Calculate the average life (in minutes ) , if the half - life of a radionuclide is 69.3 . Minutes

Options

A. 100
B. `1//100`
C. `69.3xx14.4`
D. `0.693xx69.3`

Answer

To calculate the average life (τ) of a radionuclide given its half-life (t₁/₂), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Half-Life and Decay Constant:**
The decay constant (λ) is related to the half-life (t₁/₂) by the formula:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]

2. **Substitute the Given Half-Life:**
We are given that the half-life (t₁/₂) is 69.3 minutes. Therefore, we can substitute this value into the equation for λ:
\[ \lambda = \frac{\ln 2}{69.3} \]

3. **Calculate the Value of λ:**
The natural logarithm of 2 (ln 2) is approximately 0.693. Thus, we can calculate λ:
\[ \lambda = \frac{0.693}{69.3} \approx 0.01 \text{ min}^{-1} \]

4. **Calculate the Average Life (τ):**
The average life (τ) is given by the formula:
\[ \tau = \frac{1}{\lambda} \]
Substituting the value of λ:
\[ \tau = \frac{1}{0.01} = 100 \text{ minutes} \]

5. **Final Result:**
The average life of the radionuclide is approximately 100 minutes.

### Summary:
- The average life (τ) of the radionuclide is **100 minutes**.