The frequency for a series limit of Balmer and paschen serial respectively are `f_1and f_3` if the frequency of the first line of Balmer series is then the relation between `f _1,f_2and f_3` is

Options

A. `f_1-f_2=f_3`
B. `f_1+f_3=f_2`
C. `f_1+f_2=f_3`
D. `f_2-f_3=2f_1`

Answer

To solve the problem regarding the relationship between the frequencies of the Balmer and Paschen series, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Series Limits:**
- The series limit for the Balmer series occurs when an electron transitions from an infinite energy level to \( n = 2 \).
- The series limit for the Paschen series occurs when an electron transitions from an infinite energy level to \( n = 3 \).

2. **Using the Rydberg Formula:**
- The Rydberg formula for the wavelength \( \lambda \) is given by:
\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]
- Here, \( R \) is the Rydberg constant, \( n_f \) is the final energy level, and \( n_i \) is the initial energy level.

3. **Calculating Frequencies:**
- For the Balmer series (where \( n_f = 2 \)):
\[ f_1 = R c \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = R c \left( \frac{1}{4} - 0 \right) = \frac{R c}{4} \]
- For the Paschen series (where \( n_f = 3 \)):
\[ f_3 = R c \left( \frac{1}{3^2} - \frac{1}{\infty^2} \right) = R c \left( \frac{1}{9} - 0 \right) = \frac{R c}{9} \]

4. **Finding the Frequency of the First Line of Balmer Series:**
- The first line of the Balmer series corresponds to the transition from \( n = 3 \) to \( n = 2 \):
\[ f_2 = R c \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R c \left( \frac{1}{4} - \frac{1}{9} \right) \]
- To simplify \( f_2 \):
\[ f_2 = R c \left( \frac{9 - 4}{36} \right) = R c \left( \frac{5}{36} \right) \]

5. **Establishing the Relationship:**
- Now we have:
- \( f_1 = \frac{R c}{4} \)
- \( f_3 = \frac{R c}{9} \)
- \( f_2 = \frac{5 R c}{36} \)
- We can relate these frequencies:
\[ f_1 = f_2 + f_3 \]
- Rearranging gives:
\[ f_1 - f_2 = f_3 \]

### Final Relation:
Thus, the relation between \( f_1 \), \( f_2 \), and \( f_3 \) is:
\[ f_1 - f_2 = f_3 \]