A radioactive substance has a constant activity of 2000 disintegrations/minute. The material is separated into two fractions one of which has an initial activity of 1000 disintegrations per minute while the other fraction decays with `t_((1)/(2))=24` hours. The total activity in both samples after 48 hours of separation is

Options

A. 1500
B. 1000
C. 1250
D. 2000

Answer

To solve the problem, we need to calculate the total activity of both fractions after 48 hours.

### Step 1: Understand the initial conditions
- The total initial activity of the radioactive substance is 2000 disintegrations/minute.
- After separation, one fraction has an initial activity of 1000 disintegrations/minute. Therefore, the other fraction must have an initial activity of:
\[ \text{Activity of second fraction} = 2000 - 1000 = 1000 \text{ disintegrations/minute} \]

### Step 2: Determine the decay of the first fraction
- The first fraction has an initial activity of 1000 disintegrations/minute and does not decay (it remains constant).
- After 48 hours, the activity of the first fraction remains:
\[ A_1 = 1000 \text{ disintegrations/minute} \]

### Step 3: Determine the decay of the second fraction
- The second fraction has an initial activity of 1000 disintegrations/minute and decays with a half-life (\(t_{1/2}\)) of 24 hours.
- We need to find the activity after 48 hours. Since 48 hours is 2 half-lives (48 hours / 24 hours = 2), we can use the formula for radioactive decay:
\[ A = A_0 \left(\frac{1}{2}\right)^{n} \]
where \(A_0\) is the initial activity, and \(n\) is the number of half-lives.

- For the second fraction:
\[ A_2 = 1000 \left(\frac{1}{2}\right)^{2} = 1000 \times \frac{1}{4} = 250 \text{ disintegrations/minute} \]

### Step 4: Calculate the total activity after 48 hours
- Now, we can find the total activity after 48 hours by adding the activities of both fractions:
\[ \text{Total Activity} = A_1 + A_2 = 1000 + 250 = 1250 \text{ disintegrations/minute} \]

### Final Answer
The total activity in both samples after 48 hours of separation is **1250 disintegrations/minute**.

---