Question 645: Atoms & Nuclei

Question

A radioactive sample S
1
having activity of 5 $$\mu$$Ci has twice the number of nuclei as another sample S
2
which has an activity of 10 $$\mu$$Ci. The half lives of S
1
and S
2
can be :

Accepted Answer

To solve this problem, let's begin with some fundamental concepts of radioactive decay.
The activity of a radioactive sample, which is its decay rate, is given by the equation:
$$A = \lambda N$$
where:
$$A$$ is the activity.
$$\lambda$$ is the decay constant.
$$N$$ is the number of radioactive nuclei present.
Given that sample S
1
has an activity of 5 $$\mu$$Ci and sample S
2
has an activity of 10 $$\mu$$Ci, and that sample S
1
has twice the number of nuclei as sample S
2
, we have:
$$A_1 = \lambda_1 N_1$$
$$A_2 = \lambda_2 N_2$$
We are also given that:
$$N_1 = 2N_2$$
Substitute $$N_1 = 2N_2$$ into the first equation:
$$5 \, \mu	ext{Ci} = \lambda_1 (2N_2)$$
And for the second equation:
$$10 \, \mu	ext{Ci} = \lambda_2 N_2$$
Dividing the first equation by the second equation to eliminate $$N_2$$, we get:
$$\frac{5 \, \mu	ext{Ci}}{10 \, \mu	ext{Ci}} = \frac{\lambda_1 (2N_2)}{\lambda_2 N_2}$$
Simplifying this, we obtain:
$$\frac{1}{2} = \frac{2 \lambda_1}{\lambda_2}$$
Therefore:
$$\lambda_2 = 4 \lambda_1$$
Next, we can relate the decay constant to the half-life using the equation:
$$\lambda = \frac{\ln(2)}{T_{1/2}}$$
Using the relationship between the decay constants, we have:
$$\frac{\lambda_2}{\lambda_1} = \frac{T_{1/2,1}}{T_{1/2,2}}$$
Substituting $$\lambda_2 = 4 \lambda_1$$, we get:
$$4 = \frac{T_{1/2,1}}{T_{1/2,2}}$$
Therefore:
$$T_{1/2,1} = 4 T_{1/2,2}$$
Given the options, the half-lives that satisfy this relationship are:
Option A: 20 years and 5 years, respectively.
In this case:
$$\frac{20 	ext{ years}}{5 	ext{ years}} = 4$$
Thus, option A is correct. Therefore, the half-lives of S
1
and S
2
are 20 years and 5 years, respectively.

Answer

20 years and 5 years, respectively

Answer

20 years and 10 years, respectively

Answer

10 years each

Answer

5 years each

Atoms & Nuclei

A radioactive sample S 1 having activity of 5 $\muCi has twice the number of nuclei as another sample S 2 which has an activity of 10 \mu$Ci. The half lives of S 1 and S 2 can be :

Choose the correct answer: