Question

Which of the following is not a first-order reaction?
(A) Decomposition of ${{H}_{2}}{{O}_{2}}$
(B) Decomposition of ${{N}_{2}}{{O}_{5}}$
(C) Decomposition of ${{N}_{2}}O$
(D) Decomposition of $S{{O}_{2}}C{{l}_{2}}$

Answer 1

Hint: Order of a reaction defines the relationship between the rate of a reaction and the concentration of the reactants involved. Now, if the rate of a reaction depends only on the concentration of one of the reactants, it is known as a first-order reaction.

Complete answer: The order of any reaction can be determined using the rate law equation which is given by the formula
\[r=k{{[A]}^{x}}{{[B]}^{y}}\]
Where r is the rate of the reaction, k is the rate constant, [A] is the concentration term of species A and x is its order of reaction, and [B] is the concentration term of species B and y is its order of reaction.
The order of a reaction is given by the sum of the exponents of the concentration terms.
Order of reaction = x + y
Now, the rate of first-order reactions is 1 and depends only on one reactant. The other reactants that contribute to the reaction will be of zero order.
The integrated rate law equation for first-order reactions is as
\[[A]={{[A]}_{0}}{{e}^{-kt}}\]
Where ${{[A]}_{0}}$ is the initial concentration of the reactant at a time ${{t}_{0}}$ and [A] is the concentration of reactant after time t has passed.
Now, the decomposition of ${{H}_{2}}{{O}_{2}}$ is given by the reaction
\[
    \,{{H}_{2}}{{O}_{2}}\to {{H}_{2}}O+\dfrac{1}{2}{{O}_{2}} \\
   or\text{ }2{{H}_{2}}{{O}_{2}}\to 2{{H}_{2}}O+{{O}_{2}} \\
 \]
Decomposition of ${{N}_{2}}{{O}_{5}}$ is given by the reaction
\[
    {{N}_{2}}{{O}_{5}}\to 2N{{O}_{2}}+\dfrac{1}{2}{{O}_{2}} \\
   or\text{ }2{{N}_{2}}{{O}_{5}}\to 4N{{O}_{2}}+{{O}_{2}} \\
 \]
Decomposition of ${{N}_{2}}O$ is given by the reaction
\[2{{N}_{2}}O\to 2{{N}_{2}}+{{O}_{2}}\]
Decomposition of $S{{O}_{2}}C{{l}_{2}}$ is given by the reaction
$S{{O}_{2}}C{{l}_{2}}\to S{{O}_{2}}+C{{l}_{2}}$
We can see that in all of these decomposition reactions, only one reactant, or the substance getting decomposed is involved and hence will influence the rate of the reaction.
So all of these reactions are the first-order reaction.

Note: We can determine the half-life of a first-order reaction using the following expression
\[{{t}_{\dfrac{1}{2}}}=\dfrac{0.693}{k}\]
The concentration of the reactants after a time equivalent to half-life has passed will be half of the initial concentration of the reactants.