The half-life of a radioactive nuclide is 20 hours. What fraction of original activity will remain after 40 hours?

Options

A. `1/4`
B. `1/3`
C. `1/2`
D. `1/6`

Explanation

To solve the problem of determining what fraction of the original activity of a radioactive nuclide remains after 40 hours, given that its half-life is 20 hours, we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. After one half-life, the activity (or amount) of the substance is reduced to half of its original value.

### Step 2: Determine the number of half-lives that have passed
To find out how many half-lives have elapsed in 40 hours, we can use the formula:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} \]
Substituting the values we have:
\[ \text{Number of half-lives} = \frac{40 \text{ hours}}{20 \text{ hours}} = 2 \]

### Step 3: Calculate the remaining fraction of activity
After each half-life, the remaining activity is halved. Therefore, after 2 half-lives, the remaining activity can be calculated as follows:
\[ \text{Remaining Activity} = \left(\frac{1}{2}\right)^{\text{Number of half-lives}} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 4: Conclusion
Thus, after 40 hours, the fraction of the original activity that remains is \(\frac{1}{4}\).

### Final Answer:
The fraction of original activity that will remain after 40 hours is \(\frac{1}{4}\).

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