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 begin with the formula for the inverse of the wavelength:

\( \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) \)

Given values: initial energy level \( n_i = 3 \), final level \( n_f = \infty \)

Substituting the values:

\( \frac{1}{\lambda} = 10^{-7} \times 1^2 \left( \frac{1}{3^2} - \frac{1}{\infty^2} \right) \) \( = \frac{10^{-7}}{9} \)

Thus, the wavelength is:

\( \lambda = 900 \, \text{nm} \)

Hence, the Correct Answer is (A): Paschen series, \( \infty \to 3 \)