Question

A radioactive sample $\mathrm{S}_{1}$ having an activity of $5 \mu \mathrm{Ci}$ has twice the number of nuclei as another sample $\mathrm{S}_{2}$ which has an activity of $10 \mu \mathrm{Ci}$. The half-lives of $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ can
be
A. 20 years and 5 years, respectively
B. 20 year and 10 years, respectively
C.\[10\]years each
D.$5$ years each

Answer 1

Hint: Microcurie is given to the operation of the radioactive specimen. We convert it to a curie, and one curie is $3.7 \times 10^{10}$ per second disintegrated. We need to have the sample's initial number of atoms and its half-life or decay constant to calculate the activity. Half-life, in radioactivity, the time interval required for one-half of the radioactive sample's atomic nuclei to decay (change spontaneously to other nuclear species by releasing particles and energy), or, equivalently.

Complete answer:
We know activity is given by $\dfrac{d N}{d t} .$ Let $\dfrac{d N_{1}}{d t}$ be the activity of the first sample and $\dfrac{d N_{2}}{d t}$ be the activity of the second.
Given data in the question says, $2 N_{2}=N_{1}$
$\dfrac{d N_{1}}{d t}=5 \mu \mathrm{Ci}$
$\dfrac{d N_{2}}{d t}=10 \mu \mathrm{Ci}$
We know activity is given by $\dfrac{d N}{d t}=\lambda N$
Therefore, $\dfrac{d N_{1}}{d t}=\lambda_{1} N_{1}$ and $\dfrac{d N_{2}}{d t}=\lambda_{2} N_{2}$
Substituting the values given in the question
$\Rightarrow {{\lambda }_{1}}{{N}_{1}}=5$
$\Rightarrow {{\lambda }_{2}}{{N}_{2}}=10$
Dividing we get $\dfrac{\lambda_{2} N_{2}}{\lambda_{1} N_{1}}=2$
But $2 N_{2}=N_{1}$
\[\therefore \]$\lambda_{2}=4 \lambda_{1}$
Also, the relationship between decay constant and the half-life is $T_{\dfrac{1}{2}} \lambda=.69$
So eq (1) can be modified into $T_{\left(\dfrac{1}{2}\right) 1}=4 T_{\left(\dfrac{1}{2}\right) 2}$ It is clear that the ratio of half-lives of the two
\[\therefore \] Half-lives can be 20 years and 5 years respectively.

Hence, the correct option is (A).

Note:
In this question, the radioactive sample unit of operation was not provided in standard units, but we did not bother to Convert the same so we have to use the ratio and they cancel each other, eventually. Half-life is the span of time in which That makes one-half of the initial number of atoms the number of undecayed nuclei. The time interval required for a radioactive sample's number of decays per second. Rutherford applied the concept of the half-life of a radioactive element to studies of rock age determination by calculating the decay time.