The frequency of `H_(beta)` line of lyman seris of hydrogen is

Options

A. `2.92 xx10^(15)` Hz
B. `3.07 xx10^(15)` Hz
C. `1.02 xx10^(7)` Hz
D. `9.76 xx10^(6)` Hz

Answer

To find the frequency of the H-beta line of the Lyman series of hydrogen, we can follow these steps:

### Step 1: Understand the Lyman Series
The Lyman series involves electronic transitions in hydrogen where the electron falls to the n=1 energy level from higher energy levels (n=2, 3, 4, ...). The H-beta line corresponds to the transition from n=3 to n=1.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength (λ) of the emitted light is given by:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( R_H \) is the Rydberg constant for hydrogen, \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

### Step 3: Assign Values
For the H-beta line:
- \( n_1 = 1 \) (lower level)
- \( n_2 = 3 \) (higher level)
- The Rydberg constant \( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \)

### Step 4: Calculate the Wavelength
Substituting the values into the Rydberg formula:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \]
Calculating the right side:
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \left( 1 - \frac{1}{9} \right) = 1.097 \times 10^7 \left( \frac{8}{9} \right) \]
\[ \frac{1}{\lambda} = 1.097 \times 10^7 \times \frac{8}{9} = \frac{8.776 \times 10^6}{9} \approx 9.75 \times 10^6 \, \text{m}^{-1} \]

### Step 5: Convert Wavelength to Frequency
Using the relationship between speed of light (c), wavelength (λ), and frequency (ν):
\[ c = \lambda \cdot \nu \quad \Rightarrow \quad \nu = \frac{c}{\lambda} \]
Substituting \( c = 3 \times 10^8 \, \text{m/s} \):
\[ \nu = c \cdot \frac{1}{\frac{1}{\lambda}} = c \cdot \frac{1}{R_H \left( \frac{8}{9} \right)} \]
Calculating frequency:
\[ \nu = 3 \times 10^8 \cdot 1.097 \times 10^7 \cdot \frac{8}{9} \]
\[ \nu \approx 2.925 \times 10^{15} \, \text{Hz} \]

### Final Answer
The frequency of the H-beta line of the Lyman series of hydrogen is approximately:
\[ \nu \approx 2.925 \times 10^{15} \, \text{Hz} \]