The half - life period of 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,n

Options

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

Explanation

To solve the problem, we need to analyze the relationship between the half-life and mean lifetime of the two radioactive elements, X and Y. Here’s a step-by-step solution:

### Step 1: Understand the Definitions
- **Half-life (T₁/₂)**: The time required for half of the radioactive atoms in a sample to decay.
- **Mean lifetime (τ)**: The average time a single atom exists before decaying, which is related to the decay constant (λ) by the formula τ = 1/λ.

### Step 2: Relate Half-life and Mean Lifetime
Given that the half-life of element X is the same as the mean lifetime of element Y, we can express this mathematically:
- For element X: T₁/₂ = ln(2)/λₓ
- For element Y: τ = 1/λᵧ

Setting these equal gives us:
\[ \frac{\ln(2)}{\lambdaₓ} = \frac{1}{\lambdaᵧ} \]

### Step 3: Solve for the Decay Constants
From the equation above, we can rearrange it to find the relationship between the decay constants:
\[ \lambdaₓ = \ln(2) \cdot \lambdaᵧ \]
Since ln(2) ≈ 0.693, we can say:
\[ \lambdaₓ ≈ 0.693 \cdot \lambdaᵧ \]

### Step 4: Compare the Decay Constants
From the relationship derived, we see that:
\[ \lambdaₓ < \lambdaᵧ \]
This indicates that element X has a smaller decay constant than element Y.

### Step 5: Determine the Rate of Decay
The rate of decay (R) for a radioactive element is given by:
\[ R = -\frac{dN}{dt} = \lambda \cdot N \]
Where N is the number of atoms present.

Since both elements initially have the same number of atoms (N₀), we can express the rates of decay for both elements:
- For element X: \( Rₓ = \lambdaₓ \cdot N₀ \)
- For element Y: \( Rᵧ = \lambdaᵧ \cdot N₀ \)

### Step 6: Compare the Rates of Decay
Since we established that \( \lambdaₓ < \lambdaᵧ \):
\[ Rₓ < Rᵧ \]
This means that element Y decays faster than element X.

### Conclusion
Thus, the final conclusion is that element Y decays faster than element X.