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

To determine the most suitable spectral line of the hydrogen atom for heat treatment at a wavelength of about 900 nm, we need to analyze the series and transitions available within the hydrogen emission spectrum and how they correspond to this wavelength.

The transition wavelength can be calculated using the formula:

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

where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant, and \( n_1 \) and \( n_2 \) are the principal quantum numbers with \( n_2 > n_1 \).

We are given that \( \lambda = 900 \, \text{nm} = 900 \times 10^{-9} \, \text{m} = 9000 \, \text{Å} \), and the Rydberg constant \( R_H = 10^5 \, \text{cm}^{-1} = 10^7 \, \text{m}^{-1} \).

Substituting the values into the formula:

\(\frac{1}{9000 \times 10^{-10}} = 10^7 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\)

Simplifying gives:

\(\frac{1}{9000} = \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\)

For the hydrogen atom, the spectral series have the following characteristics:

• Lyman Series: \( n_1 = 1 \) (UV region)
• Balmer Series: \( n_1 = 2 \) (Visible region)
• Paschen Series: \( n_1 = 3 \) (Infrared region)

The Paschen series transitions fit within the infrared region, typically including wavelengths near 900 nm. Hence, the transition \( \infty \to 3 \) in the Paschen series is likely to correspond to the mentioned wavelength of 900 nm.

For a spectral match near 900 nm, the most suitable transition is Paschen series, \( \infty \to 3 \), as it aligns well with the infrared wavelength provided in the question.

Answer 2

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, \( \infty \to 3 \)