Half-life and rate constant for first order reaction are related by equation,

Half-life and rate constant for first order reaction are related by equation,
  1. $t_{1 / 2}=\frac{k}{0.693}$
  2. $\mathrm{k}=\frac{[\mathrm{A}]_0-[\mathrm{A}]_{\mathrm{t}}}{\mathrm{t}_{1 / 2}}$
  3. $\mathrm{t}_{1 / 2}=\frac{0.693}{\mathrm{k}}$
  4. $\mathrm{t}_{1 / 2}=\frac{[\mathrm{A}]_{\mathrm{t}}-[\mathrm{A}]_0}{\mathrm{k}}$

Solution

For first order reaction, $\mathrm{t}=\frac{2.303}{\mathrm{k}} \cdot \log \frac{\mathrm{a}_0}{\mathrm{a}_0-\mathrm{x}}$ At half-life, $x=a_0 / 2$ $\begin{aligned} & \mathrm{t}=\frac{2.303}{\mathrm{k}} \cdot \log \frac{\mathrm{a}_0}{\mathrm{a}_0-\frac{\mathrm{a}_0}{2}} \\ & \mathrm{t}_{1 / 2}=\frac{2.303}{\mathrm{k}} \cdot \log 2=\frac{2.303 \times 0.3010}{\mathrm{k}} \\ & \mathrm{t}_{1 / 2}=\frac{0.693}{\mathrm{k}} \end{aligned}$

Asked in: MHT CET 2021 (24 Sep Shift 2)

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