Calculate the amount of `.53^( I^(128))(t_(1//2)=25 min)` left after 75 minutes.

Options

A. `1/4`
B. `1/6`
C. `1/8`
D. `1/9`

Answer

To calculate the amount of I-128 left after 75 minutes, we will use the concept of half-lives in first-order kinetics.

### Step-by-Step Solution:

1. **Identify the Half-Life (t₁/₂)**:
The half-life (t₁/₂) of I-128 is given as 25 minutes.

2. **Determine the Total Time (t)**:
We need to calculate the amount left after 75 minutes.

3. **Calculate the Number of Half-Lives (n)**:
To find out how many half-lives fit into 75 minutes, we use the formula:
\[ n = \frac{t}{t_{1/2}} = \frac{75 \text{ minutes}}{25 \text{ minutes}} = 3 \]
This means that 75 minutes is equivalent to 3 half-lives.

4. **Use the Half-Life Formula**:
The amount remaining after n half-lives can be calculated using the formula:
\[ C_t = C_0 \left(\frac{1}{2}\right)^n \]
where \(C_0\) is the initial amount and \(C_t\) is the amount remaining after time t.

5. **Substituting Values**:
Since we are not given an initial amount, we can assume \(C_0 = 1\) for simplicity. Thus:
\[ C_t = 1 \left(\frac{1}{2}\right)^3 = 1 \times \frac{1}{8} = \frac{1}{8} \]

6. **Conclusion**:
Therefore, after 75 minutes, the amount of I-128 left is \(\frac{1}{8}\) of the initial amount.

### Final Answer:
The amount of I-128 left after 75 minutes is \(\frac{1}{8}\) of the initial amount.

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