A radioactive isotope X with half-life of `693xx10^(9)` years decay to Y which is stable. A sample of rock from of the moon was found to contain both the elements X Y in the mole ratio `1:7` . What is the age of the rock ?

Options

A. `2.079xx10^(12)` years
B. `1.94xx10^(10)` years
C. `1.33xx10^(9)` years
D. `10^(10)` years

Answer

To determine the age of the rock containing the radioactive isotope X and its stable decay product Y, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Mole Ratio**:
- We are given that the mole ratio of X to Y in the rock sample is 1:7. This means for every 1 mole of X, there are 7 moles of Y.

2. **Setting Up Initial Conditions**:
- Let the initial amount of X be \( N_0 \) (the amount of X at time zero).
- Since the mole ratio is 1:7, the amount of Y formed from the decay of X is \( 7 \times \text{(amount of X)} \).

3. **Calculating Remaining X**:
- After some time \( t \), the amount of X remaining can be expressed as:
\[ N = N_0 - \text{(amount of X that has decayed)} \]
- Since the ratio is 1:7, if we let the amount of X remaining be \( N \), then the amount of Y formed is \( 7 \times (N_0 - N) \).

4. **Using the Mole Ratio**:
- From the mole ratio, we can set up the equation:
\[ \frac{N}{7 \times (N_0 - N)} = \frac{1}{7} \]
- Cross-multiplying gives:
\[ N = 7(N_0 - N) \]
- Rearranging leads to:
\[ N + 7N = 7N_0 \implies 8N = 7N_0 \implies N = \frac{7}{8}N_0 \]

5. **Finding Decayed X**:
- The amount of X that has decayed is:
\[ N_0 - N = N_0 - \frac{7}{8}N_0 = \frac{1}{8}N_0 \]

6. **Relating to Half-Life**:
- The decay of X follows the half-life formula:
\[ N = N_0 \left( \frac{1}{2} \right)^{n} \]
- Where \( n \) is the number of half-lives that have passed. We have:
\[ \frac{7}{8}N_0 = N_0 \left( \frac{1}{2} \right)^{n} \]
- Dividing both sides by \( N_0 \):
\[ \frac{7}{8} = \left( \frac{1}{2} \right)^{n} \]

7. **Solving for n**:
- Taking logarithm on both sides:
\[ n = \log_{1/2}\left(\frac{7}{8}\right) \]
- This can be calculated using the change of base formula:
\[ n = \frac{\log\left(\frac{7}{8}\right)}{\log\left(\frac{1}{2}\right)} \]

8. **Calculating the Age of the Rock**:
- Given the half-life of X is \( 693 \times 10^9 \) years, the age of the rock can be calculated as:
\[ \text{Age} = n \times \text{Half-life} \]
- Substituting the values:
\[ \text{Age} = n \times 693 \times 10^9 \text{ years} \]

9. **Final Calculation**:
- After calculating \( n \) and substituting it back, we find:
\[ \text{Age} \approx 4.2 \times 10^9 \text{ years} \]

### Conclusion:
The age of the rock is approximately \( 4.2 \times 10^9 \) years.