Q.
The composition of $N _2 O _5$ is a first order reaction represented by: $N _2 O _5 \rightarrow N _2 O _4+1 / 2 O _2$
After $20\, min$ the volume of $O _2$ produced is $10 \,mL$ and at the end of the reaction $40 \,mL$. The rate constant is equal to
Chemical Kinetics
Solution:
Consider a condition when one of the product is estimated directly or indirectly, i.e., when $V_0$ is not given
For example, in the decomposition of $NH _4 NO _2$, the volume of $N_2$ is directly measured at different intervals of time. Then, the formula used is
$k=\frac{2.303}{t} \log \frac{V_{\infty}}{V_{\infty}-V_t}$
An example is the decomposition of ammonium $\left( NH _4 NO _2\right)$ and benzene diazonium chloride $\left( C _6 H _5 N = NCl \right)$
$NH _4 NO _2( s ) \rightarrow 2 H _2 O (l)+ N _2( g )$
$ C _6 H _5- N = N - Cl (l) \rightarrow C _6 H _5- Cl (l)+ N _2( g )$
The rate of both the reactions is studied (measured) in similar manner. The volume of nitrogen $\left( N _2\right)$ is collected after a regular interval of times as follows:
Time instants
$t = 0$
$t_1$
$t_2$
$t_3$
$t_4$
$t_{\infty}$
Vol of $N_2$
$0$
$V_1$
$V_2$
$V_3$
$V_4$
$V_{\infty}$
At $t=0$, clearly the volume of $N _2=0$
Time instant $t=\infty$ means the end of a reaction, i.e., when whole of $NH _4 NO _2$ or $C _6 H _5- N = N - Cl$ is decomposed
$\Rightarrow$ At $t=\infty, V_{\infty}$ corresponds to the initial volume of $NH _4 NO _2$ or $C _6 H _5- N = N - Cl$
(Note that the ratio of stoichiometric coefficient
for both $N _2: NH _4 NO _2$ or $N _2: C _6 H _5 N = NCl$ is $1: 1$ )
$\Rightarrow c_0 \propto V_t$
At $t=t_1, t_2, t_3, \ldots$ the volume of $N _2$ corresponds to the concentration of product formed, i.e., equal to $x$
$\Rightarrow x \propto V_t$
$\Rightarrow c_0-x \propto V_{\infty}-V_t$
Hence, from first order kinetic
$k=\frac{2.303}{t} \log \frac{V_{\infty}}{V_{\infty}-V_t} $
or $k=\frac{1}{t} \operatorname{In} \frac{V_{\infty}}{V_{\infty}-V_t}$
$k=\frac{1}{20} \operatorname{In} \frac{40}{(40-10)}$
$=\frac{1}{20} \operatorname{In} \frac{40}{30}$
| Time instants | $t = 0$ | $t_1$ | $t_2$ | $t_3$ | $t_4$ | $t_{\infty}$ |
| Vol of $N_2$ | $0$ | $V_1$ | $V_2$ | $V_3$ | $V_4$ | $V_{\infty}$ |