The disintegration rate of a certain radioactive sample at any instant is 4750 disintegrations per minute. Five minutes later the rate becomes 2700 per minute. Calculate
(a) decay constant and (b) half-life of the sample

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between disintegration rates and decay constant
The disintegration rate of a radioactive sample decreases exponentially over time. The relationship can be expressed as:
\[ R(t) = R_0 e^{-\lambda t} \]
where:
- \( R(t) \) is the disintegration rate at time \( t \),
- \( R_0 \) is the initial disintegration rate,
- \( \lambda \) is the decay constant,
- \( t \) is the time elapsed.

### Step 2: Identify the given values
From the problem, we have:
- Initial disintegration rate, \( R_0 = 4750 \) disintegrations per minute,
- Disintegration rate after 5 minutes, \( R(5) = 2700 \) disintegrations per minute,
- Time elapsed, \( t = 5 \) minutes.

### Step 3: Set up the equation using the disintegration rate formula
Using the values in the formula, we can write:
\[ 2700 = 4750 e^{-\lambda \cdot 5} \]

### Step 4: Rearrange the equation to solve for the decay constant \( \lambda \)
First, divide both sides by 4750:
\[ \frac{2700}{4750} = e^{-5\lambda} \]

Now take the natural logarithm of both sides:
\[ \ln\left(\frac{2700}{4750}\right) = -5\lambda \]

### Step 5: Solve for \( \lambda \)
Rearranging gives:
\[ \lambda = -\frac{1}{5} \ln\left(\frac{2700}{4750}\right) \]

### Step 6: Calculate \( \lambda \)
Now, calculate the value:
1. Calculate \( \frac{2700}{4750} \):
\[ \frac{2700}{4750} \approx 0.5684 \]
2. Take the natural logarithm:
\[ \ln(0.5684) \approx -0.566 \]
3. Substitute into the equation for \( \lambda \):
\[ \lambda = -\frac{1}{5} \times (-0.566) \approx 0.1132 \text{ min}^{-1} \]

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

### Step 8: Substitute \( \lambda \) to find \( t_{1/2} \)
1. Calculate \( \ln(2) \):
\[ \ln(2) \approx 0.693 \]
2. Substitute \( \lambda \):
\[ t_{1/2} = \frac{0.693}{0.1132} \approx 6.12 \text{ minutes} \]

### Final Answers
(a) The decay constant \( \lambda \approx 0.1132 \text{ min}^{-1} \)
(b) The half-life \( t_{1/2} \approx 6.12 \text{ minutes} \)