Balmer gave an equation for wavelength of visible region of H-spectrum as `lambda=(Kn^(2))/(n^(2)-4)`.
Where n= principal quantum number of energy level, K=constant in terms of R (Rydberg constant).
The value of K in term of R is :

Options

A. (a)`R`
B. (b) `R/2`
C. (c) `4/R`
D. (d) `5/R`

Answer

To find the value of K in terms of the Rydberg constant R from the given equation for the wavelength of the visible region of the hydrogen spectrum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The equation provided is:
\[ \lambda = \frac{K n^2}{n^2 - 4} \]
Here, \( n \) is the principal quantum number, and \( K \) is a constant we need to express in terms of the Rydberg constant \( R \).

2. **Identify the Balmer Series**:
In the Balmer series, the lower energy level \( n_1 \) is 2. The upper energy level \( n_2 \) can take values greater than 2 (i.e., \( n_2 = 3, 4, 5, \ldots \)).

3. **Use the Rydberg Formula**:
The Rydberg formula for the wavelength is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For the Balmer series, substituting \( n_1 = 2 \):
\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{n_2^2} \right) = R \left( \frac{1}{4} - \frac{1}{n_2^2} \right) \]

4. **Rearranging the Formula**:
This can be rewritten as:
\[ \frac{1}{\lambda} = R \left( \frac{n_2^2 - 4}{4n_2^2} \right) \]
Hence, we can express \( \lambda \) as:
\[ \lambda = \frac{4 n_2^2}{R(n_2^2 - 4)} \]

5. **Comparing with the Given Equation**:
Now, we have two expressions for \( \lambda \):
- From the Rydberg formula:
\[ \lambda = \frac{4 n_2^2}{R(n_2^2 - 4)} \]
- From the given equation:
\[ \lambda = \frac{K n_2^2}{n_2^2 - 4} \]

6. **Equating the Two Expressions**:
Set the two expressions for \( \lambda \) equal to each other:
\[ \frac{4 n_2^2}{R(n_2^2 - 4)} = \frac{K n_2^2}{n_2^2 - 4} \]

7. **Solving for K**:
Cancel \( n_2^2 \) from both sides (assuming \( n_2 \neq 0 \)):
\[ \frac{4}{R} = K \]
Therefore, we find:
\[ K = \frac{4}{R} \]

### Final Answer:
The value of \( K \) in terms of the Rydberg constant \( R \) is:
\[ K = \frac{4}{R} \]