Question
Consider the reaction, $${N_2}\left( g \right) + 3{H_2}\left( g \right) \to 2N{H_3}\left( g \right)$$
The equality relationship between $$\frac{{d\left[ {N{H_3}} \right]}}{{dt}}$$ and $$ - \frac{{d\left[ {{H_2}} \right]}}{{dt}}$$ is
A.
$$\frac{{d\left[ {N{H_3}} \right]}}{{dt}} = - \frac{1}{3}\frac{{d\left[ {{H_2}} \right]}}{{dt}}$$
B.
$$ + \frac{{d\left[ {N{H_3}} \right]}}{{dt}} = - \frac{2}{3}\frac{{d\left[ {{H_2}} \right]}}{{dt}}$$
C.
$$ + \frac{{d\left[ {N{H_3}} \right]}}{{dt}} = - \frac{3}{2}\frac{{d\left[ {{H_2}} \right]}}{{dt}}$$
D.
$$\frac{{d\left[ {N{H_3}} \right]}}{{dt}} = - \frac{{d\left[ {{H_2}} \right]}}{{dt}}$$
Answer :
$$ + \frac{{d\left[ {N{H_3}} \right]}}{{dt}} = - \frac{2}{3}\frac{{d\left[ {{H_2}} \right]}}{{dt}}$$
Solution :
$$\eqalign{
& {\text{For the reaction,}} \cr
& {N_2}\left( g \right) + 3{H_2}\left( g \right) \to 2N{H_3}\left( g \right) \cr
& {\text{The rate of reaction }}wrt \cr} $$
$${N_2} = - \frac{{d\left[ {{N_2}} \right]}}{{dt}}$$ [ Rate of disappearance ]
The rate of reaction with respect to
$${H_2} = - \frac{1}{3}\frac{{d\left[ {{H_2}} \right]}}{{dt}}$$ [ Rate of disappearance ]
The rate of reaction with respect to
$$N{H_3} = + \frac{1}{2}\frac{{d\left[ {N{H_3}} \right]}}{{dt}}$$ [ Rate of appearance ]
$$\eqalign{
& {\text{Hence, at a fixed time}} \cr
& - \frac{{d\left[ {{N_2}} \right]}}{{dt}} = - \frac{1}{3}\frac{{d\left[ {{H_2}} \right]}}{{dt}} = + \frac{1}{2}\frac{{d\left[ {N{H_3}} \right]}}{{dt}} \cr
& {\text{or}}\,\,\,{\text{ + }}\frac{{d\left[ {N{H_3}} \right]}}{{dt}} = - \frac{2}{3}\frac{{d\left[ {{H_2}} \right]}}{{dt}} \cr
& {\text{or}}\,\, + \frac{{d\left[ {N{H_3}} \right]}}{{dt}} = - \frac{{2d\left[ {{N_2}} \right]}}{{dt}} \cr} $$