A radio isotope `X` with a half life `1.4xx10^(9)` yr decays of `Y` which is stable. A sample of the rock from a cave was found to contain `X` and `Y` in the ratio `1:7`. The age of the rock is

Options

A. `1.96xx xx10^(9) yr`0
B. `3.92xx xx10^(9) yr`
C. `4.20xx xx10^(9) yr`
D. `8.40xx10^(9)` yr

Explanation

To determine the age of the rock sample containing the radioisotope X and the stable isotope Y, we will follow these steps:

### Step 1: Understand the decay process
The radioisotope X decays into the stable isotope Y. Given that the half-life of X is \(1.4 \times 10^9\) years, we know that after each half-life, half of the remaining X will decay into Y.

### Step 2: Set up the initial conditions
Let the initial amount of isotope X be \(N_0\). After some time, a certain amount of X will have decayed into Y. The problem states that the current ratio of X to Y is \(1:7\). This means that for every 1 part of X, there are 7 parts of Y.

### Step 3: Express the current amounts of X and Y
If we denote the amount of X remaining as \(N\) and the amount of Y formed as \(M\), we can express the relationship as follows:
- \(N : M = 1 : 7\)

From this ratio, we can express \(M\) in terms of \(N\):
- \(M = 7N\)

### Step 4: Relate the amounts of X and Y
Since Y is formed from the decay of X, the total amount of Y formed will be equal to the initial amount of X minus the remaining amount of X:
- \(M = N_0 - N\)

Substituting \(M = 7N\) into the equation gives:
- \(7N = N_0 - N\)

### Step 5: Solve for N
Rearranging the equation:
- \(N_0 = 8N\)

This indicates that the initial amount of X was 8 times the current amount of X.

### Step 6: Determine the number of half-lives
The decay of X follows the formula:
- \(N = N_0 \left(\frac{1}{2}\right)^n\)

Where \(n\) is the number of half-lives that have passed. Since we found that \(N_0 = 8N\), we can substitute:
- \(N = N_0 \left(\frac{1}{2}\right)^n\)
- \(N_0 = 8N\)

Substituting \(N\) gives:
- \(N_0 = 8 \left(N_0 \left(\frac{1}{2}\right)^n\right)\)

This simplifies to:
- \(1 = 8 \left(\frac{1}{2}\right)^n\)

### Step 7: Solve for n
Rearranging gives:
- \(\left(\frac{1}{2}\right)^n = \frac{1}{8}\)

Recognizing that \(\frac{1}{8} = \left(\frac{1}{2}\right)^3\), we find:
- \(n = 3\)

This means 3 half-lives have passed.

### Step 8: Calculate the age of the rock
The age of the rock can be calculated using the number of half-lives and the half-life of X:
- Age = \(n \times \text{half-life} = 3 \times (1.4 \times 10^9 \text{ years})\)

Calculating this gives:
- Age = \(4.2 \times 10^9 \text{ years}\)

### Final Answer
The age of the rock is \(4.2 \times 10^9\) years.

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