Question

When the concentration of a reactant in reaction $A \rightarrow B$ is increased by $8$ times but rate increases only $2$ times, the order of the reaction would be:
(A) $2$
(B) $\dfrac{1}{3}$
(C) $4$
(D) $\dfrac{1}{2}$

Answer 1

This question is based on the concept of rate and order of a reaction. The order of a reaction is the power to which the concentration of a reactant is raised. It is the stoichiometric coefficient of the reactant in a reaction. A reaction involving only a single reactant is called a first-order reaction. In a first-order reaction, the rate is dependent on the concentration of only one reactant with its stoichiometric coefficient raised to its power.

Complete step by step answer:
The rate of a reaction involving a single reactant is given by the following formula:
$r = k[R]^n$
Where $r$ is the rate of the equation;
$k$ is the rate constant or the proportionality constant;
$[R]$ is the concentration of the reactant
$n$ is the order of the reaction.
For the reaction, $A \rightarrow B$, the reactant is $A$. So, the rate equation for the given reaction is as follows:
$r = k[A]^n$ …$(i)$
When the concentration of the reaction is increased by $8$ times, it becomes $[8A]$.
When the rate increases two times, the new rate becomes $2r$.
Therefore, according to the question, the new rate equation can be represented as:
$2r = k[8A]^n$ …$(ii)$
Dividing equation $(ii)$ by $(i)$, we get;
$\dfrac{2r}{r} = \dfrac{k[8A]^n}{k[A]^n}$
$\Rightarrow 2 = 8^n$
$\Rightarrow 2^1 = (2^3)^n$
$\Rightarrow 2^1 = (2)^3n$
$\Rightarrow 1 = 3n$
$\Rightarrow n = \dfrac{1}{3}$
Therefore, the value of the order of the reaction, $n$ is equal to $\dfrac{1}{3}$.

So, the correct answer is Option B .

Note: The difference between the rate and rate constant of a reaction is that the rate $(r)$ of a reaction depends upon the concentration of the reactant at a particular temperature, and is variable, whereas, the rate constant $(k)$ of a reaction is constant (fixed) at a particular temperature, and is invariable. According to the rule of exponentiation, the powers (exponents) of two common bases are the same. For example – If $a^x = a^y$, then $x = y$.