Assuming f to be the frequency of the electromagnetic wave corresponding to the first line in Balmer series, the frequency of the immediate next line is

Options

A. 0.5 f
B. 1.35 f
C. 2.05 f
D. 2.70 f

Answer

To solve the problem, we need to find the frequency of the immediate next line in the Balmer series after the first line. The Balmer series corresponds to transitions of electrons in a hydrogen atom from higher energy levels to the n=2 level.

### Step-by-Step Solution:

1. **Identify the first line of the Balmer series**:
The first line in the Balmer series corresponds to the transition from n=3 to n=2. The frequency (f) of this transition can be expressed using the Rydberg formula:
\[ f = R \cdot c \left( \frac{1}{n_2^2} - \frac{1}{n_1^2} \right) \]
where \( R \) is the Rydberg constant, \( c \) is the speed of light, \( n_2 = 2 \) (final level), and \( n_1 = 3 \) (initial level).

2. **Calculate the frequency for the first line**:
Substituting \( n_1 = 3 \) and \( n_2 = 2 \):
\[ f = R \cdot c \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \cdot c \left( \frac{1}{4} - \frac{1}{9} \right) \]
Finding a common denominator (36):
\[ f = R \cdot c \left( \frac{9}{36} - \frac{4}{36} \right) = R \cdot c \cdot \frac{5}{36} \]

3. **Express \( R \cdot c \)**:
From the equation above, we can express \( R \cdot c \):
\[ R \cdot c = \frac{36f}{5} \]

4. **Identify the next line in the Balmer series**:
The next line corresponds to the transition from n=4 to n=2. We will calculate the frequency for this transition:
\[ f' = R \cdot c \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = R \cdot c \left( \frac{1}{4} - \frac{1}{16} \right) \]
Finding a common denominator (16):
\[ f' = R \cdot c \left( \frac{4}{16} - \frac{1}{16} \right) = R \cdot c \cdot \frac{3}{16} \]

5. **Substituting \( R \cdot c \)**:
Now substitute \( R \cdot c = \frac{36f}{5} \) into the equation for \( f' \):
\[ f' = \frac{36f}{5} \cdot \frac{3}{16} = \frac{108f}{80} = \frac{27f}{20} \]

6. **Calculate the ratio of the new frequency to the original**:
To express this in terms of the original frequency \( f \):
\[ f' = 1.35f \]

### Final Answer:
The frequency of the immediate next line in the Balmer series is approximately \( 1.35f \).