JEE Main 2025 — Structure Of Atom Question with Solution

To determine which spectral line of the hydrogen atom suits the wavelength for heat treatment of muscular pain (900 nm), we begin by applying the Rydberg equation:

$\frac{1}{\lambda }={\mathrm{R}}_{\mathrm{H}}{\mathrm{Z}}^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

Given:

$\lambda =900\text{ nm}=9\times 10^{-5}\text{ cm}$

${\mathrm{R}}_{\mathrm{H}}=10^5{\text{ cm}}^{-1}$

Hydrogen atom (Z = 1)

We proceed by solving the equation:

$\frac{1}{\lambda \times {\mathrm{R}}_{\mathrm{H}}}=\frac{1}{n_1^2}-\frac{1}{n_2^2}$

Substituting the known values:

$\frac{1}{9\times 10^{-5}\text{ cm}\times 10^5{\text{ cm}}^{-1}}=\frac{1}{n_1^2}-\frac{1}{n_2^2}$

This simplifies to:

$\frac{1}{9}=\frac{1}{n_1^2}-\frac{1}{n_2^2}$

This condition is satisfied when $n_1=3$ and $n_2=\infty$.

Therefore, the appropriate spectral transition is from $n_2=\infty$ to $n_1=3$, which corresponds to the Paschen series transition $\infty \to 3$.