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

Answer

To find the age of the rock based on the decay of the radioisotope \(X\) into the stable isotope \(Y\), we can follow these steps:

### Step 1: Understand the Initial Conditions
At time \(t = 0\), we assume that the initial amount of isotope \(X\) is \(N_0\) and the amount of isotope \(Y\) is \(0\).

### Step 2: Establish the Current Ratio
We are given that the current ratio of \(X\) to \(Y\) is \(1:7\). This means that for every part of \(X\), there are 7 parts of \(Y\).

### Step 3: Set Up the Equation
Let the remaining amount of \(X\) at time \(t\) be \(N_X\) and the amount of \(Y\) produced be \(N_Y\). From the ratio, we can express:
\[ \frac{N_X}{N_Y} = \frac{1}{7} \]
If we let \(N_Y = 7N_X\), then the total amount of \(X\) and \(Y\) can be expressed as:
\[ N_X + N_Y = N_0 \]
Substituting \(N_Y\) gives:
\[ N_X + 7N_X = N_0 \implies 8N_X = N_0 \implies N_X = \frac{N_0}{8} \]

### Step 4: Calculate the Remaining Amount of \(X\)
The remaining amount of \(X\) is:
\[ N_X = N_0 - N_Y = N_0 - 7N_X \implies N_X = N_0 - 7 \left(\frac{N_0}{8}\right) = N_0 - \frac{7N_0}{8} = \frac{1N_0}{8} \]

### Step 5: Determine the Number of Half-Lives
The amount of \(X\) remaining is \(\frac{N_0}{8}\). The decay of \(X\) can be described by the half-life formula:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
where \(n\) is the number of half-lives. Setting \(N = \frac{N_0}{8}\):
\[ \frac{N_0}{8} = N_0 \left(\frac{1}{2}\right)^n \]
Dividing both sides by \(N_0\) (assuming \(N_0 \neq 0\)):
\[ \frac{1}{8} = \left(\frac{1}{2}\right)^n \]
Since \(\frac{1}{8} = \left(\frac{1}{2}\right)^3\), we find that \(n = 3\). This means that 3 half-lives have passed.

### Step 6: Calculate the Age of the Rock
The half-life of \(X\) is given as \(1.4 \times 10^9\) years. Therefore, the age of the rock is:
\[ \text{Age} = n \times \text{Half-life} = 3 \times 1.4 \times 10^9 \text{ years} = 4.2 \times 10^9 \text{ years} \]

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

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