# Reaction rate and rate expressions

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

This module covers core concepts for IB HL kinetics: quantifying reaction rate, deriving empirical rate expressions from experimental data, calculating reaction order and the rate constant, and determining the correct units for $k$.

**Prerequisites:** [Collision theory and factors affecting reaction rate](https://www.owlsprep.com/study/ib-chemistry-hl-u5-collision-theory/); [Molar quantities and concentration calculations](https://www.owlsprep.com/study/ib-chemistry-hl-u1-molar-concepts/)

## Learning objectives

- Define reaction rate and correctly state its units
- Write empirical rate expressions from experimental initial rate data
- Determine reaction order with respect to each reactant and overall order
- Calculate the rate constant $k$ and determine its correct units

## Defining Reaction Rate

**Reaction rate** — The change in concentration of a reactant or product per unit of time, always reported as a positive value.

*Example:* For the generic reaction $A \rightarrow B$, rate is defined as: $rate = -\frac{\Delta[A]}{\Delta t} = \frac{\Delta[B]}{\Delta t}$

The negative sign for reactants accounts for the fact that reactant concentration decreases over time, so the final rate value remains positive. Reaction rate can be measured as an average rate over a time interval, or an instantaneous rate at a specific time.

**Worked example:** In the first 15 seconds of the decomposition reaction $2N_2O_5 \rightarrow 4NO_2 + O_2$, the concentration of $N_2O_5$ decreases from 0.25 mol dm⁻³ to 0.16 mol dm⁻³. Calculate the average reaction rate with respect to $N_2O_5$.

1. Calculate the change in concentration of $N_2O_5$:
2. $$\Delta[N_2O_5] = 0.16 - 0.25 = -0.09 \ mol \ dm^{-3}$$
3. Apply the rate definition, substituting values:
4. $$rate = -\frac{\Delta[N_2O_5]}{\Delta t} = -\frac{(-0.09 \ mol \ dm^{-3})}{15 \ s} = 0.006 \ mol \ dm^{-3} \ s^{-1}$$

> **Exam tip:** Always include units for reaction rate. Examiners consistently penalize missing units in kinetics questions.

## Rate Expressions and Reaction Order

**Rate expression (rate law)** — An experimentally derived equation that relates reaction rate to the concentration of reactants, where $k$ is the rate constant, $m$ is the order with respect to $A$, and $n$ is the order with respect to $B$.

*Notation:* For $aA + bB \rightarrow \text{products}$, $\text{rate} = k[A]^m[B]^n$

> **warning**
>
> Reaction orders are not derived from the stoichiometric coefficients of the balanced overall reaction. They can only be determined experimentally. Only elementary reaction steps have orders matching stoichiometric coefficients.

The overall order of a reaction is the sum of the individual orders with respect to each reactant ($\text{overall order} = m + n + ...$).

**Worked example:** Experimental data for the reaction $2NO + Cl_2 \rightarrow 2NOCl$ shows rate is proportional to $[NO]^2$ and $[Cl_2]^1$. Write the rate expression and state the overall order.

1. Write the general form of the rate expression:
2. $$rate = k[NO]^m[Cl_2]^n$$
3. Substitute the experimentally determined orders:
4. $$rate = k[NO]^2[Cl_2]^1$$
5. Calculate the overall order by summing the individual orders:
6. $$\text{overall order} = 2 + 1 = 3$$

## Deducing Rate Expression from Initial Rate Data

The most common IB exam question provides a table of initial concentrations and initial rates for multiple experimental runs. To find the order for each reactant, compare two runs where all other reactant concentrations are constant, then use the rate ratio method.

**Worked example:** Use the data below to deduce the rate expression for the reaction $A + B \rightarrow C$:
Run 1: $[A] = 0.10$, $[B] = 0.10$, rate = $2.0 \times 10^{-5}$
Run 2: $[A] = 0.20$, $[B] = 0.10$, rate = $4.0 \times 10^{-5}$
Run 3: $[A] = 0.10$, $[B] = 0.20$, rate = $1.6 \times 10^{-4}$
(Concentrations in mol dm⁻³, rates in mol dm⁻³ s⁻¹)

1. Find order with respect to $A$, comparing runs 1 and 2 where $[B]$ is constant:
2. $$\frac{rate_2}{rate_1} = \left(\frac{[A]_2}{[A]_1}\right)^m \rightarrow \frac{4.0 \times 10^{-5}}{2.0 \times 10^{-5}} = \left(\frac{0.20}{0.10}\right)^m \rightarrow 2 = 2^m \rightarrow m=1$$
3. Find order with respect to $B$, comparing runs 1 and 3 where $[A]$ is constant:
4. $$\frac{rate_3}{rate_1} = \left(\frac{[B]_3}{[B]_1}\right)^n \rightarrow \frac{1.6 \times 10^{-4}}{2.0 \times 10^{-5}} = \left(\frac{0.20}{0.10}\right)^n \rightarrow 8 = 2^n \rightarrow n=3$$
5. Write the final rate expression:
6. $$rate = k[A]^1[B]^3$$

**Check your understanding**

Test your understanding of the rate ratio method:

1. Doubling the concentration of reactant X does not change the initial rate of reaction. What is the order with respect to X?

   - 0
   - 1
   - 2
   - 3

   *Why:* If doubling concentration leaves rate unchanged: $2^m = 1 = 2^0$, so $m=0$. A zero order reactant does not affect the reaction rate.

## Units of the Rate Constant $k$

The units of the rate constant $k$ depend on the overall order of the reaction. You can always derive units by rearranging the rate expression to solve for $k$, then substituting the standard units of rate (mol dm⁻³ s⁻¹) and concentration (mol dm⁻³).

**Worked example:** Find the units of $k$ for a reaction with rate expression $rate = k[A]^2[B]$, overall order 3.

1. Rearrange the rate expression to isolate $k$:
2. $$k = \frac{rate}{[A]^2[B]}$$
3. Substitute standard units for each quantity:
4. $$\text{Units of } k = \frac{mol \ dm^{-3} \ s^{-1}}{(mol \ dm^{-3})^2 (mol \ dm^{-3})} = \frac{mol \ dm^{-3} \ s^{-1}}{mol^3 \ dm^{-9}} = mol^{-2} \ dm^6 \ s^{-1}$$

| Overall order of reaction | Common units of $k$ |
| --- | --- |
| 0 | mol dm⁻³ s⁻¹ |
| 1 | s⁻¹ |
| 2 | dm³ mol⁻¹ s⁻¹ |
| 3 | dm⁶ mol⁻² s⁻¹ |
| 4 | dm⁹ mol⁻³ s⁻¹ |

> **tip**
>
> A general formula for units of $k$ is $(mol \ dm^{-3})^{(1-n)} s^{-1}$, where $n$ is the overall order of the reaction. This works for any overall order, including fractional orders.

> **Exam tip:** Checking your units for $k$ is a quick way to verify you calculated the overall order correctly. If your units don't match the expected pattern, you made a mistake in finding the order.

## Common pitfalls

- **Wrong:** Assuming reaction orders match the stoichiometric coefficients of the balanced overall equation.
  - Why it fails: Only elementary reaction steps have orders matching stoichiometry; overall reactions are almost never elementary.
  - Correct: Always determine reaction orders from the experimental data provided in the question, never from the balanced equation alone.
- **Wrong:** Getting a negative reaction rate when calculating from reactant concentration change.
  - Why it fails: Reactant concentration decreases over time, so $\Delta[\text{reactant}]$ is inherently negative.
  - Correct: Always add a negative sign to the $\Delta[\text{reactant}]$ term to get a positive reaction rate.
- **Wrong:** Flipping the ratio of rates or concentrations when calculating reaction order.
  - Why it fails: Mixing up the order of runs leads to incorrect values for the reaction order.
  - Correct: Keep the ratio order consistent: $\frac{rate_2}{rate_1} = \left(\frac{[X]_2}{[X]_1}\right)^m$, with run 2 on top for both sides.
- **Wrong:** Memorizing units of $k$ instead of deriving them, leading to incorrect units for higher overall orders.
  - Why it fails: Different overall orders have different units, and memorization often leads to mistakes.
  - Correct: Always derive units by rearranging the rate expression and substituting units for rate and concentration.
- **Wrong:** Using average rate from the end of the reaction instead of initial rate to calculate $k$.
  - Why it fails: Initial rate data uses the rate at $t=0$, when product concentration is zero and reactant concentrations are the starting values given.
  - Correct: Always use the initial rates provided in the experimental table to calculate orders and the rate constant.

## Cheatsheet

| Concept | Key Formula/Rule | Common Units |
| --- | --- | --- |
| Average reaction rate | $rate = -\frac{\Delta[\text{reactant}]}{\Delta t} = \frac{\Delta[\text{product}]}{\Delta t}$ | $mol \ dm^{-3} \ s^{-1}$ |
| General rate expression | $rate = k[A]^m[B]^n$ | $k$ units depend on order |
| Overall reaction order | $n = m + n + ...$ | Unitless |
| Rate constant units (n = overall order) | $(mol \ dm^{-3})^{(1-n)} \ s^{-1}$ | Varies by order |
| Order from rate ratio | $\frac{rate_2}{rate_1} = \left(\frac{[X]_2}{[X]_1}\right)^m$ | Solve for $m$ |

## What's next

Reaction rate and rate expressions are the foundation for all further topics in IB HL kinetics. Mastery of these concepts is required to identify the rate-determining step in reaction mechanisms, derive rate laws from reaction mechanisms, use the Arrhenius equation to calculate activation energy, and analyze integrated rate law graphs for zero, first, and second order reactions. Rate expressions are also a common theme in extended response questions across both papers 1 and 2, so solid understanding of this sub-topic will directly contribute to your overall exam score.

- [Collision theory](https://www.owlsprep.com/study/ib-chemistry-hl-u5-collision-theory/)
- [Dynamic equilibrium](https://www.owlsprep.com/study/ib-chemistry-hl-u5-dynamic-equilibrium/)
- [The equilibrium constant](https://www.owlsprep.com/study/ib-chemistry-hl-u5-the-equilibrium-constant/)

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