The extreme wavelengths of Paschen series are

Options

A. `0.365 mum` and 0.565 mum`
B. `0.818 mum and 1.89 mum`
C. `1.45 mum and 4.04 mum`
D. `2.27 mum and 7.43 mum`

Answer

To find the extreme wavelengths of the Paschen series in the hydrogen atom, we will use the Rydberg formula:

\[ \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

where:
- \( \lambda \) is the wavelength,
- \( R \) is the Rydberg constant (\( R \approx 1.1 \times 10^7 \, \text{m}^{-1} \)),
- \( n_1 \) 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 \( n_1 \) and \( n_2 \) for the Paschen series
For the Paschen series, the lower energy level corresponds to \( n_1 = 3 \). The higher energy level \( n_2 \) can take values starting from 4 and can go to infinity (4, 5, 6, ...).

### Step 2: Calculate the maximum wavelength (\( \lambda_{\text{max}} \))
The maximum wavelength occurs when \( n_2 \) is at its lowest value, which is 4.

Using the Rydberg formula:
\[ \frac{1}{\lambda_{\text{max}}} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \]

Calculating the right side:
\[ \frac{1}{\lambda_{\text{max}}} = R \left( \frac{1}{9} - \frac{1}{16} \right) \]

Finding a common denominator (144):
\[ \frac{1}{\lambda_{\text{max}}} = R \left( \frac{16 - 9}{144} \right) = R \left( \frac{7}{144} \right) \]

Substituting \( R \):
\[ \frac{1}{\lambda_{\text{max}}} = 1.1 \times 10^7 \left( \frac{7}{144} \right) \]

Calculating \( \lambda_{\text{max}} \):
\[ \lambda_{\text{max}} = \frac{144}{7R} = \frac{144}{7 \times 1.1 \times 10^7} \]
\[ \lambda_{\text{max}} \approx 1.89 \times 10^{-6} \, \text{m} \, \text{or} \, 1.89 \, \mu m \]

### Step 3: Calculate the minimum wavelength (\( \lambda_{\text{min}} \))
The minimum wavelength occurs when \( n_2 \) approaches infinity.

Using the Rydberg formula:
\[ \frac{1}{\lambda_{\text{min}}} = R \left( \frac{1}{3^2} - 0 \right) = R \left( \frac{1}{9} \right) \]

Substituting \( R \):
\[ \frac{1}{\lambda_{\text{min}}} = 1.1 \times 10^7 \left( \frac{1}{9} \right) \]

Calculating \( \lambda_{\text{min}} \):
\[ \lambda_{\text{min}} = \frac{9}{1.1 \times 10^7} \]
\[ \lambda_{\text{min}} \approx 0.818 \times 10^{-6} \, \text{m} \, \text{or} \, 0.818 \, \mu m \]

### Final Answers
- Maximum Wavelength (\( \lambda_{\text{max}} \)): \( 1.89 \, \mu m \)
- Minimum Wavelength (\( \lambda_{\text{min}} \)): \( 0.818 \, \mu m \)