Question

The decomposition of $N_2{O}_5$ dissolved in carbontetrachloride occurs as follows.
$N_2{O}_{5_{\left(Solution\right)}}\rightleftharpoons 2N{O}_{2_{\left(Solution\right)}}+\frac{1}{2}{O}_{2\left(g\right)}$.
This reaction is of first order and its rate constant is $5.0\times {10}^{-4}se{c}^{-1}$. If initial concentration of $N_2{O}_5$ for this reaction is $0.50molelitr{e}^{-1}$, then
a) What will be the initial reaction rate?
b) What will be half life period of this reaction?
c) What will be the concentrations of $N_2{O}_5$ and $N{O}_2$ at the end of $50$ minutes after the starting of reaction?

Answer 1

First-order reaction means

a) $rate=\frac{-d\left[N_2{O}_5\right]}{dt}=k[N_2{O}_5{]}^1$

Given: $k=5\times {10}^{-4}{s}^{-1}$ and$\left[N_2{O}_5\right]=0.5M$

On substiution we get:

$Rate(initial=5\times {10}^{-4}\times 0.5=2.5\times {10}^{-4}M/s$

b) Half-life for FIrst order reaction:

$t=t_{1/2},\left[N_2{O}_5\right]=\frac{[N_2{O}_5{]}_0}{2}$

$t_{1/2}=\frac{0.69}{k}$

$t_{1/2}=1380s=23min$

c) Rate law of First order reaction is:

$\ln \left(\frac{{\left[N_2{O}_5\right]}_0}{{\left[N_2{O}_5\right]}_t}\right)=kt$

Given $[N_2O_5]_0=0.5\ M$,$t=3000\ s$

Substitute the values we get:

$\ln \left(\frac{0.5}{x}\right)=5\times {10}^{-4}\times 3000$

$x=\left[N_2{O}_5\right]=0.11M$

$\left[N{O}_2\right]=2\times (0.50-0.11)=0.78M$