Half life of a radioactive elemets is 12.5 hour and its quantity is 256 gm. After how much time is quantity will remain 1 gm ?

Options

A. 50 Hrs
B. 100 Hrs
C. 150 Hrs
D. 200 Hrs

Answer

To solve the problem, we need to determine how many half-lives it takes for a quantity of a radioactive element to decay from 256 grams to 1 gram, given that the half-life of the element is 12.5 hours.

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life of a radioactive substance is the time required for half of the substance to decay. After one half-life, half of the original quantity remains.

2. **Set Up the Equation**:
We start with the formula for the remaining quantity after a certain number of half-lives:
\[ \text{Remaining Quantity} = \frac{\text{Original Quantity}}{2^n} \]
where \( n \) is the number of half-lives.

3. **Substitute the Known Values**:
We know the original quantity is 256 grams and we want to find out when it will be 1 gram:
\[ 1 = \frac{256}{2^n} \]

4. **Rearranging the Equation**:
Multiply both sides by \( 2^n \):
\[ 2^n = 256 \]

5. **Express 256 as a Power of 2**:
We know that:
\[ 256 = 2^8 \]
Therefore, we can equate:
\[ 2^n = 2^8 \]

6. **Solve for \( n \)**:
Since the bases are the same, we can set the exponents equal to each other:
\[ n = 8 \]

7. **Calculate the Total Time**:
Now that we know it takes 8 half-lives to go from 256 grams to 1 gram, we can calculate the total time:
\[ \text{Total Time} = n \times \text{Half-Life} = 8 \times 12.5 \text{ hours} \]
\[ \text{Total Time} = 100 \text{ hours} \]

### Final Answer:
The quantity will remain 1 gram after **100 hours**.

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