Question

Half-lives of a first order and a zero order reaction are the same. The ratio of initial rates of the first order reaction to that of the zero order reaction is :

• A. $\frac{2}{0.693}$
• B. $0.693$
• C. $2\times 0.693$
• D. $\frac{1}{0.693}$

Answer 1

The correct option is C

$2\times 0.693$

$t_{\frac{1}{2}}$ of first order reaction $=\frac{0.693}{K}$

$t_{\frac{1}{2}}$ of zero order reaction $=\frac{[A{]}_0}{2K^{\prime }}$

Initial rate of $1^{st}$ order $\left(r_1\right)=K[A{]}_0$

Initial rate of zero order $\left(r_0\right)=K^{\prime }$

$\Rightarrow \frac{r_1}{r_0}=\frac{K[A{]}_0}{K^{\prime }}but\frac{0.693}{K}=\frac{[A{]}_0}{2K^{\prime }}$

$\Rightarrow \frac{K[A{]}_0}{K^{\prime }}=2\times 0.693=\frac{r_1}{r_0}$

$\therefore \frac{r_1}{r_0}=2\times 0.693$