 0 | Rate unchanged | Rate unchanged | Rate unchanged |
| 1 | Rate × 2 | Rate × 3 | Rate × ½ |
| 2 | Rate × 4 | Rate × 9 | Rate × ¼ |
| 3 | Rate × 8 | Rate × 27 | Rate × ⅛ |
How do you find the rate law?
The most common approach is the method of initial rates. A series of experiments is carried out with different initial concentrations, and the initial rate is measured in each. By comparing two experiments in which only one concentration changes, you can see the effect of that concentration on the rate. Another approach uses the integrated rate laws: concentrations are measured as a function of time and plotted in different ways to see which gives a straight line.
Worked example: the method of initial rates
The reaction A + B → products was studied at constant temperature:
| Experiment | [A] (M) | [B] (M) | Initial rate (M/s) |
|---|---|---|---|
| 1 | 0.10 | 0.10 | 2.0 × 10−3 |
| 2 | 0.20 | 0.10 | 8.0 × 10−3 |
| 3 | 0.10 | 0.20 | 4.0 × 10−3 |
- Order in A: compare experiments 1 and 2, where [B] is constant and [A] doubles. The rate increases by 8.0 / 2.0 = 4 = 2m, so m = 2.
- Order in B: compare experiments 1 and 3, where [A] is constant and [B] doubles. The rate increases by 4.0 / 2.0 = 2 = 2n, so n = 1.
- Rate law: rate = k[A]2[B], with an overall order of 3.
- Rate constant: use experiment 1: k = rate / ([A]2[B]) = 2.0 × 10−3 / (0.102 × 0.10) = 2.0 × 10−3 / 1.0 × 10−3 = 2.0 M−2s−1.
- Check: experiment 2 gives 2.0 × 0.202 × 0.10 = 8.0 × 10−3 M/s, which matches the data.
Using the rate law for a prediction
What is the rate when [A] = 0.50 M and [B] = 0.20 M? Rate = 2.0 × 0.502 × 0.20 = 2.0 × 0.25 × 0.20 = 0.10 M/s.
What are the units of the rate constant?
Since the rate has the units M/s, the units of k depend on the overall order:
| Overall order | Rate law | 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 |
The general rule is that the units of k are M1−n s−1 for overall order n. The units of k can therefore be used to find the order of a reaction.
Real examples of rate laws
- 2N2O5 → 4NO2 + O2: rate = k[N2O5] (first order).
- 2NO + O2 → 2NO2: rate = k[NO]2[O2] (third order).
- H2 + I2 → 2HI: rate = k[H2][I2] (second order, in the classic view).
- NO2 + CO → NO + CO2 (below about 225 °C): rate = k[NO2]2, zero order in CO.
Where do rate laws come from?
The law of mass action of Cato Guldberg and Peter Waage (1864) was the first attempt to formulate how the rate depends on concentration. Jacobus van ‘t Hoff introduced the idea of reaction order in his work of 1884, and Wilhelm Ostwald, who received the Nobel Prize in Chemistry in 1909 for his work on catalysis and rates, developed methods for determining orders. The method of initial rates is a standard technique in modern kinetics.
Which mistakes are common?
- Taking the orders from the coefficients: only elementary steps follow this rule.
- Forgetting that k depends on temperature: a rate constant is valid only for one temperature.
- Using the wrong experiments: compare runs where only one concentration changes.
- Wrong units for k: check them against the overall order.
- Including solids and catalysts without thinking: catalysts can appear in the rate law, and pure solids usually do not.