Question

The half-life period of a first-order reaction is 15 minutes. The amount of substance left after one hour will be:
(A) $\dfrac{1}{4}$ of the original amount
(B) $\dfrac{1}{8}$ of the original amount
(C) $\dfrac{1}{16}$ of the original amount
(D) $\dfrac{1}{32}$ of the original amount

Answer 1

It is given that half-life of first order reaction is 15 minutes. So, in 15 minutes the concentration of reactant is reduced to half its original concentration. The reactant undergoes 4 half-lives in 60 minutes. The amount of reactant left over after 60 minutes can be calculated by the below formula:
$\text{Amount left = }\dfrac{\dfrac{{{[\text{A }\\!\\!]\\!\\!\text{ }}_{0}}}{{{2}^{n}}}}{{{[\text{A }\\!\\!]\\!\\!\text{ }}_{0}}}$
Where,
${{[\text{A }\\!\\!]\\!\\!\text{ }}_{0}}$ is the initial concentration of the radioactive element,
n is the number of half-lives spent by the radioactive element.

Complete step by step solution:
We know that the total number of half-lives spent is 4.
Substituting the values in the formula,
$\text{Amount left = }\dfrac{\dfrac{{{[\text{A }\\!\\!]\\!\\!\text{ }}_{0}}}{{{2}^{n}}}}{{{[\text{A }\\!\\!]\\!\\!\text{ }}_{0}}}$
$\text{Amount left = }\dfrac{\dfrac{{{[\text{A }\\!\\!]\\!\\!\text{ }}_{0}}}{{{2}^{4}}}}{{{[\text{A }\\!\\!]\\!\\!\text{ }}_{0}}}$
$\text{Amount left = }\dfrac{1}{16}$
The amount of substance left after one hour will be $\dfrac{1}{16}$ of the original amount.

Therefore, the correct answer is option (C).

Additional information:
Most 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 ${{\min }^{-1}}$.