The rate of radioactive decay of a sample are `3 xx 10^(8) dps` and `3 xx 10^(7) dps` after time `20 min` and `43.03 min` respectively. The fraction of radio atom decaying per second is equal to

Options

A. `A. (1)/(600)`
B. `B. 1`
C. `C. 0.5`
D. `D. 0.001`

Explanation

To solve the problem, we need to find the fraction of radioactive atoms decaying per second, which is represented by the decay constant \( k \). The decay of radioactive materials follows first-order kinetics, and we can use the rates of decay at two different times to find \( k \).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Rate of decay at time \( T_1 \) (20 minutes): \( R_1 = 3 \times 10^8 \) dps
- Rate of decay at time \( T_2 \) (43.03 minutes): \( R_2 = 3 \times 10^7 \) dps
- Time \( T_1 = 20 \) minutes
- Time \( T_2 = 43.03 \) minutes

2. **Calculate the Time Difference:**
\[ \Delta T = T_2 - T_1 = 43.03 \, \text{min} - 20 \, \text{min} = 23.03 \, \text{min} \]

3. **Use the First-Order Rate Equation:**
The first-order rate constant \( k \) can be calculated using the formula:
\[ k = \frac{2.303}{\Delta T} \log\left(\frac{R_1}{R_2}\right) \]

4. **Calculate the Logarithm:**
\[ \frac{R_1}{R_2} = \frac{3 \times 10^8}{3 \times 10^7} = 10 \]
Therefore,
\[ \log(10) = 1 \]

5. **Substitute the Values into the Equation for \( k \):**
\[ k = \frac{2.303}{23.03} \cdot 1 \]

6. **Calculate \( k \):**
\[ k \approx \frac{2.303}{23.03} \approx 0.1 \, \text{min}^{-1} \]

7. **Convert \( k \) to per second:**
Since \( k \) is currently in per minute, we convert it to per second:
\[ k = 0.1 \, \text{min}^{-1} \cdot \frac{1 \, \text{min}}{60 \, \text{s}} = \frac{0.1}{60} \, \text{s}^{-1} \approx \frac{1}{600} \, \text{s}^{-1} \]

8. **Final Answer:**
The fraction of radioactive atoms decaying per second is:
\[ k \approx \frac{1}{600} \, \text{s}^{-1} \]