The half-life of radium is about `1600 yr`. Of `100g` of radium existing now, `25 g` will remain unchanged after

Options

A. `4800 yr`
B. `6400 yr`
C. `2400 yr`
D. `3200 yr`

Answer

To solve the problem, we need to determine how long it will take for 100 grams of radium to decay to 25 grams, given that the half-life of radium is 1600 years.

### Step-by-step Solution:

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

2. **Setting Up the Equation**:
We start with an initial amount \( N_0 = 100 \) grams of radium. We want to find out how many half-lives it takes for this amount to reduce to \( N = 25 \) grams.

3. **Using the Half-Life Formula**:
The relationship between the initial amount, the remaining amount, and the number of half-lives can be expressed as:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
where \( n \) is the number of half-lives.

4. **Substituting the Values**:
Substituting the known values into the equation:
\[ 25 = 100 \left( \frac{1}{2} \right)^n \]

5. **Simplifying the Equation**:
Dividing both sides by 100 gives:
\[ \frac{25}{100} = \left( \frac{1}{2} \right)^n \]
This simplifies to:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^n \]

6. **Recognizing Powers of 2**:
We know that \( \frac{1}{4} = \left( \frac{1}{2} \right)^2 \). Thus, we can equate the exponents:
\[ n = 2 \]

7. **Calculating the Total Time**:
Now that we know it takes 2 half-lives to go from 100 grams to 25 grams, we can calculate the total time taken:
\[ \text{Total Time} = n \times \text{Half-life} = 2 \times 1600 \text{ years} = 3200 \text{ years} \]

### Final Answer:
The time taken for 100 grams of radium to decay to 25 grams is **3200 years**.

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