Question

The reaction mechanism for the reaction $P \to R$ is as follows:
$P\xrightarrow{{{K_1}}}2Q(fast)$ ; $2Q + P\xrightarrow{{{K_2}}}R(slow)$
The rate law for the main reaction $(P \to R)$ is [where ${K_1}$ is an equilibrium constant]
A. ${K_1}[P][Q]$
B. ${K_1}{K_2}[Q]$
C. ${K_1}{K_2}{[P]^2}$
D. ${K_1}{K_2}[P]$

Answer 1

Rate law depicts a relationship of reactant concentration with the rate of reaction. The slow step determines the rate equation for the reaction. Use the rate law of mass action to determine the rate of the equation with the formula is given as, $rate = K{[A]^a}{[B]^b}$. Determine the equation in terms of reactant concentration.

Complete answer:
The rate law is an equation that for a balanced chemical equation is expressed by the concentration of reactants raised to its coefficient each. For a chemical reaction as follows:
$aA + bB\xrightarrow{K}cC + dD$, the rate law will be written as $rate = K{[A]^a}{[B]^b}$and the rate constant, $K = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}}$
The rate constant is constant and it is independent of the concentration of reactants. But, it depends upon the temperature of the reaction.
The order of the reaction is calculated by the summation of the power of reactant concentration.

For the given equation, $P \to R$ this reaction follows two steps as follows:
Step 1: $P\xrightarrow{{{K_1}}}2Q(fast)$
Step 2: $2Q + P\xrightarrow{{{K_2}}}R(slow)$
The rate of reaction is always dependent on the slow step of the reaction, here step 2 will determine the rate of the reaction. Thus, the rate of reaction = rate of step 2.
From step 1, rate constant will be, ${K_1} = \dfrac{{{{[Q]}^2}}}{{[P]}} \Rightarrow {[Q]^2} = {K_1}[P]$.
And from step 2, the rate constant will be, ${K_2} = {[Q]^2}[P]$.
Now, the rate equation for the overall reaction according to the rate law will be as follows;
$\Rightarrow rate = {K_2}[P]{[Q]^2}$, from the rate constants equation we can substitute the value of ${[Q]^2}$
$\Rightarrow rate = {K_2}[P]({K_1}[P])$
$\Rightarrow rate = {K_1}{K_2}{[P]^2}$
Thus, the rate equation of reaction is $rate = {K_1}{K_2}{[P]^2}$.

Thus, the correct option is (C).

Note:

The rate of the constant unit can be used to determine the order of the reaction.
Order of reaction is defined as the sum of powers of concentration terms in the rate law.
Order of reaction can be determined with a rate equation but the molecularity of reaction can only be determined experimentally.