Rate equations, order of reaction and the rate constant, determining orders from initial rates and concentration-time graphs, half-life, the rate-determining step, and the effect of temperature on the rate constant.

What this dot point is asking

CCEA wants you to write rate equations, find orders and the rate constant from initial-rate data and concentration-time graphs, use half-life, link the rate equation to the rate-determining step, and describe how temperature affects the rate constant.

Rate, and what a rate equation tells you

The rate of reaction is the change in concentration of a reactant or product per unit time, in $\text{mol dm}^{-3}\,\text{s}^{-1}$. It is measured by following any property that changes as the reaction proceeds: gas volume, mass loss, colour (colorimetry), conductivity, or by sampling and titrating. The rate is fastest at the start, when concentrations are highest, and falls as reactants are used up, so the rate at any instant is the gradient of a concentration-time graph at that point. The initial rate is the gradient of the tangent at $t = 0$.

The orders cannot be read off the balanced equation because they depend on the mechanism, not the overall stoichiometry. A reactant can even be zero order, meaning its concentration has no effect on the rate at all. The orders are usually $0$, $1$ or $2$ at this level.

A first-order reactant gives a curved concentration-time graph with a constant half-life. A zero-order reactant gives a straight-line concentration-time graph (the rate is constant until the reactant runs out). A rate-concentration graph, plotted from initial-rate data, is horizontal for zero order, a straight line through the origin for first order, and an upward curve for second order.

The rate constant and its units

The rate constant $k$ is found by substituting one experiment into the rate equation and rearranging. Its units are not fixed: they are whatever makes the equation balance, and they depend on the overall order. Because rate always has units $\text{mol dm}^{-3}\,\text{s}^{-1}$, working through the algebra gives: overall order $0$, units $\text{mol dm}^{-3}\,\text{s}^{-1}$; order $1$, $\text{s}^{-1}$; order $2$, $\text{mol}^{-1}\,\text{dm}^3\,\text{s}^{-1}$; order $3$, $\text{mol}^{-2}\,\text{dm}^6\,\text{s}^{-1}$. A reliable method is to write rate $= k \times [\,]^{\text{overall order}}$, substitute the units, and rearrange for the units of $k$.

The rate-determining step

This link between the rate equation and the mechanism is the most powerful idea in kinetics: by measuring orders we can deduce which molecules collide in the slow step. If a reactant appears first order, one molecule of it is in the slow step (or a step before it); if second order, two molecules are. The orders in the rate equation must add up to the molecularity of the steps up to and including the slow one, which lets you propose and test a mechanism.

Effect of temperature

Raising temperature increases the rate constant $k$, and so increases the rate even though the concentrations are unchanged. There are two reasons: a larger fraction of molecules have energy above the activation energy $E_a$ (the dominant effect), and collisions are more frequent and more energetic. The quantitative relationship is the Arrhenius equation, $k = Ae^{-E_a/RT}$, which shows $k$ rising steeply with temperature; a rough rule of thumb is that the rate roughly doubles for every $10\ \text{K}$ rise near room temperature. Taking logarithms gives $\ln k = \ln A - \dfrac{E_a}{RT}$, so a plot of $\ln k$ against $1/T$ is a straight line of gradient $-E_a/R$, from which the activation energy can be found.

Examples in context

The hydrolysis of a tertiary haloalkane such as 2-bromo-2-methylpropane is first order overall (rate $= k[\text{RBr}]$, zero order in $\text{OH}^-$), telling us the slow step is the ionisation of $\text{RBr}$ to a carbocation, an SN1 mechanism. A primary haloalkane hydrolyses by SN2 with rate $= k[\text{RBr}][\text{OH}^-]$, so both species are in the single slow step. CCEA practicals follow such reactions by sampling and titrating or by using a colorimeter to plot concentration against time and read off half-lives.

Try this

Q1. State what a constant half-life tells you about the order of a reaction. [1 mark]

• Cue. The reaction is first order with respect to that reactant.

Q2. A reaction has $\text{rate} = k[\text{A}]^2$. State the units of $k$. [1 mark]

• Cue. $\text{mol}^{-1}\,\text{dm}^3\,\text{s}^{-1}$.

Q3. A reaction is first order in P and zero order in Q. Write the rate equation and give the overall order. [2 marks]

• Cue. $\text{rate} = k[\text{P}]$; overall order $1$.