Chemistry · Ch 3 — Chemical Kinetics

Half-life of a Reaction

3.3.3

Half-life of a Reaction

Defining half-life

The half-life of a reaction, written $t_{1/2}$ , is the time taken for the concentration of a reactant to fall to exactly one-half of its initial value. It is a convenient single number for comparing how fast different reactions (or the same reaction under different conditions) run, without having to quote a full concentration-vs-time curve.

Half-life of a zero order reaction

For a zero order reaction the rate constant is related to concentration and time by

k = \frac{[\text{R}]_0 - [\text{R}]}{t}

At $t = t_{1/2}$ , by definition $[\text{R}] = \tfrac{1}{2}[\text{R}]_0$ . Substituting,

k = \frac{[\text{R}]_0 - \tfrac{1}{2}[\text{R}]_0}{t_{1/2}} = \frac{[\text{R}]_0}{2t_{1/2}}

which rearranges to

t_{1/2} = \frac{[\text{R}]_0}{2k}

So for a zero order reaction, $t_{1/2}$ is directly proportional to the initial concentration of the reactant and inversely proportional to the rate constant — starting with more reactant simply takes proportionally longer to consume half of it.

Half-life of a first order reaction

For a first order reaction, start from the base-10 rate-constant expression

k = \frac{2.303}{t}\log\frac{[\text{R}]_0}{[\text{R}]}

Again set $t = t_{1/2}$ and $[\text{R}] = \tfrac{1}{2}[\text{R}]_0$ :

k = \frac{2.303}{t_{1/2}}\log\frac{[\text{R}]_0}{[\text{R}]_0/2} = \frac{2.303}{t_{1/2}}\log 2

Solving for $t_{1/2}$ and using $\log 2 = 0.301$ ,

t_{1/2} = \frac{2.303}{k}\log 2 = \frac{2.303 \times 0.301}{k}

t_{1/2} = \frac{0.693}{k}

Comparing the two orders

For a zero order reaction, $t_{1/2} \propto [\text{R}]_0$ — halving the amount of reactant present halves the time needed to consume the next half.

For a first order reaction, $t_{1/2}$ is a constant, completely independent of $[\text{R}]_0$ — it depends only on $k$ . Equal further intervals always halve whatever concentration remains, which is why first order half-life can be calculated from $k$ alone (and vice versa), and why measuring a constant half-life across different starting concentrations is itself experimental evidence that a reaction is first order.

The differential and integrated rate laws, straight-line plots, half-life expressions, and units of $k$ for both zero and first order reactions are collected together for quick reference in the accompanying summary table.

Table 3.4Integrated Rate Laws for the Reactions of Zero and First Order

Order	Reaction type	Differential rate law	Integrated rate law	Straight line plot	Half-life	Units of k
0	R → P	$d[R]/dt = -k$	$kt = [R]_0 - [R]$	$[R]$ vs $t$	$[R]_0/2k$	conc time⁻¹ or mol L⁻¹ s⁻¹

When a higher-order reaction behaves like first order

The apparent order of a reaction can shift under particular experimental conditions, even though its true order (fixed by how many species genuinely control the rate) does not change. The hydrolysis of ethyl acetate is a good illustration:

$\text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O} \xrightarrow{\text{H}^+} \text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH}$ …

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