Question

At a certain temperature, the half-life period for the thermal decomposition of a gaseous substance depends on the initial partial pressure of the substance as follows

${\text{P}}\left( {{\text{mmHg}}} \right)$	${{\text{t}}_{1/2}}\;$ (in minutes)
500	235
250	950

Find the order of reaction [Given${\text{log(23}}{\text{.5) = 1}}{\text{.37; log(95) = 1}}{\text{.97; log2 = 0}}{\text{.30}}$]
A.1
B.2
C.$2.5$
D.3

Answer 1

To answer this question, you should recall the concept of the half-life of the decomposition reaction. The term half-life refers to the time taken for the concentration of a given reactant to reach 50% of its initial concentration in any chemical reaction.

Formula used: ${\text{log}}\dfrac{{{{\text{t}}_{\text{1}}}}}{{{{\text{t}}_{\text{2}}}}} = (1 - {\text{n)log}}\dfrac{{{{\text{c}}_{\text{1}}}}}{{{{\text{c}}_{\text{2}}}}}$
where ${{\text{t}}_1}$=Initial half-life, ${{\text{t}}_2}$=Final half-life, ${{\text{c}}_1}$=Initial pressure and ${{\text{c}}_2}$=Final pressure

Complete step by step solution:
We know that half-life is the time required to reduce the original concentration in half. In this time interval, the number of disintegrations reduces to half. Substituting the given values in the formula we have:
$1.37 - 1.97 = (1 - n)\log 2$
$\Rightarrow - 0.60 = (1 - n)(0.30)$
Solving this for n:
$\Rightarrow n = 3$

Hence, the correct answer is option D.

Note: Consider a thought experiment where about a 1000 people are inside a hall with each one among the group being given a coin. This coin is an expression of the capacity to decay and each individual represents the atom with radioactivity. Each of the 1000 individuals was asked to toss their coins per minute once. Now for the results generated if the outcome is heads, the individual could be asked to leave the hall, this denotes atom decay and in the second case, if the outcome is tails, there is no need to take action except waiting for one minute more for another attempt for the toss. The important formulas for the half-life of a reaction which varies with the order of the reaction:
A.In the case of a zero-order reaction, the expression for the half-life is: ${{\text{t}}_{1/2}}\; = {\text{ }}\dfrac{{{{\left[ {\text{R}} \right]}_0}}}{{2{\text{k}}}}$
B.In the case of the first-order reaction, the expression for the half-life is given by: ${{\text{t}}_{1/2}}\; = \dfrac{{{\text{ }}0.693}}{{\text{k}}}$
C.In case of a second-order reaction, the expression for the half-life of the reaction is ${{\text{t}}_{1/2}} = \;\dfrac{1}{{{\text{k}}{{[{\text{R}}]}_0}}}$