Question

Which of the following graphs is correct for a first order reaction ?

Answer 1

(a, d)

For the first order reaction

$k=\frac{2.303}{t}log\frac{[R{]}_0}{\left[R\right]}\frac{kt}{2.303}=log\frac{[R{]}_0}{\left[R\right]}log\frac{[R{]}_0}{\left[R\right]}=\left(\frac{k}{2.303}\right)\times t+0\underset{y}{\uparrow }=\underset{m}{\uparrow }\underset{x}{\uparrow }+\underset{c}{\uparrow }$

Correct plot of log $log\frac{[R{]}_0}{\left[R\right]}$ can be represented by (d)

where, slope= $\frac{k}{2.303}$

The time taken for any fraction of the reaction to complete is independent of the initial concentration. Let, us consider it for half of the reaction to complete.

$t=\frac{2.303}{k}log\frac{a}{a-x}$

For half-life $t=t_{1/2}andx=\frac{a}{2}$

$t_{1/2}=\frac{2.303}{k}log\frac{a}{a-\frac{a}{2}}$

$t_{1/2}=\frac{2.303}{k}log2$

Half-life time $t_{1/2}=\frac{0.693}{k}$

$t_{1/2}$ is independent of initial concentration. Hence, correct plot of $t_{1/2}$ and $[R{]}_0$ can be represneted by a.