Question

Describe the half-life method for determining the order of a first order reaction.

Answer 1

The half-life of a reaction is usually denoted as t_1/2. We have already seen that for a first-order reaction,

\[\ce{k = \frac{2.303}{t} log_10 \frac{[A]_0}{[A]}}\]

The equation can be written as 

\[\ce{t = \frac{2.303}{k} log_10 \frac{[A]_0}{[A]}}\]

When half of the reaction is complete,

\[\ce{[A] = \frac{1}{2} [A]_0}\] and t = t_1/2 (i.e., half-life)

Therefore, for a first-order reaction to reach halfway, we have

\[\ce{t_{1/2} = \frac{2.303}{k} log_10 \frac{[A]_0}{\frac{1}{2} [A]_0}}\]

or, \[\ce{t_{1/2} = \frac{2.303}{k} log_10 2}\]

= \[\ce{\frac{2.303}{k} \times 0.3010}\]

= \[\ce{\frac{0.693}{k}}\]

∴ t_1/2 = \[\ce{\frac{0.693}{k}}\]

If t_1/2 remains constant for different [A]₀​, the reaction is first order, and k can be calculated from t_1/2​ using

\[\ce{k = \frac{0.0693}{t_{1/2}}}\]