The wavelength `lambda` of the `K_(a)` line of characteristic `X` - ray spectra varies with atomic number approximately

Options

A. (Z)^-2
B. Z
C. 1/Z
D. `(1)/(Z)^(2)`

Answer

To solve the problem regarding the variation of the wavelength \( \lambda \) of the \( K_{\alpha} \) line of characteristic X-ray spectra with atomic number \( Z \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand Moseley's Law**:
Moseley's law states that the frequency \( \nu \) of the X-ray emission lines is related to the atomic number \( Z \) of the element. The relationship can be expressed as:
\[ \nu = a(Z - b)^2 \]
where \( a \) and \( b \) are constants.

2. **Express Frequency in Terms of Atomic Number**:
From the equation above, we can see that frequency \( \nu \) is proportional to \( (Z - b)^2 \). For large \( Z \), we can simplify this to:
\[ \nu \propto Z^2 \]
This means that the frequency increases with the square of the atomic number.

3. **Relate Frequency to Wavelength**:
The relationship between frequency \( \nu \) and wavelength \( \lambda \) is given by:
\[ \nu = \frac{c}{\lambda} \]
where \( c \) is the speed of light. This implies that frequency is inversely proportional to wavelength:
\[ \lambda \propto \frac{1}{\nu} \]

4. **Combine the Relationships**:
Since we have established that \( \nu \propto Z^2 \), we can substitute this into the wavelength relationship:
\[ \lambda \propto \frac{1}{Z^2} \]
This indicates that the wavelength \( \lambda \) is inversely proportional to the square of the atomic number \( Z \).

5. **Final Relation**:
Therefore, we can conclude that:
\[ \lambda \propto \frac{1}{Z^2} \]
or equivalently,
\[ \lambda \text{ is inversely proportional to } Z^2. \]

### Conclusion:
The correct relation for the variation of the wavelength \( \lambda \) of the \( K_{\alpha} \) line of characteristic X-ray spectra with atomic number \( Z \) is:
\[ \lambda \propto \frac{1}{Z^2} \]