The constant `'a'` Mosely's law `sqrt(v) = a(Z-b)` for `k_(alpha)X-`rays is related to Rydberg constant R as (`'c'` is the speed of light)

Options

A. `R = (4a^(2))/(3c)`
B. `R = (3c)/(4a)`
C. `R = (3c^(2))/(4)`
D. `R = (2c)/(a)`

Explanation

To solve the problem, we will derive the relationship between the constant 'a' in Moseley's law and the Rydberg constant 'R'.

### Step-by-Step Solution:

1. **Start with Moseley's Law**:
Moseley's law states that for K-alpha X-rays, the relationship can be expressed as:
\[ \sqrt{v} = a(Z - b) \]
where \(v\) is the frequency, \(Z\) is the atomic number, \(a\) is a constant, and \(b\) is a constant related to the screening effect.

2. **Square Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ v = a^2(Z - b)^2 \]

3. **Relate Frequency and Wavelength**:
We know that the frequency \(v\) and wavelength \(\lambda\) are related by the speed of light \(c\):
\[ v = \frac{c}{\lambda} \]
Therefore, we can substitute this into our equation:
\[ \frac{c}{\lambda} = a^2(Z - b)^2 \]

4. **Express Wavelength**:
Rearranging the equation gives us:
\[ \lambda = \frac{c}{a^2(Z - b)^2} \]

5. **Use the Rydberg Formula**:
The Rydberg formula for the wavelength of emitted radiation is given by:
\[ \frac{1}{\lambda} = RZ^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]
For K-alpha X-rays, we take \(n_1 = 1\) and \(n_2 = 2\):
\[ \frac{1}{\lambda} = RZ^2\left(1 - \frac{1}{4}\right) = RZ^2\left(\frac{3}{4}\right) \]

6. **Combine the Two Expressions**:
From the Rydberg formula, we have:
\[ \lambda = \frac{4}{3RZ^2} \]
Setting the two expressions for \(\lambda\) equal gives:
\[ \frac{c}{a^2(Z - b)^2} = \frac{4}{3RZ^2} \]

7. **Cross Multiply**:
Cross-multiplying leads to:
\[ 3RZ^2c = 4a^2(Z - b)^2 \]

8. **Solve for Rydberg Constant**:
Rearranging this equation to isolate \(R\):
\[ R = \frac{4a^2(Z - b)^2}{3cZ^2} \]

9. **Assuming b = 0**:
If we assume \(b = 0\) for simplification, we get:
\[ R = \frac{4a^2Z^2}{3cZ^2} = \frac{4a^2}{3c} \]

### Final Result:
Thus, the relationship between the constant 'a' in Moseley's law and the Rydberg constant \(R\) is:
\[ R = \frac{4a^2}{3c} \]