Ratio of longest wavelengths corresponding to Lyman and Balmer series in hydrogen spectrum is

Options

A. `(5)/(27)`
B. `(3)/(23)`
C. `(7)/(29)`
D. `(9)/(31)`

Explanation

To find the ratio of the longest wavelengths corresponding to the Lyman and Balmer series in the hydrogen spectrum, we can follow these steps:

### Step 1: Understand the Rydberg Formula
The Rydberg formula for the wavelengths of spectral lines in hydrogen 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, \( n_1 \) is the principal quantum number of the lower energy level, and \( n_2 \) is the principal quantum number of the higher energy level.

### Step 2: Determine the Longest Wavelength for Lyman Series
For the Lyman series, the transition occurs to the ground state (n=1). The longest wavelength corresponds to the transition from \( n_2 = 2 \) to \( n_1 = 1 \):
\[ \frac{1}{\lambda_L} = R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R_H \left( 1 - \frac{1}{4} \right) = R_H \left( \frac{3}{4} \right) \]
Thus, the longest wavelength for the Lyman series is:
\[ \lambda_L = \frac{4}{3 R_H} \]

### Step 3: Determine the Longest Wavelength for Balmer Series
For the Balmer series, the transition occurs to the first excited state (n=2). The longest wavelength corresponds to the transition from \( n_2 = 3 \) to \( n_1 = 2 \):
\[ \frac{1}{\lambda_B} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{9} \right) = R_H \left( \frac{9 - 4}{36} \right) = R_H \left( \frac{5}{36} \right) \]
Thus, the longest wavelength for the Balmer series is:
\[ \lambda_B = \frac{36}{5 R_H} \]

### Step 4: Calculate the Ratio of Longest Wavelengths
Now, we can find the ratio of the longest wavelengths of the Lyman and Balmer series:
\[ \frac{\lambda_L}{\lambda_B} = \frac{\frac{4}{3 R_H}}{\frac{36}{5 R_H}} = \frac{4}{3} \cdot \frac{5}{36} = \frac{20}{108} = \frac{5}{27} \]

### Final Answer
The ratio of the longest wavelengths corresponding to the Lyman and Balmer series in the hydrogen spectrum is:
\[ \frac{5}{27} \]