Question

Heat treatment of muscular pain involves radiation of wavelength of about 900 nm. Which spectral line of H atom is suitable for this? Given: Rydberg constant \( R_H = 10^5 \, \text{cm}^{-1} \), \( h = 6.6 \times 10^{-34} \, \text{J s} \), and \( c = 3 \times 10^8 \, \text{m/s} \)

• A. Paschen series, \( \infty \to 3 \)
• B. Lyman series, \( \infty \to 1 \)
• C. Balmer series, \( \infty \to 2 \)
• D. Paschen series, 5 \( \to \) 3

Hint:

For spectral lines in hydrogen atoms, use the Rydberg formula and identify the correct \( n_1 \) and \( n_2 \) values based on the wavelength to determine the suitable series.

Answer 1

The Correct Option is A

We are given: \[ \lambda = 900 \, \text{nm} = 9 \times 10^{-5} \, \text{cm}, \quad R_H = 10^5 \, \text{cm}^{-1}, \quad Z = 1 \, (\text{for H-atom}) \] We use the Rydberg formula to determine the suitable spectral line: \[ \frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] Substitute the given values into the equation: \[ \frac{1}{9 \times 10^{-5} \, \text{cm} \times 10^5 \, \text{cm}^{-1}} = \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] Simplifying: \[ \frac{1}{9} = \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] It is possible when \( n_1 = 3 \) and \( n_2 = \infty \). This corresponds to the Paschen series with \( \infty \to 3 \).
Thus, the suitable spectral line is the Paschen series, and the correct answer is option (1).