Atomic numbers of two element are 31 and 41. Find the ratio of the wavelength of the `K_(alpha)` are

Options

A. `4:3`
B. `9:16`
C. `16:27`
D. `16:9`

Answer

To find the ratio of the wavelengths of the K-alpha lines for two elements with atomic numbers 31 and 41, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Mosley's Law**:
According to Mosley's law, the frequency (ν) of the emitted X-ray is given by:
\[ \nu = A(Z - B)^2 \]
where \( Z \) is the atomic number, \( A \) and \( B \) are constants specific to the K-alpha line.

2. **Relating Frequency to Wavelength**:
The relationship between frequency (ν) and wavelength (λ) is given by:
\[ c = \nu \lambda \]
where \( c \) is the speed of light. Rearranging gives:
\[ \nu = \frac{c}{\lambda} \]

3. **Substituting for Wavelength**:
From the above, we can express wavelength in terms of atomic number:
\[ \lambda \propto \frac{1}{(Z - B)^2} \]
This means that the wavelength is inversely proportional to the square of \( (Z - B) \).

4. **Determining the Value of B for K-alpha**:
For K-alpha radiation, the value of \( B \) is typically taken as 1.

5. **Calculating Wavelengths for Each Element**:
- For element with atomic number \( Z_1 = 31 \):
\[ \lambda_1 \propto \frac{1}{(31 - 1)^2} = \frac{1}{30^2} \]
- For element with atomic number \( Z_2 = 41 \):
\[ \lambda_2 \propto \frac{1}{(41 - 1)^2} = \frac{1}{40^2} \]

6. **Finding the Ratio of Wavelengths**:
We need to find the ratio \( \frac{\lambda_1}{\lambda_2} \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{1}{30^2}}{\frac{1}{40^2}} = \frac{40^2}{30^2} = \frac{1600}{900} \]
Simplifying this gives:
\[ \frac{1600}{900} = \frac{16}{9} \]

7. **Final Result**:
Therefore, the ratio of the wavelengths of the K-alpha lines for the two elements is:
\[ \text{Ratio} = 16 : 9 \]