Question

For the first order reaction, \[{t_{99\% }} = x \times {t_{90\% }}\], the value of x will be:
(A) 10
(B) 6
(C) 3
(D) 2

Answer 1

Hint: \[{t_{99\% }}\] is the time when 1% of the initial reactant is left and 99% of the reactant is consumed in the reaction, and \[{t_{90\% }}\] is the time when 10% of the reactant is left and 90% of the reactant is consumed.

Complete Solution:
A reaction that proceeds at a rate which depends only on the concentration of one reactant is called first order reaction. The unit for the first order reaction is given as \[{s^{ - 1}}\].
The relation in time of reaction and rate constant of reaction is given as follows:
\[k = \dfrac{{2.303}}{t}\log \dfrac{a}{{a - x}}\]
Where,
k is the rate constant,
t is the time taken,
a is the initial concentration,
x is the concentration of product at time t
a - x is the concentration of a left at time t.

Half-life is the time of reaction when half of the reactant is consumed, for first order reaction, the formula for half-life is:
\[{t_{\dfrac{1}{2}}} = \dfrac{{0.693}}{k}\]
Where, \[{t_{\dfrac{1}{2}}}\] is the time when half of the reaction is completed.
For the given time in the question, for \[{t_{99\% }}\] the formula for rate constant will be,
\[k = \dfrac{{2.303}}{{{t_{99\% }}}}\log \dfrac{{100}}{1}\]
\[{t_{99\% }} = \dfrac{{2.303}}{k}\log \dfrac{{100}}{1}\]
\[{t_{99\% }} = \dfrac{{2.303}}{k} \times 2\]

And for \[{t_{90\% }}\], the formula will be written as:
\[{t_{90\% }} = \dfrac{{2.303}}{k}\log \dfrac{{100}}{{10}}\]
\[{t_{90\% }} = \dfrac{{2.303}}{k}\]

Comparing the two expression of \[{t_{99\% }}\] and \[{t_{90\% }}\], we will get,
\[{t_{99\% }} = x \times {t_{90\% }}\]
\[\dfrac{{{t_{99\% }}}}{{{t_{90\% }}}} = x\]
\[\dfrac{{\dfrac{{2.303}}{k} \times 2}}{{\dfrac{{2.303}}{k}}} = x\]
\[x = 2\]
Therefore, the value of x is 2.

So, the correct answer is “Option D”.

Note: First order reaction is dependent on only one reactant, so the first order reaction is unimolecular reaction. Decomposition of Hydrogen peroxide is an example of first order reaction. To find the overall order of a reaction, we have to sum or add off all the exponents of the concentration terms in the rate equation.