The count rate of a radioactive sample was 1600 count/s at t=0 and 100 count/s at t=8s.Select the correct option.

Options

A. Its count rate was 400 count/s at t = 2s
B. Half life of the sample is 2.88 s
C. Mean life of the sample is 4 s
D. Its count rate was 200 count/s at t = 6 s

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a radioactive sample with an initial count rate of 1600 counts/s at time \( t = 0 \) and a count rate of 100 counts/s at \( t = 8 \) seconds. We need to find the decay constant \( \lambda \) and then use it to determine the half-life and mean life of the sample.

### Step 2: Write the decay formula
The count rate \( N(t) \) at any time \( t \) can be expressed using the formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where:
- \( N_0 \) is the initial count rate,
- \( N(t) \) is the count rate at time \( t \),
- \( \lambda \) is the decay constant.

### Step 3: Substitute the known values
At \( t = 0 \):
\[ N_0 = 1600 \, \text{counts/s} \]
At \( t = 8 \):
\[ N(8) = 100 \, \text{counts/s} \]
Substituting these values into the decay formula gives:
\[ 100 = 1600 e^{-\lambda \cdot 8} \]

### Step 4: Simplify the equation
Dividing both sides by 1600:
\[ \frac{100}{1600} = e^{-8\lambda} \]
This simplifies to:
\[ \frac{1}{16} = e^{-8\lambda} \]

### Step 5: Take the natural logarithm
Taking the natural logarithm of both sides:
\[ \ln\left(\frac{1}{16}\right) = -8\lambda \]
Using the property of logarithms:
\[ \ln(16) = \ln(2^4) = 4\ln(2) \]
Thus:
\[ -8\lambda = -4\ln(2) \]
This leads to:
\[ 8\lambda = 4\ln(2) \]
Dividing both sides by 8 gives:
\[ \lambda = \frac{\ln(2)}{2} \]

### Step 6: Calculate the value of \( \lambda \)
Using \( \ln(2) \approx 0.693 \):
\[ \lambda \approx \frac{0.693}{2} \approx 0.3465 \, \text{s}^{-1} \]

### Step 7: Calculate the half-life \( t_{1/2} \)
The half-life is given by:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]
Substituting the value of \( \lambda \):
\[ t_{1/2} = \frac{\ln(2)}{\frac{\ln(2)}{2}} = 2 \, \text{s} \]

### Step 8: Calculate the mean life \( \tau \)
The mean life is given by:
\[ \tau = \frac{1}{\lambda} \]
Substituting the value of \( \lambda \):
\[ \tau = \frac{1}{0.3465} \approx 2.88 \, \text{s} \]

### Step 9: Verify the options
Now we need to check the options provided:
1. Half-life of the sample is 2.88 seconds (incorrect, we found it to be 2 seconds).
2. Mean life of the sample is 4 seconds (incorrect, we found it to be approximately 2.88 seconds).
3. Count rate at \( t = 2 \) seconds is 800 counts/s (correct).
4. Count rate at \( t = 6 \) seconds is 200 counts/s (correct).

The correct option is the one that states the count rate at \( t = 6 \) seconds is 200 counts/s.