Question

The decomposition of $N_2{O}_5$ dissolved in carbon tetra chloride occurs followingly at constant temperature.
$N_2{O}_{5\left(solution\right)}\rightleftharpoons 2N{O}_{2\left(solution\right)}+\frac{1}{2}{O}_{2\left(g\right)}$
This reaction is of first order and its rate constant is $5\times {10}^{-4}{\sec }^{-1}$? If initial concentration of $N_2{O}_5$ is $0.4$ mol $litr{e}^{-1}$ then
(i) What will be the initial reaction rate?
(ii) What will be the half-life period of this reaction?
(iii) What time will be taken to complete $75\%$ reaction?

Answer 1

Here the reaction is first order reaction.

$K=5\times {10}^{-4}se{c}^{-1}$

i) Initial concentration of $N_2{O}_5=0.4mollitr{e}^{-1}$

$\therefore$initial rate of reaction$=K\left[N_2{O}_5\right]$

$\therefore$ initial rate of reaction$=5\times {10}^{-4}\times 0.4$

Thus, initial rate of reaction $=2\times {10}^{-4}mollitr{e}^{-1}se{c}^{-1}$

ii) Half-life period for the first order reaction is given as follow:

$t_{1/2}=\frac{0.693}{K}$

$=\frac{0.693}{5\times {10}^{-4}}$

$=0.1386\times {10}^{4}sec$

iii) The rate constant can be given as:

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

where a= initial amount of substance and a-x= amount of substance remaining

Thus, time taken to complete 75% of the reaction can be given as follow:

$t_{75\%}=\frac{2.303}{K}log\frac{100}{100-75}$

$t_{75\%}=\frac{2.303}{K}log\frac{100}{25}$

$t_{75\%}=\frac{2.303}{K}log\left(4\right)$

$t_{75\%}=\frac{2.303}{5\times {10}^{-4}}\times \left(0.602\right)$

$t_{75\%}=2.762\times {10}^{3}sec$