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 identify the most appropriate hydrogen atom spectral line for heat treatment at approximately 900 nm, an analysis of hydrogen's emission spectrum series and transitions is required to determine their correspondence to this wavelength.

The formula for calculating transition wavelengths is:

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

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

Given are \( \lambda = 900 \, \text{nm} = 900 \times 10^{-9} \, \text{m} = 9000 \, \text{Å} \) and \( R_H = 10^5 \, \text{cm}^{-1} = 10^7 \, \text{m}^{-1} \).

Substituting these values into the formula yields:

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

Simplification results in:

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

Hydrogen atom spectral series are characterized as follows:

• 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 fall within the infrared spectrum, commonly encompassing wavelengths around 900 nm. Consequently, the transition from \( \infty \) to 3 within the Paschen series is likely to correspond to the specified 900 nm wavelength.

For a spectral match near 900 nm, the optimal transition identified is the Paschen series, \( \infty \to 3 \), due to its alignment with the infrared wavelength provided.