The atomic weight of `Cu` is `63.546`. There are only two naturally occurring isotopes of copper `.^(63)Cu` and `.^(65)Cu`. The natural abundance of the`.^(63)Cu` isotope must be approximately.

Options

A. `10%`
B. `30%`
C. `50%`
D. `72.7%`

Explanation

To find the natural abundance of the isotope \(\text{Cu}^{63}\), we can use the concept of weighted averages based on the atomic weights of the isotopes and their respective abundances.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Atomic weight of copper (\(Cu\)): \(63.546\)
- Isotopes:
- \(\text{Cu}^{63}\) with atomic weight \(63\)
- \(\text{Cu}^{65}\) with atomic weight \(65\)

2. **Define Variables:**
- Let \(x\) be the natural abundance (percentage) of \(\text{Cu}^{63}\).
- Therefore, the natural abundance of \(\text{Cu}^{65}\) will be \(100 - x\).

3. **Set Up the Equation:**
The average atomic weight can be calculated using the formula:
\[ \text{Average Atomic Weight} = \frac{(m_1 \cdot x) + (m_2 \cdot (100 - x))}{100} \]
Where:
- \(m_1 = 63\) (weight of \(\text{Cu}^{63}\))
- \(m_2 = 65\) (weight of \(\text{Cu}^{65}\))
- The average atomic weight is given as \(63.546\).

Plugging in the values, we get:
\[ 63.546 = \frac{(63 \cdot x) + (65 \cdot (100 - x))}{100} \]

4. **Multiply Through by 100:**
To eliminate the denominator, multiply both sides by 100:
\[ 6354.6 = 63x + 6500 - 65x \]

5. **Combine Like Terms:**
Rearranging the equation gives:
\[ 6354.6 = 6500 - 2x \]
\[ 2x = 6500 - 6354.6 \]
\[ 2x = 145.4 \]

6. **Solve for \(x\):**
Dividing both sides by 2:
\[ x = \frac{145.4}{2} = 72.7 \]

7. **Conclusion:**
The natural abundance of the isotope \(\text{Cu}^{63}\) is approximately \(72.7\%\).

### Final Answer:
The natural abundance of \(\text{Cu}^{63}\) is approximately \(72.7\%\).

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