The half life of a radioactive isotope is 3 hours. If the initial mass of the isotope were 256 g, the mass of it remaining undecayed after 18 hours would be:

Options

A. 12g
B. 16g
C. 4g
D. 8g

Explanation

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

### Step 1: Identify the given values
- Half-life of the isotope (t_half) = 3 hours
- Initial mass of the isotope (N0) = 256 g
- Total time elapsed (t) = 18 hours

### Step 2: Calculate the number of half-lives (n)
The number of half-lives can be calculated using the formula:
\[ n = \frac{t}{t_{half}} \]

Substituting the given values:
\[ n = \frac{18 \text{ hours}}{3 \text{ hours}} = 6 \]

### Step 3: Use the formula for remaining mass
The remaining mass (Nt) after n half-lives can be calculated using the formula:
\[ N_t = N_0 \left(\frac{1}{2}\right)^n \]

### Step 4: Substitute the values into the formula
Now, substituting the values we have:
\[ N_t = 256 \text{ g} \left(\frac{1}{2}\right)^6 \]

### Step 5: Calculate \( \left(\frac{1}{2}\right)^6 \)
Calculating \( \left(\frac{1}{2}\right)^6 \):
\[ \left(\frac{1}{2}\right)^6 = \frac{1}{64} \]

### Step 6: Calculate the remaining mass
Now substituting back into the equation:
\[ N_t = 256 \text{ g} \times \frac{1}{64} \]
\[ N_t = \frac{256}{64} \text{ g} \]
\[ N_t = 4 \text{ g} \]

### Final Answer
The mass of the radioactive isotope remaining undecayed after 18 hours is **4 g**.

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