Question

The first member of the Paschen series in the hydrogen spectrum is of wavelength \( 18800 \, \text{Å} \). What is the short wavelength limit of the Paschen series?

• A. \( 8225 \, \text{Å} \)
• B. \( 8220 \, \text{Å} \)
• C. \( 8300 \, \text{Å} \)
• D. None of these

Hint:

The Paschen series corresponds to transitions where electrons fall to the \( n = 3 \) energy level. The short wavelength limit occurs when \( n \to \infty \).

Answer 1

The Correct Option is A

Step 1: Recall the formula for the Paschen series.
The wavelength \( \lambda \) of a spectral line in the Paschen series is given by: \[ \frac{1}{\lambda} = R \left( \frac{1}{3^2} - \frac{1}{n^2} \right), \quad n = 4, 5, 6, \dots \] where: \begin{itemize} \( R \) is the Rydberg constant (\( 1.097 \times 10^7 \, \text{m}^{-1} \)), \( n \) is the principal quantum number (\( n > 3 \)). \end{itemize} Step 2: Calculate the wavelength for the first member (\( n = 4 \)).
For \( n = 4 \): \[ \frac{1}{\lambda_1} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right). \] Simplify: \[ \frac{1}{\lambda_1} = R \left( \frac{1}{9} - \frac{1}{16} \right). \] Find the common denominator: \[ \frac{1}{\lambda_1} = R \cdot \frac{16 - 9}{144} = R \cdot \frac{7}{144}. \] Given \( R = 1.097 \times 10^7 \, \text{m}^{-1} \), calculate: \[ \frac{1}{\lambda_1} = 1.097 \times 10^7 \cdot \frac{7}{144}. \] Simplify: \[ \frac{1}{\lambda_1} \approx 5.33 \times 10^5 \, \text{m}^{-1}. \] Convert to \( \lambda_1 \): \[ \lambda_1 = \frac{1}{5.33 \times 10^5} \approx 18800 \, \text{Å}. \] Step 3: Calculate the short wavelength limit (\( n = \infty \)).
For \( n = \infty \), the formula becomes: \[ \frac{1}{\lambda_{\text{min}}} = R \left( \frac{1}{3^2} \right). \] Substitute \( R = 1.097 \times 10^7 \, \text{m}^{-1} \): \[ \frac{1}{\lambda_{\text{min}}} = 1.097 \times 10^7 \cdot \frac{1}{9}. \] Simplify: \[ \frac{1}{\lambda_{\text{min}}} = \frac{1.097 \times 10^7}{9} \approx 1.22 \times 10^6 \, \text{m}^{-1}. \] Convert to \( \lambda_{\text{min}} \): \[ \lambda_{\text{min}} = \frac{1}{1.22 \times 10^6} \approx 8.225 \times 10^{-7} \, \text{m}. \] Convert to angstroms: \[ \lambda_{\text{min}} = 8225 \, \text{Å}. \] Thus, the short wavelength limit of the Paschen series is \( 8225 \, \text{Å} \).