The wavelength of `K_(alpha)` –line characteristic X–rays emitted by an element is 0.32Å. The wavelength of `K_(beta)`–line emitted by the same element will be :–

Options

A. 0.27Å
B. 0.32Å
C. 0.39Å
D. 0.2Å

Answer

To solve the problem of finding the wavelength of the K-beta line emitted by the same element given that the wavelength of the K-alpha line is 0.32 Å, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Wavelength Relation**:
The wavelength of X-rays can be related to the atomic number (Z) and the transition states. The formula for the wavelength (λ) of the characteristic X-rays is given by:
\[ \frac{1}{\lambda} = K \left( \frac{1}{\sqrt{n_i}} - \frac{1}{\sqrt{n_f}} \right)^2 \]
where \( n_i \) is the initial state and \( n_f \) is the final state.

2. **Identify the States for K-alpha and K-beta**:
- For K-alpha transition (L to K), the initial state \( n_i = 2 \) and the final state \( n_f = 1 \).
- For K-beta transition (M to K), the initial state \( n_i = 3 \) and the final state \( n_f = 1 \).

3. **Calculate the Wavelength for K-alpha**:
Given \( \lambda_{K\alpha} = 0.32 \, \text{Å} \), we can express this in the formula:
\[ \frac{1}{0.32} = K \left( \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}} \right)^2 \]
Simplifying this gives:
\[ \frac{1}{0.32} = K \left( 1 - \frac{1}{\sqrt{2}} \right)^2 \]

4. **Calculate the Wavelength for K-beta**:
For K-beta, we use:
\[ \frac{1}{\lambda_{K\beta}} = K \left( \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{3}} \right)^2 \]
This simplifies to:
\[ \frac{1}{\lambda_{K\beta}} = K \left( 1 - \frac{1}{\sqrt{3}} \right)^2 \]

5. **Setting Up the Ratios**:
We can set up a ratio using the two equations:
\[ \frac{\frac{1}{\lambda_{K\beta}}}{\frac{1}{\lambda_{K\alpha}}} = \frac{\left( 1 - \frac{1}{\sqrt{3}} \right)^2}{\left( 1 - \frac{1}{\sqrt{2}} \right)^2} \]

6. **Substituting Values**:
We know \( \lambda_{K\alpha} = 0.32 \, \text{Å} \), thus:
\[ \lambda_{K\beta} = \lambda_{K\alpha} \cdot \frac{\left( 1 - \frac{1}{\sqrt{2}} \right)^2}{\left( 1 - \frac{1}{\sqrt{3}} \right)^2} \]

7. **Calculating the Values**:
- Calculate \( 1 - \frac{1}{\sqrt{2}} \) and \( 1 - \frac{1}{\sqrt{3}} \).
- Substitute these values into the ratio and solve for \( \lambda_{K\beta} \).

8. **Final Calculation**:
After performing the calculations, we find:
\[ \lambda_{K\beta} = 0.27 \, \text{Å} \]

### Final Answer:
The wavelength of the K-beta line emitted by the same element is \( 0.27 \, \text{Å} \).