Question

For a first-order reaction which one from the folllowing is correct:

• A. The pre-exponential factor in the Arrhenius equation has the dimension of time, $T^{-1}$
• B. The degree of dissociation is equal to $(1-e^{-kt})$
• C. A plot of reciprocal of concentration of the reactant vs time gives a straight line
• D. The time taken for the completion of $75\%$ reaction is thrice the $t_{1/2}$ of the reaction

Answer 1

Answer: (A), (B)

The degree of dissociation is equal to $(1-e^{-kt})$

In first order reaction, if $\alpha$ is the degree of dissociation then,
$kt=l{n}_e\frac{1}{(1-\alpha )}=-l{n}_e(1-\alpha )$
$\Rightarrow e^{-kt}=1-\alpha$
$\therefore \alpha =1-e^{-kt}$

For first order reaction, plot of $ln\left[A\right]vs\text{time}$ gives a straight line with slope $=-k$.

For a first order reaction, the time taken for $50\%$ completion of the reaction is, $t=\frac{1}{k}ln\frac{100}{50}=\frac{1}{k}ln2$
Therefore, time taken for $75\%$ completion of the reaction is, $t=\frac{1}{k}ln\frac{100}{25}=\frac{2}{k}ln2$. Thus the time taken for the completion of $75\%$ reaction is twice the $t_{1/2}$ of the reaction.

The arrhenius equation is, $k=A{e}^{-Ea/RT}$
Dimensions of pre exponential factor ‘A’ are equivalent to dimensions of $k$, which is $T^{-1}$ for a first order reaction.