Question

For the reaction
${N_2}{O_5} \to 2N{O_2} + \dfrac{1}{2}{O_2}$ , given that
$\dfrac{{ - d[{N_2}{O_5}]}}{{dt}} = {K_1}[{N_2}{O_5}],$$\dfrac{{d[N{O_2}]}}{{dt}} = {K_2}[{N_2}{O_5}]$ and $\dfrac{{d[{O_2}]}}{{dt}} = {K_3}[{N_2}{O_5}]$
The relation between ${K_1},{K_2}$ and ${K_3}$ is:
A.) $2{K_1} = {K_2} = 4{K_3}$
B.) ${K_1} = {K_2} = {K_3}$
C.) $2{K_1} = 4{K_2} = {K_3}$
D.) None of these.

Answer 1

Hint:In a chemical reaction, the rate of change of reactants is given as the negative of the rate of change of product also we divide the stoichiometric coefficient of the reactant or product respectively with their rates of change in the given reaction with the rate of change of the reactant or product.

Complete step by step answer:
As we know that the rate of reaction or reaction rate can be defined as the change in concentration of a reactant or a product with respect to the time. In a reaction, like:
Reactants $ \to $ Products
We know that during the reaction process, the reactants molecules are used up and product molecules are formed. That is, we can say that during the reaction, the reactants are disappearing and the products are forming with the time.
For a general chemical reaction:
$aA + bB \to cC + dD$
The rate of reaction \[ = - \dfrac{1}{a}\dfrac{{d[A]}}{{dt}} = - \dfrac{1}{b}\dfrac{{d[B]}}{{dt}} = \dfrac{1}{c}\dfrac{{d[C]}}{{dt}} = \dfrac{1}{d}\dfrac{{d[D]}}{{dt}}\]
Here as we can see that the rate of disappearance of reactant is given as negative this is because the concentration of reactant is decreasing with time and rate of formation of product is positive because the concentration of reactant is increasing with time. Also, the coefficient of every reactant or product is divided with its rate.
For the given reaction that is,
${N_2}{O_5} \to 2N{O_2} + \dfrac{1}{2}{O_2}$
The rate of reaction can be given as:
Rate of reaction $ = - \dfrac{{d[{N_2}{O_5}]}}{{dt}} = \dfrac{1}{2}\dfrac{{d[N{O_2}]}}{{dt}} = 2\dfrac{{d[{O_2}]}}{{dt}}$ $ - (1)$
According to question,
$\dfrac{{ - d[{N_2}{O_5}]}}{{dt}} = {K_1}[{N_2}{O_5}],$$\dfrac{{d[N{O_2}]}}{{dt}} = {K_2}[{N_2}{O_5}]$ and $\dfrac{{d[{O_2}]}}{{dt}} = {K_3}[{N_2}{O_5}]$
Now, by putting all above values in equation $ - (1)$, we get:
${K_1}[{N_2}{O_5}] = \dfrac{1}{2}{K_2}[{N_2}{O_5}] = 2{K_3}[{N_2}{O_5}]$
$2{K_1} = {K_2} = 4{K_3}$

Hence, option A.) is the correct answer.

Note:
Always remember that the rate of change of reactants is taken as negative because the concentration of reactant is decreasing with the time and rate of change of products is taken as positive because the concentration of products is increasing with the time.