Radioactive decay follows first-order kinetic. The mean life and half-life of nuclear decay process are `tau = 1// lambda` and `t_(1//2) = 0.693//lambda`. Therefore are a number of radioactive elements in nature, their abundance is directly proportional to half life. The amount remaining after `n` half lives of radioactive elements can be calculated using the relation:
`N = N_(0) ((1)/(2))^(n)`
Select the correct relation.

Options

A. `t_(1//2) = (0.693)/(lambda)`
B. `tau = (1)/(lambda)`
C. `tau = 1.44 xx t_(1//2)`
D. `tau = (t_(1//2))/(0.693)`

Explanation

To solve the question regarding the relationships between the mean life (τ), half-life (t_(1/2)), and the decay constant (λ) in radioactive decay, we can follow these steps:

### Step 1: Understand the Definitions
- **Mean Life (τ)**: This is the average lifetime of a radioactive particle before it decays.
- **Half-Life (t_(1/2))**: This is the time required for half of the radioactive substance to decay.
- **Decay Constant (λ)**: This is a probability rate at which a radioactive particle decays.

### Step 2: Use the Relationships
From the problem statement, we know:
- Mean life (τ) is given by the formula:
\[ τ = \frac{1}{λ} \]
- Half-life (t_(1/2)) is given by the formula:
\[ t_{1/2} = \frac{0.693}{λ} \]

### Step 3: Deriving Relationships
1. **From the Half-Life to Mean Life**:
- Rearranging the half-life formula gives:
\[ λ = \frac{0.693}{t_{1/2}} \]
- Substituting this into the mean life formula:
\[ τ = \frac{1}{λ} = \frac{t_{1/2}}{0.693} \]

2. **Relating Mean Life and Half-Life**:
- We can express τ in terms of t_(1/2):
\[ τ = \frac{t_{1/2}}{0.693} \quad \text{(this shows that τ is directly proportional to t_(1/2))} \]

3. **Finding the Constant**:
- To find the constant relating τ and t_(1/2):
\[ τ = 1.44 \cdot t_{1/2} \quad \text{(since } \frac{1}{0.693} \approx 1.44\text{)} \]

### Step 4: Conclusion
Based on the relationships derived:
- The correct relationships are:
- \( t_{1/2} = \frac{0.693}{λ} \)
- \( τ = \frac{1}{λ} \)
- \( τ = 1.44 \cdot t_{1/2} \)
- \( τ = \frac{t_{1/2}}{0.693} \)

Thus, all options presented in the question are correct.