# AHL: Rate laws and reaction order

> IB Chemistry HL · R2: How much / how fast / how far?
> Source: https://www.owlsprep.com/study/ib-chemistry-hl-u5-ahl-rate-laws-and-reaction/

This sub-topic covers how to construct and interpret rate laws, determine reaction order from experimental data, and relate order to reaction rate behavior. You will learn to calculate rate constants and overall reaction order.

**Prerequisites:** [Collision theory and rate measurement](https://www.owlsprep.com/study/ib-chemistry-hl-introduction-to-kinetics/)

## Learning objectives

- Distinguish between rate expressions, rate constants and reaction order
- Calculate individual and overall reaction order from experimental rate data
- Deduce rate laws from initial rate experimental results
- Relate reaction order to the effect of concentration on reaction rate

## Definition and structure of a rate law

**Rate law** — A mathematical expression that relates reaction rate to the concentration of reactants raised to their individual reaction orders, where $k$ is the rate constant.

*Notation:* \text{rate} = k [A]^m [B]^n

*Example:* For $2N_2O_5 \rightarrow 4NO_2 + O_2$, the rate law is $\text{rate} = k[N_2O_5]$

Rate laws can **only be determined experimentally**; they cannot be deduced from the stoichiometry of the overall balanced reaction. Order with respect to a reactant is independent of its stoichiometric coefficient, unless the reaction is a single elementary step.

> **tip**
>
> Only elementary (single-step) reactions have rate laws that match their overall stoichiometry. For multi-step reactions, the rate law depends on the rate-determining step.

**Worked example:** Identify the order with respect to each reactant in the rate law: $\text{rate} = k[X]^1[Y]^0[Z]^2$

1. The order of a reactant is equal to the power it is raised to in the rate law.
2. Order with respect to X = 1, order with respect to Y = 0, order with respect to Z = 2.

**Check your understanding**

Test your basic understanding

1. True or false: The rate law of a reaction can always be determined from the balanced overall equation

   - True
   - False

   *Answer:* False

   *Why:* Correct: Rate laws are experimental, and depend on the reaction mechanism, not just overall stoichiometry.

## Determining order from initial rate data

The initial rate method is the most common experimental approach to find reaction order. You change the concentration of one reactant at a time, measure the initial rate, and compare how rate changes with concentration:

- If $[A]$ doubles, rate stays the same → order 0
- If $[A]$ doubles, rate doubles → order 1
- If $[A]$ doubles, rate quadruples → order 2

**Worked example:** For the reaction $A + B \rightarrow C$, use the data below to deduce the order with respect to A and B, then write the rate law:

| Experiment | [A] / mol dm⁻³ | [B] / mol dm⁻³ | Initial rate / mol dm⁻³ s⁻¹ |
|------------|----------------|----------------|------------------------------|
| 1          | 0.1            | 0.1            | 0.002                        |
| 2          | 0.2            | 0.1            | 0.008                        |
| 3          | 0.1            | 0.2            | 0.004                        |

1. Compare experiments 1 and 2 ([B] is constant):
2. $$\frac{[A]_2}{[A]_1} = 2, \quad \frac{\text{rate}_2}{\text{rate}_1} = 4$$
3. For order $m$: $2^m = 4 \rightarrow m=2$. Order with respect to A = 2.
4. Compare experiments 1 and 3 ([A] is constant):
5. $$\frac{[B]_3}{[B]_1} = 2, \quad \frac{\text{rate}_3}{\text{rate}_1} = 2$$
6. For order $n$: $2^n = 2 \rightarrow n=1$. Order with respect to B = 1.
7. Final rate law:
8. $$\text{rate} = k[A]^2[B]^1$$

> **Exam tip:** Always compare experiments where only one concentration changes. If both concentrations change, you cannot isolate the effect of each reactant to find order.

## Properties of zero, first and second order reactions

**Reaction order** — The power to which the concentration of a reactant is raised in the rate law, describing how the reactant's concentration affects reaction rate.

Order can be zero, a positive integer, or even negative for inhibition. IB Chemistry HL only assesses positive zero, first and second order behavior.

**Worked example:** A reactant has zero order in the rate law. What happens to the reaction rate if the concentration of this reactant is tripled?

1. Zero order means the reactant concentration is raised to the power of 0:
2. $$[A]^0 = 1$$
3. Substitute into the rate law:
4. $$\text{rate} = k [A]^0 = k$$
5. Rate is independent of the concentration of a zero order reactant. Tripling the concentration has no effect on rate.

| Order | Effect of doubling [A] on rate | Units of k |
| --- | --- | --- |
| 0 | No change | mol dm⁻³ time⁻¹ |
| 1 | Rate doubles | time⁻¹ |
| 2 | Rate quadruples | dm³ mol⁻¹ time⁻¹ |

## Overall order and rate constant calculation

The overall order of a reaction is the sum of all individual reaction orders for reactants in the rate law. Once order is known, you can calculate the value and units of the rate constant $k$ from experimental data.

**Worked example:** Using the rate law $\text{rate} = k[A]^2[B]$ and data from experiment 1 ($[A] = 0.1$ mol dm⁻³, $[B] = 0.1$ mol dm⁻³, rate = 0.002 mol dm⁻³ s⁻¹), calculate $k$ and its units.

1. Substitute the values into the rate law:
2. $$0.002 = k \times (0.1)^2 \times (0.1)$$
3. Rearrange to solve for k:
4. $$k = \frac{0.002}{(0.1)^2 (0.1)} = 2$$
5. Calculate units: Overall order = 2 + 1 = 3. Units of k are $(mol \ dm^{-3})^{1-\text{overall order}} \ time^{-1}$:
6. $$\text{Units} = dm^6 \ mol^{-2} \ s^{-1}$$
7. Final result: $k = 2 \ dm^6 \ mol^{-2} \ s^{-1}$

## Common pitfalls

- **Wrong:** Deducing reaction order from stoichiometric coefficients of the overall balanced equation
  - Why it fails: Rate laws depend on reaction mechanism, not overall stoichiometry. Only elementary steps follow stoichiometric order
  - Correct: Always use experimental rate data to deduce order; never rely on overall reaction coefficients
- **Wrong:** Claims k changes when reactant concentration changes
  - Why it fails: The rate constant k is only affected by temperature and catalysts, not concentration
  - Correct: Recognize k is constant at fixed temperature, changing concentration changes rate but not k
- **Wrong:** Memorizing units of k incorrectly for different overall orders
  - Why it fails: Memorization often leads to errors, especially for higher overall orders
  - Correct: Derive units by rearranging the rate law to isolate k, then substitute concentration and rate units
- **Wrong:** Calculating order from two experiments where both reactant concentrations change
  - Why it fails: You cannot isolate the effect of each reactant if both change, leading to incorrect order values
  - Correct: Always select pairs of experiments where only one reactant concentration changes to find individual order

## Cheatsheet

| Property | Zero order | First order | Second order |
| --- | --- | --- | --- |
| Rate dependence | rate = k, independent of [A] | rate ∝ [A], rate = k[A] | rate ∝ [A]², rate = k[A]² |
| Units of k | mol dm⁻³ t⁻¹ | t⁻¹ | dm³ mol⁻¹ t⁻¹ |
| Half-life behavior | Decreases as [A] decreases | Constant, independent of [A] | Increases as [A] decreases |
| Overall order | Sum of all individual orders | Sum of all individual orders | Sum of all individual orders |

## What's next

Understanding rate laws and reaction order is the foundation for further study of reaction mechanisms and integrated rate laws in IB Chemistry AHL. This knowledge allows you to connect experimental kinetic data to proposed reaction mechanisms, identify the rate-determining step, and predict how changes to reaction conditions will impact reaction rate. Mastery of this sub-topic is essential for tackling complex kinetics problems that frequently appear in Paper 2 and Paper 3 of IB Chemistry HL exams. Next, you will build on this to study integrated rate equations, half-life properties, and connect reaction order to full reaction mechanisms.

- [AHL: Activation energy and Arrhenius equation](https://www.owlsprep.com/study/ib-chemistry-hl-u5-ahl-activation-energy-and-arrhenius/)
- [AHL: Extended Le Chatelier's principle](https://www.owlsprep.com/study/ib-chemistry-hl-u5-ahl-extended-le-chatelier-s/)
- [AHL: Reaction quotient](https://www.owlsprep.com/study/ib-chemistry-hl-u5-ahl-reaction-quotient/)

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