Which element has a ` K_(alpha)` line of wavlength 1.785Å ?

Options

A. Copper
B. cobalt
C. Sodium
D. Aluminium

Explanation

To determine which element has a K-alpha line with a wavelength of 1.785 Å, we can use the formula related to the wavelength of the emitted X-rays during electronic transitions in atoms. The K-alpha line corresponds to an electronic transition from the second energy level (n=2) to the first energy level (n=1).

### Step-by-step Solution:

1. **Understand the K-alpha Transition**:
The K-alpha transition involves an electron falling from the second energy level (n=2) to the first energy level (n=1).

2. **Use the Formula**:
The formula for the wavelength (λ) of the emitted X-ray is given by:
\[ \frac{1}{\lambda} = R \cdot (Z - 1)^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
where:
- \( R \) is the Rydberg constant, approximately \( 1.1 \times 10^7 \, \text{m}^{-1} \)
- \( Z \) is the atomic number of the element
- \( n_f \) is the final energy level (1 for K-alpha)
- \( n_i \) is the initial energy level (2 for K-alpha)

3. **Substitute the Known Values**:
We know:
- \( \lambda = 1.785 \, \text{Å} = 1.785 \times 10^{-10} \, \text{m} \)
- \( n_f = 1 \)
- \( n_i = 2 \)

Substituting these values into the formula:
\[ \frac{1}{1.785 \times 10^{-10}} = R \cdot (Z - 1)^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

4. **Calculate the Right Side**:
The term \( \frac{1}{1^2} - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} \).

Therefore, we can rewrite the equation as:
\[ \frac{1}{1.785 \times 10^{-10}} = R \cdot (Z - 1)^2 \cdot \frac{3}{4} \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ (Z - 1)^2 = \frac{4}{3} \cdot \frac{1}{R} \cdot \frac{1}{1.785 \times 10^{-10}} \]

6. **Substituting the Rydberg Constant**:
Now substituting \( R = 1.1 \times 10^7 \):
\[ (Z - 1)^2 = \frac{4}{3} \cdot \frac{1}{1.1 \times 10^7} \cdot \frac{1}{1.785 \times 10^{-10}} \]

7. **Calculating the Value**:
Performing the calculations:
\[ (Z - 1)^2 = \frac{4}{3} \cdot \frac{1}{1.1 \times 10^7} \cdot \frac{1}{1.785 \times 10^{-10}} \approx 26 \]

8. **Finding Z**:
Taking the square root gives:
\[ Z - 1 = 5 \quad \Rightarrow \quad Z = 6 \]
However, we need to check again as we have an approximation.

9. **Final Calculation**:
After recalculating correctly, we find:
\[ Z - 1 = 26 \quad \Rightarrow \quad Z = 27 \]

10. **Identify the Element**:
The atomic number \( Z = 27 \) corresponds to the element Cobalt (Co).

### Conclusion:
The element with a K-alpha line of wavelength 1.785 Å is **Cobalt (Co)**.