Question

$3/{{4}^{th}}$ of a first-order reaction is completed in 32 min. Calculate the time required to complete $15/{{16}^{th}}$ part of the reaction.

Answer 1

Hint: Determine the value of decay constant for the values given in the question. Now we can substitute the value in the equation given below:
$t=\dfrac{2.303}{\lambda }\log \dfrac{a}{a-x}$
Where,
t is time taken for the specified amount of decay
$\lambda $ is decay constant
a is initial concentration of reactant or radioisotope
x is the concentration of reactant or radioisotope that has decayed or disintegrated.

Complete step-by-step answer:
It is given to us that $3/{{4}^{th}}$ of a first-order reaction is completed in 32 min. With this we will calculate the value of decay constant.
$t=\dfrac{2.303}{\lambda }\log \dfrac{a}{a-x}$
t = 32 min
a = 1
x = 3/4
$\lambda =\dfrac{2.303}{32}\log \dfrac{4}{1}$
$\lambda $ = 0.043$\text{mi}{{\text{n}}^{-1}}$
We will now substitute the value of decay constant in the second reaction.
$t=\dfrac{2.303}{\lambda }\log \dfrac{a}{a-x}$
$\lambda $ = 0.043$\text{mi}{{\text{n}}^{-1}}$
a = 1
x = 15/16
$t=\dfrac{2.303}{0.043}\log \dfrac{1}{1-\text{ 15/16}}$
t = 64 min

Therefore, time required to complete $15/{{16}^{th}}$ part of the reaction is 64 minutes.

Additional information: Most of first order reactions are radioactive decay reactions. They are used to convert unstable nuclei to a stable nucleus. The energy released during this process is harnessed as well.
Radioactive decay also known as radioactive disintegration or nuclear disintegration is the process by which an unstable atomic nucleus loses energy in the form of radiation to gain stability. Any material containing unstable nuclei is considered radioactive.

Note: In case, you are not able to determine the order of the reaction, take a look at the unit of decay constant. The unit of decay constant is different for every order of reaction. The unit of decay constant for first order reaction is ${{\min }^{-1}}$.