The half - line period a radioactive element `X` is same as the mean life time of another radioactive element `Y`. Initially both of them have the same number of atoms. Then:

Options

A. X and Y have the same decay rate initially
B. X andY decay at the same rate always
C. Y will decay at a faster rate than X
D. X will decay at a faster rate than Y

Explanation

To solve the problem, we need to analyze the relationship between the half-life of element X and the mean lifetime of element Y.

### Step-by-Step Solution:

1. **Understanding Half-Life and Mean Lifetime**:
- The half-life (T₁/₂) of a radioactive element is the time required for half of the radioactive atoms to decay.
- The mean lifetime (τ) of a radioactive element is the average time that a single atom of the substance will exist before decaying.

2. **Formulas**:
- The relationship between half-life and decay constant (λ) is given by:
\[ T_{1/2} = \frac{0.693}{\lambda} \]
- The mean lifetime is related to the decay constant by:
\[ \tau = \frac{1}{\lambda} \]

3. **Given Condition**:
- We are given that the half-life of element X is equal to the mean lifetime of element Y:
\[ T_{1/2}^X = \tau^Y \]

4. **Substituting Formulas**:
- From the above relationships, we can substitute:
\[ \frac{0.693}{\lambda_X} = \frac{1}{\lambda_Y} \]

5. **Rearranging the Equation**:
- Rearranging gives us:
\[ \lambda_Y = \frac{1}{0.693} \lambda_X \]
- This implies:
\[ \lambda_Y > \lambda_X \]

6. **Interpreting Decay Rates**:
- The decay rate of a radioactive substance is directly proportional to its decay constant (λ). Therefore, since λ_Y > λ_X, it follows that element Y will decay faster than element X.

7. **Conclusion**:
- Since element Y has a greater decay constant, it will decay at a faster rate compared to element X.

### Final Answer:
- The correct option is: **Y will decay at a faster rate than X**.