A radioactive isotope has a half-life of 27 days. Starting with 4g of the isotope, what will be mass remaining after 75 days

Options

A. 100g
B. 50g
C. 0.58 g
D. 1.58 g

Explanation

To solve the problem of determining the remaining mass of a radioactive isotope after a certain period, we can follow these steps:

### Step 1: Understand the Half-Life Concept
The half-life of a radioactive isotope is the time required for half of the isotope to decay. In this case, the half-life is given as 27 days.

### Step 2: Calculate the Number of Half-Lives
We need to find out how many half-lives fit into the total time of 75 days.
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{75 \text{ days}}{27 \text{ days}} \approx 2.78 \]

### Step 3: Determine the Remaining Mass
The remaining mass of a radioactive substance after a certain number of half-lives can be calculated using the formula:
\[ \text{Remaining mass} = \text{Initial mass} \times \left(\frac{1}{2}\right)^{n} \]
where \( n \) is the number of half-lives.

Substituting the values:
\[ \text{Remaining mass} = 4 \text{ g} \times \left(\frac{1}{2}\right)^{2.78} \]

### Step 4: Calculate \(\left(\frac{1}{2}\right)^{2.78}\)
Using a calculator:
\[ \left(\frac{1}{2}\right)^{2.78} \approx 0.174 \]

### Step 5: Calculate the Remaining Mass
Now, we can find the remaining mass:
\[ \text{Remaining mass} \approx 4 \text{ g} \times 0.174 \approx 0.696 \text{ g} \]

### Conclusion
After 75 days, approximately 0.696 grams of the radioactive isotope will remain.