A radioactive sample X has thrice the number of nuclei and activity one-third as compared to other radioactive sample Y. The ratio of half-lives of X and Y is

Options

A. 6
B. 9
C. 12
D. 16

Explanation

To solve the problem, we need to find the ratio of half-lives of two radioactive samples, X and Y, given their number of nuclei and activity. Let's break down the solution step by step.

### Step 1: Understand the relationship between activity, decay constant, and number of nuclei
The activity \( A \) of a radioactive sample is given by the formula:
\[ A = \lambda N \]
where:
- \( A \) is the activity,
- \( \lambda \) is the decay constant,
- \( N \) is the number of radioactive nuclei.

### Step 2: Set up the known values for samples X and Y
From the problem, we know:
- Sample X has three times the number of nuclei of sample Y:
\[ N_X = 3N_Y \]
- Sample X has one-third the activity of sample Y:
\[ A_X = \frac{1}{3} A_Y \]

### Step 3: Express the activities in terms of decay constants and nuclei
Using the activity formula for both samples:
For sample X:
\[ A_X = \lambda_X N_X = \lambda_X (3N_Y) \]
For sample Y:
\[ A_Y = \lambda_Y N_Y \]

### Step 4: Substitute the expressions into the activity relationship
From the relationship \( A_X = \frac{1}{3} A_Y \), we can substitute:
\[ \lambda_X (3N_Y) = \frac{1}{3} (\lambda_Y N_Y) \]

### Step 5: Simplify the equation
Dividing both sides by \( N_Y \) (assuming \( N_Y \neq 0 \)):
\[ 3\lambda_X = \frac{1}{3} \lambda_Y \]
Multiplying both sides by 3:
\[ 9\lambda_X = \lambda_Y \]

### Step 6: Relate decay constants to half-lives
The half-life \( T_{1/2} \) is related to the decay constant by the formula:
\[ T_{1/2} = \frac{\ln 2}{\lambda} \]
Thus, for samples X and Y:
\[ T_{1/2,X} = \frac{\ln 2}{\lambda_X}, \quad T_{1/2,Y} = \frac{\ln 2}{\lambda_Y} \]

### Step 7: Find the ratio of half-lives
To find the ratio of the half-lives:
\[ \frac{T_{1/2,X}}{T_{1/2,Y}} = \frac{\frac{\ln 2}{\lambda_X}}{\frac{\ln 2}{\lambda_Y}} = \frac{\lambda_Y}{\lambda_X} \]
Substituting \( \lambda_Y = 9\lambda_X \):
\[ \frac{T_{1/2,X}}{T_{1/2,Y}} = \frac{9\lambda_X}{\lambda_X} = 9 \]

### Conclusion
The ratio of the half-lives of samples X and Y is:
\[ \frac{T_{1/2,X}}{T_{1/2,Y}} = 9 \]

### Final Answer
The ratio of half-lives of X and Y is \( 9:1 \).

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