The activity per ml of a solution of radioactive substances is x.How much water be added to 200 ml of this so that the acitivity falls to x//20 per ml after 4 half-lives ?

Explanation

To solve the problem, we need to determine how much water should be added to a 200 ml solution of a radioactive substance so that its activity per ml falls to \( \frac{x}{20} \) after 4 half-lives.

### Step-by-step Solution:

1. **Determine Initial Activity:**
The initial activity of the solution can be calculated as:
\[ \text{Initial Activity} = \text{Volume} \times \text{Activity per ml} = 200 \, \text{ml} \times x = 200x \]

**Hint:** Remember that the total activity is the product of the volume of the solution and the activity per unit volume.

2. **Determine Final Activity After 4 Half-Lives:**
After 4 half-lives, the activity can be expressed using the formula:
\[ A = A_0 \left( \frac{1}{2} \right)^n \]
where \( A_0 \) is the initial activity, \( n \) is the number of half-lives, and \( A \) is the final activity. Here, \( n = 4 \):
\[ A = 200x \left( \frac{1}{2} \right)^4 = 200x \times \frac{1}{16} = \frac{200x}{16} = 12.5x \]

**Hint:** The activity decreases exponentially with each half-life, so use the half-life formula to find the final activity.

3. **Set Up the Equation for Final Activity:**
We know that the final activity per ml after dilution is \( \frac{x}{20} \). If we denote the final volume of the solution as \( V \), then:
\[ \text{Final Activity} = V \times \frac{x}{20} \]
Setting the two expressions for final activity equal gives:
\[ V \times \frac{x}{20} = 12.5x \]

**Hint:** Equate the expression for final activity derived from the half-life decay to the diluted activity expression.

4. **Solve for Final Volume \( V \):**
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ V \times \frac{1}{20} = 12.5 \implies V = 12.5 \times 20 = 250 \, \text{ml} \]

**Hint:** When solving for \( V \), ensure to isolate it properly by multiplying both sides by 20.

5. **Calculate the Volume of Water to be Added:**
The amount of water to be added is the difference between the final volume and the initial volume:
\[ \text{Volume of Water} = V - \text{Initial Volume} = 250 \, \text{ml} - 200 \, \text{ml} = 50 \, \text{ml} \]

**Hint:** The volume of water added is simply the final volume minus the initial volume.

### Final Answer:
The amount of water that must be added is **50 ml**.