Question

If the half-life for a reaction is 20. Seconds, what would be the second half-life, assuming the reaction is either zero, first, or second-order?

Short answer

The second half-life for the zero, first, and second-order reactions is 10, 40 and 40 s, respectively.

Step-by-step solution

Calculating the second half-life for the first-order reaction

The first half-life $t^{\frac{1}{2}}$is 20s.

The second half-life is the time needed for the conversion from 50% to 25%.

For the second half-life, $\left[A\right]={\left[A\right]}_0/4$

For zero-order,

$\begin{array}{rcl}{t}_{\frac{1}{2}}& =& \frac{{\left[A\right]}_0}{2k}\\ t_{2\mathrm{nd}\mathrm{half}\mathrm{life}}& =& \frac{\frac{{\left[A\right]}_0}{2}}{2k}\end{array}$

Second half-life

_{$$\begin{gathered} {\mathrm{t}}_{2\mathrm{nd} \mathrm{half} \mathrm{life}} = \frac{20}{2} \\ {\mathrm{t}}_{2\mathrm{nd} \mathrm{half} \mathrm{life}} = 10 \mathrm{s} \end{gathered}$$}

The second half-life for the zero-order reaction is the 10 seconds.

Calculating the second half-life for a first-order reaction

For the first order,

$$\begin{gathered} t_{\frac{1}{2}}=\frac{0.693}{k} \\ k=\frac{0.693}{t_{\frac{1}{2}}} \end{gathered}$$

$$\begin{gathered} k = \frac{0.693}{20} \\ k = 0.03465 \end{gathered}$$

Second half-life,

$\begin{array}{rcl}{\mathrm{t}}_{2\mathrm{nd} \mathrm{half} \mathrm{life}} & =& \frac{1.386}{\mathrm{k}}\\ {\mathrm{t}}_{2\mathrm{nd} \mathrm{half} \mathrm{life}} & =& \frac{1.386}{0.03465}\\ {\mathrm{t}}_{2\mathrm{nd} \mathrm{half} \mathrm{life}} & =& 40\mathrm{s}\end{array}$

The second half-life for the first-order reaction is 40 seconds.

Calculating the second half-life for a second-order reaction

For second-order,

_{$t_{\frac{1}{2}}=\frac{1}{k{\left[A\right]}_0}$}

Second half-life

role="math" localid="1663761517108" $\begin{array}{rcl}{t}_{2\mathrm{nd}\mathrm{half}\mathrm{life}}& =& \frac{2}{k{\left[A\right]}_0}\\ & =& 2\left(\frac{1}{k{\left[A\right]}_0}\right)\\ & =& 2t_{\frac{1}{2}}\\ & =& 2\left(20\right)\\ & =& 40s\end{array}$

The second half-life for the second-order reaction is 40 seconds.