According to Moseley's law, the ratio of the slope of graph between `sqrtf` and Z for `K_beta` and `K_alpha` is

Options

A. `sqrt((32)/(27))`
B. `sqrt((27)/(32))`
C. `sqrt((33)/(22))`
D. `sqrt((22)/(33))`

Explanation

To solve the problem regarding Moseley's law and the ratio of the slope of the graph between \( \sqrt{f} \) and \( Z \) for \( K_{\beta} \) and \( K_{\alpha} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Moseley's Law**: According to Moseley's law, the frequency \( f \) of the X-ray emitted by an atom can be expressed as:
\[ f \propto (Z - 1)^2 \]
where \( Z \) is the atomic number.

2. **Frequency Expressions for \( K_{\alpha} \) and \( K_{\beta} \)**:
- For \( K_{\alpha} \) transition (where \( n = 2 \)):
\[ f_{\alpha} = R \cdot (Z - 1)^2 \cdot \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \cdot (Z - 1)^2 \cdot \left( 1 - \frac{1}{4} \right) = R \cdot (Z - 1)^2 \cdot \frac{3}{4} \]
Thus, we can write:
\[ \sqrt{f_{\alpha}} = \sqrt{R} \cdot (Z - 1) \cdot \sqrt{\frac{3}{4}} \]

- For \( K_{\beta} \) transition (where \( n = 3 \)):
\[ f_{\beta} = R \cdot (Z - 1)^2 \cdot \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = R \cdot (Z - 1)^2 \cdot \left( 1 - \frac{1}{9} \right) = R \cdot (Z - 1)^2 \cdot \frac{8}{9} \]
Thus, we can write:
\[ \sqrt{f_{\beta}} = \sqrt{R} \cdot (Z - 1) \cdot \sqrt{\frac{8}{9}} \]

3. **Finding the Slopes**:
- The slope of the graph of \( \sqrt{f} \) vs \( Z \) for \( K_{\alpha} \) is:
\[ m_{\alpha} = \sqrt{R} \cdot \sqrt{\frac{3}{4}} \]
- The slope of the graph of \( \sqrt{f} \) vs \( Z \) for \( K_{\beta} \) is:
\[ m_{\beta} = \sqrt{R} \cdot \sqrt{\frac{8}{9}} \]

4. **Calculating the Ratio of Slopes**:
- The ratio of the slopes \( \frac{m_{\beta}}{m_{\alpha}} \) is:
\[ \frac{m_{\beta}}{m_{\alpha}} = \frac{\sqrt{R} \cdot \sqrt{\frac{8}{9}}}{\sqrt{R} \cdot \sqrt{\frac{3}{4}}} \]
- Simplifying this gives:
\[ = \frac{\sqrt{\frac{8}{9}}}{\sqrt{\frac{3}{4}}} = \frac{\sqrt{8} \cdot \sqrt{4}}{\sqrt{9} \cdot \sqrt{3}} = \frac{4\sqrt{2}}{3\sqrt{3}} = \frac{4\sqrt{2}}{3\sqrt{3}} \]

5. **Final Ratio**:
- The final ratio of the slopes is:
\[ \frac{m_{\beta}}{m_{\alpha}} = \frac{8}{9} \div \frac{3}{4} = \frac{8 \cdot 4}{9 \cdot 3} = \frac{32}{27} \]

### Conclusion:
The ratio of the slope of the graph between \( \sqrt{f} \) and \( Z \) for \( K_{\beta} \) and \( K_{\alpha} \) is \( \frac{32}{27} \).