The half-life of a radioactive isotope `X` is `50` yr. It decays to an other element `Y` which is stable. The two elements `X` and `Y` were found to be in the ratio of `1 : 15` in a sample of a give rock. The age of the rock was estimated to be

Options

A. `200 yr`
B. `250 yr`
C. `100 yr`
D. `150 yr`

Answer

To determine the age of the rock based on the decay of radioactive isotope `X` to stable element `Y`, we can follow these steps:

### Step 1: Understand the Given Information
- The half-life of isotope `X` is `50 years`.
- The ratio of `X` to `Y` in the rock sample is `1:15`.

### Step 2: Set Up the Relationship Between `X` and `Y`
Let the number of nuclei of `X` be `n_X` and the number of nuclei of `Y` be `n_Y`. According to the given ratio:
\[ n_Y = 15 \cdot n_X \]

### Step 3: Determine the Total Initial Nuclei
The total number of initial nuclei (before decay) can be expressed as:
\[ n_0 = n_X + n_Y = n_X + 15n_X = 16n_X \]

### Step 4: Use the Radioactive Decay Formula
The number of remaining nuclei of `X` after time `t` can be expressed using the radioactive decay formula:
\[ n_X = n_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
Where:
- \( n_0 \) is the initial amount of `X`.
- \( T_{1/2} \) is the half-life of `X` (50 years).
- \( t \) is the age of the rock.

### Step 5: Substitute the Expression for `n_0`
Substituting \( n_0 = 16n_X \) into the decay formula gives:
\[ n_X = 16n_X \cdot \left(\frac{1}{2}\right)^{\frac{t}{50}} \]

### Step 6: Simplify the Equation
Dividing both sides by \( n_X \) (assuming \( n_X \neq 0 \)):
\[ 1 = 16 \cdot \left(\frac{1}{2}\right)^{\frac{t}{50}} \]

### Step 7: Rearranging the Equation
This can be rearranged to:
\[ \left(\frac{1}{2}\right)^{\frac{t}{50}} = \frac{1}{16} \]

### Step 8: Express \( \frac{1}{16} \) as a Power of \( \frac{1}{2} \)
Since \( \frac{1}{16} = \left(\frac{1}{2}\right)^4 \), we can write:
\[ \left(\frac{1}{2}\right)^{\frac{t}{50}} = \left(\frac{1}{2}\right)^4 \]

### Step 9: Equate the Exponents
From the equality of the bases, we can equate the exponents:
\[ \frac{t}{50} = 4 \]

### Step 10: Solve for \( t \)
Multiplying both sides by 50 gives:
\[ t = 4 \cdot 50 = 200 \text{ years} \]

### Conclusion
The age of the rock is estimated to be **200 years**.

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