Half-life of a radioactive substance is `12.5 h` and its mass is `256 g`. After what time the amount of remaining substance is `1g`?

Options

A. `75 h`
B. `100 h`
C. `125 h`
D. `150 h`

Answer

To solve the problem step by step, we can follow these guidelines:

### Step 1: Identify the given values
- Initial mass (M₀) = 256 g
- Final mass (M) = 1 g
- Half-life (t₁/₂) = 12.5 h

### Step 2: Use the formula for radioactive decay
The formula for the remaining mass of a radioactive substance after a certain number of half-lives is given by:
\[ M = M_0 \times \left(\frac{1}{2}\right)^n \]
where \( n \) is the number of half-lives.

### Step 3: Rearrange the formula to find n
We can rearrange the formula to solve for \( n \):
\[ n = \log_2\left(\frac{M_0}{M}\right) \]

### Step 4: Substitute the known values into the equation
Substituting the values we have:
\[ n = \log_2\left(\frac{256 \text{ g}}{1 \text{ g}}\right) \]
\[ n = \log_2(256) \]

### Step 5: Calculate \( n \)
Since \( 256 = 2^8 \):
\[ n = 8 \]

### Step 6: Calculate the total time
The total time (T) can be calculated using the formula:
\[ T = n \times t_{1/2} \]
Substituting the values:
\[ T = 8 \times 12.5 \text{ h} \]
\[ T = 100 \text{ h} \]

### Final Answer
The time after which the amount of remaining substance is 1 g is **100 hours**.

---