# Rate equations and order of reaction
> Chemistry · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9701-u20-rate-equations-and-order-of/
This sub-topic introduces the core relationship between reactant concentration and reaction rate. You will learn to determine reaction orders, construct rate equations, and interpret their meaning for reaction mechanisms, a frequent exam topic.
**Prerequisites:** [Basic reaction rates and experimental rate measurement](https://www.owlsprep.com/study/cie-9701-u10-rate-of-reaction/)
## Learning objectives
- Write the general rate equation for a reaction
- Define and calculate individual and overall order of reaction
- Deduce reaction order from initial rate and graphical data
- Interpret rate equations in terms of reaction mechanism
## 1. Core Definitions and the General Rate Equation
**Rate Equation** — An expression that describes how reaction rate depends on reactant concentrations, determined experimentally (not from reaction stoichiometry)
*Notation:* For the general reaction $aA + bB \rightarrow cC + dD$
$$rate = k[A]^m[B]^n$$
Where $k$ = the rate constant, $[A], [B]$ = concentrations of A and B, $m$ = order with respect to A, $n$ = order with respect to B.
**Order of Reaction** — The power to which a reactant's concentration term is raised in the rate equation. Orders can be 0, 1, 2, or rarely fractions/ higher values
*Example:* If $m=1$, the reaction is first order with respect to A
**Worked example:** For the reaction $2HI(g) \rightarrow H_2(g) + I_2(g)$, the rate equation is $rate = k[HI]^2$. State the order with respect to HI and the overall order of reaction.
1. 1. The order with respect to a reactant equals the power of its concentration term:
2. $$[HI]^2$$
3. The power is 2, so the order of reaction with respect to HI is 2.
4. 2. Overall order is the sum of all individual orders. There is only one reactant term, so overall order = $2$.
## 2. Deducing Order from Initial Rate Data
The most common exam question asks you to deduce the rate equation from a table of experimental initial rate data. The method compares two experiments where only one concentration is changed, to find its effect on rate:
- Zero order: If [X] doubles and rate stays the same, order = 0 ($2^0 = 1$, rate × 1)
- First order: If [X] doubles and rate doubles, order = 1 ($2^1 = 2$, rate × 2)
- Second order: If [X] doubles and rate quadruples, order = 2 ($2^2 = 4$, rate × 4)
**Worked example:** Deduce the rate equation for the reaction $A + 2B \rightarrow C + D$ using the data below:
1: [A] = 0.1, [B] = 0.1, rate = 0.001
2: [A] = 0.2, [B] = 0.1, rate = 0.002
3: [A] = 0.1, [B] = 0.2, rate = 0.004 (all units mol dm⁻³/s)
1. 1. Find order with respect to A: compare Experiments 1 and 2 (constant [B])
2. [A] doubles, rate doubles, so order with respect to A = 1.
3. 2. Find order with respect to B: compare Experiments 1 and 3 (constant [A])
4. [B] doubles, rate quadruples. $2^n = 4 \rightarrow n=2$, so order with respect to B = 2.
5. 3. Substitute orders into the general rate equation:
6. $$rate = k[A]^1[B]^2 = k[A][B]^2$$
> **Exam tip:** Always confirm only one reactant concentration changes between the two experiments you compare. If two change, you cannot directly calculate order.
## 3. Deducing Order from Rate-Concentration Graphs
Order can also be found from continuous rate data plotted as rate against concentration. Each order gives a characteristic graph shape:
| Order | Shape of rate vs [X] graph | Key property |
| --- | --- | --- |
| 0 | Horizontal line | Gradient = 0, rate independent of [X] |
| 1 | Straight line through origin | Gradient equals the rate constant $k$ |
| 2 | Upwards curving parabola | Gradient increases as [X] increases |
> **tip**
>
> Do not confuse this with concentration vs time graphs: gradient of a concentration vs time graph for zero order equals $-k$, not the gradient of a rate-concentration graph.
**Worked example:** A rate-concentration graph is a straight line through the origin with gradient 0.032 s⁻¹. State the order of reaction and the value of the rate constant.
1. 1. A straight line through the origin for a rate-concentration graph is the characteristic shape of a first order reaction.
2. 2. For first order reactions, the gradient of the rate-concentration graph equals the rate constant $k$.
3. Therefore: order = 1, $k = 0.032$ s⁻¹
## 4. Rate Equations and Reaction Mechanisms
The rate equation gives information about the rate-determining step (RDS), the slowest step in a reaction mechanism that controls the overall reaction rate. The order with respect to a reactant equals the number of molecules of that reactant that participate in the rate-determining step.
**Worked example:** The reaction $NO_2(g) + CO(g) \rightarrow NO(g) + CO_2(g)$ has the rate equation $rate = k[NO_2]^2$. What does this tell us about the rate-determining step?
1. 1. The exponent of each reactant equals the number of its molecules in the RDS.
2. $[NO_2]$ is raised to the power 2, so 2 molecules of $NO_2$ are present in the rate-determining step.
3. CO does not appear in the rate equation, so its order is 0. This means CO is not involved in the rate-determining step, and reacts in a faster, later step after the RDS.
## Common pitfalls
- **Wrong:** Assuming order equals the stoichiometric coefficient in the balanced equation
- Why it fails: Rate equations are determined experimentally, not from reaction stoichiometry. Only elementary single-step reactions have orders matching coefficients.
- Correct: Always deduce order from experimental data, never assume it equals the balancing number.
- **Wrong:** Treating the rate constant $k$ as constant regardless of temperature
- Why it fails: The rate constant always increases with increasing temperature, even for the same reaction.
- Correct: Remember that $k$ is temperature dependent, and always quote the temperature when giving a value of $k$.
- **Wrong:** Comparing two experiments where two reactant concentrations change to find order
- Why it fails: If two concentrations change, you cannot isolate the effect of each reactant on the rate.
- Correct: Always select two experiments where all concentrations except one are constant to calculate each order individually.
- **Wrong:** Miscalculating order when concentration changes by a factor other than 2, e.g. calling order 3 when concentration triples and rate increases 9x
- Why it fails: Order is the exponent, not the factor of increase in concentration.
- Correct: Calculate order as $n = \frac{\ln(rate_2/rate_1)}{\ln([X]_2/[X]_1)}$. For 3x concentration / 9x rate, $n = \frac{\ln 9}{\ln 3} = 2$.
## Cheatsheet
| Change in [X] | Change in rate | Order of X |
| --- | --- | --- |
| [X] doubles | No change | 0 |
| [X] doubles | Rate doubles | 1 |
| [X] doubles | Rate × 4 | 2 |
| [X] triples | Rate × 3 | 1 |
| [X] triples | Rate × 9 | 2 |
| X not in rate equation | No change | 0 |
| Overall order | | Sum of all individual orders |
## What's next
Rate equations are the foundation of all further work in reaction kinetics for CIE A-Level Chemistry. Mastering how to find order and construct rate equations is critical for scoring high marks in kinetics questions, which appear regularly across all papers (1, 2 and 4). The relationship between order and the rate-determining step that you learned here also forms the basis for understanding how reaction mechanisms are determined from experimental data. Next, you will build on this core knowledge to explore half-life for first order reactions, how to identify the rate-determining step, and how to calculate activation energy using the Arrhenius equation.
- [Rate-determining step and mechanism](https://www.owlsprep.com/study/cie-9701-u20-rate-determining-step/)
- [Arrhenius equation and activation energy](https://www.owlsprep.com/study/cie-9701-u20-arrhenius-equation/)
- [Rate constant calculations](https://www.owlsprep.com/study/cie-9701-u20-rate-constant-calculations/)
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