The ratio of the shortest wavelength of two spectral series of hydrogen spectrum is found to be about 9. The spectral series are:

Options

A. Lymann and Paschen
B. Balmer and Brackett
C. Brackett and Pfund
D. Paschen and Pfund

Answer

To solve the problem of finding the ratio of the shortest wavelengths of two spectral series of the hydrogen spectrum, we will use 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:
- \( \lambda \) is the wavelength,
- \( R_H \) is the Rydberg constant,
- \( n_1 \) is the principal quantum number of the lower energy level,
- \( n_2 \) is the principal quantum number of the higher energy level.

### Step 1: Identify the series and their quantum numbers
The spectral series of hydrogen include:
- Lyman series: \( n_1 = 1 \)
- Balmer series: \( n_1 = 2 \)
- Paschen series: \( n_1 = 3 \)
- Brackett series: \( n_1 = 4 \)
- Pfund series: \( n_1 = 5 \)

For the shortest wavelength in each series, \( n_2 \) approaches infinity (\( n_2 \to \infty \)).

### Step 2: Calculate the shortest wavelengths for Lyman and Paschen series
1. **Lyman Series**:
- For Lyman, \( n_1 = 1 \) and \( n_2 = \infty \):
\[ \frac{1}{\lambda_L} = R_H \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R_H \left( 1 - 0 \right) = R_H \]
Thus, \( \lambda_L = \frac{1}{R_H} \).

2. **Paschen Series**:
- For Paschen, \( n_1 = 3 \) and \( n_2 = \infty \):
\[ \frac{1}{\lambda_P} = R_H \left( \frac{1}{3^2} - \frac{1}{\infty^2} \right) = R_H \left( \frac{1}{9} - 0 \right) = \frac{R_H}{9} \]
Thus, \( \lambda_P = \frac{9}{R_H} \).

### Step 3: Find the ratio of the wavelengths
Now, we can find the ratio of the shortest wavelengths of the Lyman and Paschen series:
\[ \frac{\lambda_L}{\lambda_P} = \frac{\frac{1}{R_H}}{\frac{9}{R_H}} = \frac{1}{9} \]

### Step 4: Invert the ratio to find the ratio of the wavelengths
Since the problem states the ratio of the shortest wavelengths, we take the reciprocal:
\[ \frac{\lambda_P}{\lambda_L} = 9 \]

### Conclusion
The ratio of the shortest wavelengths of the Lyman series to the Paschen series is 9. Therefore, the spectral series are Lyman and Paschen.