If 400 gm radioactive Iodine is taken initially. Then calculate the amount the `""(53)I^(128)(t_((1)/(2))="25 min")` left after 75 minutes

Answer

To solve the problem of how much radioactive iodine \(^{128}I\) is left after 75 minutes, we can follow these steps:

### Step 1: Identify the initial amount and half-life
- The initial amount of radioactive iodine \(N_0\) is 400 grams.
- The half-life \(t_{1/2}\) of \(^{128}I\) is given as 25 minutes.

### Step 2: Determine the number of half-lives in 75 minutes
- To find out how many half-lives fit into 75 minutes, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{t_{1/2}} = \frac{75 \text{ min}}{25 \text{ min}} = 3 \]

### Step 3: Calculate the remaining amount after 3 half-lives
- After each half-life, the amount of radioactive substance is halved. Therefore, after 3 half-lives, the remaining amount \(N\) can be calculated using the formula:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
where \(n\) is the number of half-lives.

Substituting the values:
\[ N = 400 \text{ g} \left(\frac{1}{2}\right)^3 \]
\[ N = 400 \text{ g} \times \frac{1}{8} = 50 \text{ g} \]

### Final Answer
The amount of radioactive iodine \(^{128}I\) left after 75 minutes is **50 grams**.

---