Chemistry · General chemistry II · Concept
Finding reaction orders and the rate constant 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.
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.
| 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.
| 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.
| 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.
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.
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 the orders and k for 2NO + O₂ from initial rates →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