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((5)/(36))`
D. `sqrt((36)/(5))`

Explanation

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

### Step-by-Step Solution:

1. **Understand Moseley's Law**:
Moseley's law states that the square root of the frequency (\( \sqrt{f} \)) of the emitted X-ray is proportional to the atomic number (\( Z \)) of the element. The general formula is given by:
\[ \sqrt{f} = \sqrt{R} \cdot (Z - 1) \cdot \sqrt{\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)} \]
where \( R \) is a constant, and \( n_1 \) and \( n_2 \) are principal quantum numbers.

2. **Identify Quantum Numbers for \( K_\beta \) and \( K_\alpha \)**:
- For \( K_\beta \): \( n_1 = 1 \), \( n_2 = 3 \)
- For \( K_\alpha \): \( n_1 = 1 \), \( n_2 = 2 \)

3. **Write the Equation for \( K_\beta \)**:
\[ \sqrt{f_\beta} = \sqrt{R} \cdot (Z - 1) \cdot \sqrt{\left(\frac{1}{1^2} - \frac{1}{3^2}\right)} \]
Calculate the term:
\[ \frac{1}{1^2} - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9} \]
Thus,
\[ \sqrt{f_\beta} = \sqrt{R} \cdot (Z - 1) \cdot \sqrt{\frac{8}{9}} = \sqrt{R} \cdot (Z - 1) \cdot \frac{2\sqrt{2}}{3} \]

4. **Write the Equation for \( K_\alpha \)**:
\[ \sqrt{f_\alpha} = \sqrt{R} \cdot (Z - 1) \cdot \sqrt{\left(\frac{1}{1^2} - \frac{1}{2^2}\right)} \]
Calculate the term:
\[ \frac{1}{1^2} - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} \]
Thus,
\[ \sqrt{f_\alpha} = \sqrt{R} \cdot (Z - 1) \cdot \sqrt{\frac{3}{4}} = \sqrt{R} \cdot (Z - 1) \cdot \frac{\sqrt{3}}{2} \]

5. **Find the Ratio \( \frac{\sqrt{f_\beta}}{\sqrt{f_\alpha}} \)**:
\[ \frac{\sqrt{f_\beta}}{\sqrt{f_\alpha}} = \frac{\sqrt{R} \cdot (Z - 1) \cdot \frac{2\sqrt{2}}{3}}{\sqrt{R} \cdot (Z - 1) \cdot \frac{\sqrt{3}}{2}} \]
The common terms \( \sqrt{R} \) and \( (Z - 1) \) cancel out:
\[ = \frac{2\sqrt{2}}{3} \cdot \frac{2}{\sqrt{3}} = \frac{4\sqrt{2}}{3\sqrt{3}} \]

6. **Simplify the Ratio**:
\[ = \frac{4\sqrt{2}}{3\sqrt{3}} = \frac{4\sqrt{6}}{9} \]

### Final Answer:
The ratio of the slope of the graph between \( \sqrt{f} \) and \( Z \) for \( K_\beta \) and \( K_\alpha \) is:
\[ \frac{4\sqrt{6}}{9} \]