The half-life of a radioactive sample A is same as the mean life of sample B. The number of atoms present initially in both the samples is same.

Options

A. After 10 days, the number of atoms is same in A and B.
B. A decays at a faster rate than B.
C. A and B decay at same rate.
D. The decay rate of B is greater than A.

Explanation

To solve the problem, we need to analyze the relationship between the half-life and mean life of the two samples, A and B, and their decay rates.

### Step-by-Step Solution:

1. **Understanding Half-Life and Mean Life**:
- The half-life (T₁/₂) of a radioactive sample is the time taken for half of the radioactive nuclei to decay. It is related to the decay constant (λ) by the formula:
\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]
- The mean life (τ) of a radioactive sample is the average lifetime of the nuclei and is given by:
\[ \tau = \frac{1}{\lambda} \]

2. **Setting Up the Problem**:
- We are given that the half-life of sample A is equal to the mean life of sample B:
\[ T_{1/2}^A = \tau^B \]
- Using the formulas:
\[ \frac{\ln(2)}{\lambda_A} = \frac{1}{\lambda_B} \]

3. **Relating the Decay Constants**:
- Rearranging the equation gives us:
\[ \lambda_A = \ln(2) \cdot \lambda_B \]
- Since \(\ln(2) \approx 0.693\), we can conclude that:
\[ \lambda_A \approx 0.693 \cdot \lambda_B \]
- This means that the decay constant of sample A is less than that of sample B.

4. **Comparing Decay Rates**:
- Since \(\lambda_B > \lambda_A\), sample B decays faster than sample A.

5. **Calculating the Number of Atoms After a Given Time**:
- The number of atoms remaining after time \(t\) for both samples can be expressed as:
\[ N_A = N_0 e^{-\lambda_A t} \]
\[ N_B = N_0 e^{-\lambda_B t} \]
- After 10 days, since \(\lambda_B > \lambda_A\), it follows that:
\[ N_A > N_B \]
- Thus, the number of atoms in sample A will be greater than that in sample B after 10 days.

6. **Conclusion**:
- Based on the analysis, we can conclude that sample B decays faster than sample A, and the number of atoms in both samples will not be the same after 10 days.

### Final Answer:
- The correct option is that the decay rate of sample B is greater than that of sample A.