Question

A second order reaction where $a = b$ is $20\% $ completed in $500{\text{ seconds}}$. How long will the reaction take to be $60\% $ complete?

Answer 1

Hint: To solve this question we must know the equation for the rate constant of second order reaction. A reaction in which the rate of the reaction is directly proportional to the concentration of each of the two reactant species is known as the second order reaction. Using the equation deduce the expression for rate constant at $20\% $ completion and $60\% $ completion.

Complete answer:
We know the equation for the rate constant of a second order reaction is,
$k = \dfrac{1}{t}\dfrac{x}{{a\left( {a - x} \right)}}$
Where $k$ is the rate constant of a second order reaction,
 $t$ is time,
 $a$ is the initial concentration of the reactant,
 $x$ is the final concentration of the reactant.
The expression when reaction is $20\% $ complete is as follows:
Let the initial concentration of the reactant be 100. When the reaction is $20\% $ complete the final concentration will be 20. Thus,
\[k = \dfrac{1}{{500{\text{ seconds}}}} \times \dfrac{{20}}{{100\left( {100 - 20} \right)}}\] …… (1)
\[k = 5 \times {10^{ - 6}}{\text{ }}{{\text{M}}^{ - 2}}{\text{ }}{{\text{s}}^{ - 1}}\]
Thus, the rate constant when the reaction is $20\% $ complete is \[5 \times {10^{ - 6}}{\text{ }}{{\text{M}}^{ - 2}}{\text{ }}{{\text{s}}^{ - 1}}\].
The expression when reaction is $60\% $ complete is as follows:
Let the initial concentration of the reactant be 100. When the reaction is $60\% $ complete the final concentration will be 60. Thus,
\[k = \dfrac{1}{t} \times \dfrac{{60}}{{100\left( {100 - 60} \right)}}\] …… (2)
Rearrange the equation for time as follows:
\[t = \dfrac{1}{k} \times \dfrac{{60}}{{100\left( {100 - 60} \right)}}\]
Substitute \[5 \times {10^{ - 6}}{\text{ }}{{\text{M}}^{ - 2}}{\text{ }}{{\text{s}}^{ - 1}}\] for the rate constant. Thus,
\[t = \dfrac{1}{{5 \times {{10}^{ - 6}}{\text{ }}{{\text{M}}^{ - 2}}{\text{ }}{{\text{s}}^{ - 1}}}} \times \dfrac{{60}}{{100\left( {100 - 60} \right)}}\]
\[t = 3000{\text{ seconds}}\]

Thus, the reaction will be $60\% $ complete in \[3000{\text{ seconds}}\].

Note: The unit of rate constant for second order reaction is \[{{\text{M}}^{ - 2}}{\text{ }}{{\text{s}}^{ - 1}}\]. The unit contains concentration terms. Thus, we can say that the rate constant of a second order reaction is dependent on the concentration of the reactant.