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 solve the problem of calculating the amount of I-53 left after 75 minutes, we can use the radioactive decay law. Here's a step-by-step solution:

### Step 1: Understand the Given Information
- The half-life (t_half) of I-53 is given as 25 minutes.
- The total time elapsed (t) is 75 minutes.

### Step 2: Determine the Number of Half-Lives
To find out how many half-lives fit into the total time elapsed, we can use the formula:
\[ \text{Number of half-lives} = \frac{t}{t_{1/2}} \]
Substituting the values:
\[ \text{Number of half-lives} = \frac{75 \text{ minutes}}{25 \text{ minutes}} = 3 \]

### Step 3: Calculate the Remaining Quantity
The amount of substance remaining after a certain number of half-lives can be calculated using the formula:
\[ n = n_0 \left(\frac{1}{2}\right)^n \]
Where:
- \( n \) is the remaining quantity,
- \( n_0 \) is the initial quantity,
- \( n \) is the number of half-lives.

Since we want the ratio of the remaining quantity to the initial quantity, we can express it as:
\[ \frac{n}{n_0} = \left(\frac{1}{2}\right)^n \]
Substituting \( n = 3 \):
\[ \frac{n}{n_0} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

### Step 4: Conclusion
The amount of I-53 left after 75 minutes is \( \frac{1}{8} \) of the initial quantity.

### Final Answer
The amount of I-53 left after 75 minutes is \( \frac{1}{8} \).

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