A radioactive substance emit 100 beta particfles in the first 2 s and 50 beta particles in the next 2s. The mean life of the sample is

Options

A. 4 s
B. 2 s
C. `(2)/(0.693)s`
D. `2xx0.693 s`

Explanation

To find the mean life of the radioactive substance based on the given information, we can follow these steps:

### Step 1: Understand the decay process
The problem states that a radioactive substance emits 100 beta particles in the first 2 seconds and 50 beta particles in the next 2 seconds. This indicates that the substance is undergoing radioactive decay.

### Step 2: Determine the half-life
From the information given:
- In the first 2 seconds, 100 particles are emitted.
- In the next 2 seconds, 50 particles are emitted.

This suggests that after the first 2 seconds, the number of particles remaining is:
\[ N_0 - 100 = N \]
Where \( N_0 \) is the initial number of particles.

After the next 2 seconds, the number of particles remaining is:
\[ N - 50 = N' \]

Since 100 particles were emitted in the first 2 seconds, we can infer that the initial count of particles was 100. After 2 seconds, there are 0 particles left, indicating that the half-life is indeed 2 seconds.

### Step 3: Relate half-life to decay constant
The relationship between half-life (\( t_{1/2} \)) and the decay constant (\( \lambda \)) is given by:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]

### Step 4: Calculate the decay constant
From the half-life, we can rearrange the formula to find \( \lambda \):
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
Substituting \( t_{1/2} = 2 \) seconds:
\[ \lambda = \frac{\ln(2)}{2} \]

### Step 5: Calculate the mean life
The mean life (\( \tau \)) is related to the decay constant by:
\[ \tau = \frac{1}{\lambda} \]
Substituting for \( \lambda \):
\[ \tau = \frac{1}{\frac{\ln(2)}{2}} = \frac{2}{\ln(2)} \]

### Step 6: Substitute the value of \( \ln(2) \)
We know that \( \ln(2) \approx 0.693 \):
\[ \tau = \frac{2}{0.693} \]

### Step 7: Calculate the final result
Now, we can compute the mean life:
\[ \tau \approx \frac{2}{0.693} \approx 2.885 \text{ seconds} \]

Thus, the mean life of the sample is approximately **2.885 seconds**.

### Summary of the Solution Steps:
1. Understand the decay process and the emitted particles.
2. Determine the half-life based on the emitted particles.
3. Relate half-life to the decay constant.
4. Calculate the decay constant using the half-life.
5. Use the decay constant to find the mean life.
6. Substitute the value of \( \ln(2) \) to find the mean life.