Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- Measure one reactant, A, many times during one run of the reaction.
- The order in A shows in the shape of the data. Turn it into a straight line to read it.
- Make three columns: [A], ln[A], 1/[A]. With equal time steps, the column with constant differences is the straight one.
- The slope of that straight line gives the rate constant k.
- Half-life (t½) is the time for [A] to fall to half. It is constant only for first order.
- Radioactive decay is first order: each isotope has a fixed half-life.
Key relationships
| Order | Integrated rate law | Straight plot | Slope | Units of k | Half-life |
|---|---|---|---|---|---|
| 0 | [A]ₜ − [A]₀ = −kt | [A] vs t | −k | M s⁻¹ | gets shorter |
| 1 | ln[A]ₜ − ln[A]₀ = −kt | ln[A] vs t | −k | s⁻¹ | constant: t½ = 0.693/k |
| 2 | 1/[A]ₜ − 1/[A]₀ = kt | 1/[A] vs t | +k | M⁻¹ s⁻¹ | gets longer |
Fraction left after n half-lives (first order): (½)ⁿ.
Assumptions
- Temperature is constant, so k stays the same during the run.
- The rate depends only on the reactant you are watching; any other reactant is absent from the rate law or in large excess.
- The quantity you plot is proportional to concentration (for example absorbance, if Beer’s law holds).
- The data cover a large enough fall in [A] to tell the plots apart.
Mistakes to avoid
- “Gone after two half-lives.” No: one quarter is left.
- Constant half-life for every order. Only first order.
- Wrong sign. Second-order slope is +k; k is always positive.
- log instead of ln. Use the natural log.
- Mixed time units. Match t to the units of k.
- Plotting the product instead of the reactant.
Quick self-check
- A plot of ln[A] against time is straight. What is the order in A? (first order)
- A first-order reaction has t½ = 45 s. What is k? (0.693 ÷ 45 s = 0.0154 s⁻¹)
- What fraction of a first-order reactant is left after three half-lives? (one eighth, 0.125)
- A ln[A] against time plot has slope −0.0350 min⁻¹. What is the half-life? (0.693 ÷ 0.0350 min⁻¹ = 19.8 min)
- Which units of k mean second order? (M⁻¹ s⁻¹)
Next: practice questions.