The half life period of radium is 1600 years . In how many years 1 g radium will be reduced to `0.125` g ?

Options

A. 1600
B. 3200
C. 4800
D. 800

Explanation

To solve the problem of how long it takes for 1 g of radium to reduce to 0.125 g, we can use the concept of half-life.

### Step-by-step Solution:

1. **Understanding Half-Life**: The half-life of a substance is the time required for half of the substance to decay. For radium, the half-life is given as 1600 years.

2. **Determine the Number of Half-Lives**: We need to find out how many half-lives it takes for 1 g of radium to reduce to 0.125 g.
- Starting amount (n₀) = 1 g
- Remaining amount (n) = 0.125 g

We can express this relationship using the formula:
\[ n = n_0 \left(\frac{1}{2}\right)^n \]
Rearranging gives:
\[ \frac{n}{n_0} = \left(\frac{1}{2}\right)^n \]
Plugging in the values:
\[ \frac{0.125}{1} = \left(\frac{1}{2}\right)^n \]
This simplifies to:
\[ 0.125 = \left(\frac{1}{2}\right)^n \]

3. **Convert 0.125 to a Power of 1/2**:
- We know that \(0.125 = \frac{1}{8} = \frac{1}{2^3}\).
- Therefore, we can equate:
\[ \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 \]
This implies that:
\[ n = 3 \]

4. **Calculate Total Time**: Now that we know it takes 3 half-lives to reduce from 1 g to 0.125 g, we can calculate the total time.
- Total time \(T\) is given by:
\[ T = n \times t_{1/2} \]
Where \(t_{1/2} = 1600\) years (the half-life of radium).
- Thus:
\[ T = 3 \times 1600 = 4800 \text{ years} \]

### Final Answer:
The time taken for 1 g of radium to reduce to 0.125 g is **4800 years**.

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