Question

If the wavelength of Kα2 X-ray line of an element is 1.544 Å, then the atomic number \( Z \) of the element is ................

Hint:

To find the atomic number from the wavelength of X-ray lines, use the Rydberg formula and rearrange to solve for \( Z \).

Answer 1

Correct Answer: 29

Step 1: Use the Rydberg formula.
The wavelength \( \lambda \) of the Kα X-ray line is related to the atomic number \( Z \) by the following formula: \[ \lambda = \frac{R}{Z^2} \left( 1 - \frac{1}{n^2} \right) \] where \( R = 1.097 \times 10^7 \, \text{m}^{-1} \) is the Rydberg constant, and \( n \) is the principal quantum number (for the Kα line, \( n = 2 \) and \( n = 1 \)).
Step 2: Rearrange the formula.
Since the given wavelength corresponds to \( \lambda = 1.544 \, \text{Å} = 1.544 \times 10^{-10} \, \text{m} \), we can substitute the values and solve for \( Z \). Solving the equation for \( Z \), we get: \[ Z = \sqrt{\frac{R}{\lambda} \left( 1 - \frac{1}{2^2} \right)} \]
Step 3: Calculate the atomic number \( Z \).
Substitute the known values: \[ Z = \sqrt{\frac{1.097 \times 10^7}{1.544 \times 10^{-10}} \left( 1 - \frac{1}{4} \right)} \] \[ Z = \sqrt{\frac{1.097 \times 10^7}{1.544 \times 10^{-10}} \times \frac{3}{4}} \] \[ Z = 18 \]
Step 4: Conclusion.
Thus, the atomic number \( Z \) of the element is 18.