The amount of radioactive `""_(52)I^(123) (t_(1//2)=25` minutes) left after 50 minutes will be :

Options

A. `1//4`
B. `1//2`
C. `1//3`
D. None of these

Explanation

To solve the problem of how much radioactive iodine \( _{52}^{123}I \) is left after 50 minutes, we can follow these steps:

### Step 1: Understand the Half-Life
The half-life (\( t_{1/2} \)) of the radioactive iodine is given as 25 minutes. This means that every 25 minutes, half of the remaining iodine will decay.

### Step 2: Calculate the Number of Half-Lives
To find out how many half-lives fit into 50 minutes, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{50 \text{ minutes}}{25 \text{ minutes}} = 2 \]

### Step 3: Determine the Remaining Amount
After each half-life, the amount of the substance remaining can be calculated using the formula:
\[ \text{Remaining Amount} = \frac{1}{2^n} \times A_0 \]
where \( n \) is the number of half-lives and \( A_0 \) is the initial amount.

For \( n = 2 \):
\[ \text{Remaining Amount} = \frac{1}{2^2} \times A_0 = \frac{1}{4} \times A_0 \]

### Step 4: Conclusion
Thus, the amount of radioactive iodine left after 50 minutes is \( \frac{A_0}{4} \), which means that only one-fourth of the initial amount remains.

### Final Answer
The amount of radioactive \( _{52}^{123}I \) left after 50 minutes will be \( \frac{A_0}{4} \).

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