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 \(.^{63}Cu\), we can use the average atomic weight of copper and the known weights of its isotopes. Here’s a step-by-step solution:

### Step 1: Define Variables
Let:
- \( x \) = natural abundance of \(.^{63}Cu\) (in percentage)
- \( 100 - x \) = natural abundance of \(.^{65}Cu\) (in percentage)

### Step 2: Write the Average Atomic Weight Equation
The average atomic weight of copper can be expressed as:
\[ \text{Average atomic weight} = \left(\frac{x}{100} \times 63\right) + \left(\frac{100 - x}{100} \times 65\right) \]
Given that the average atomic weight of copper is \( 63.546 \), we can set up the equation:
\[ 63.546 = \left(\frac{x}{100} \times 63\right) + \left(\frac{100 - x}{100} \times 65\right) \]

### Step 3: Simplify the Equation
Multiply through by 100 to eliminate the fraction:
\[ 6354.6 = 63x + 65(100 - x) \]
Expanding the right side:
\[ 6354.6 = 63x + 6500 - 65x \]
Combine like terms:
\[ 6354.6 = -2x + 6500 \]

### Step 4: Solve for \( x \)
Rearranging gives:
\[ -2x = 6354.6 - 6500 \]
\[ -2x = -145.4 \]
Dividing both sides by -2:
\[ x = \frac{145.4}{2} = 72.7 \]

### Step 5: Conclusion
The natural abundance of the \(.^{63}Cu\) isotope is approximately \( 72.7\% \).