Q.(a) A hydrogen atom is excited by radiation of wavelength 97.5 nm. Find the principal quantum number of the excited state.

When an individual gaseous element such as hydrogen is excited (by heating or an electrical discharge) and its emitted light is examined through a spectrometer, the result is not a smooth, continuous rainbow of colours but rather a set of sharp, discrete bright lines at very specific wavelengths - a line spectrum, unique to that particular element, functioning almost like a fingerprint that lets scientists identify a gas's composition from its light alone (including the composition of distant stars and the Sun).

For hydrogen specifically, every observed spectral-line wavelength is given by a single Rydberg-type formula,

where $R=1.09737\times10^7\ \text{m}^{-1}$ is the Rydberg constant. Fixing the lower level $n$ and letting $m$ range over every larger integer generates one complete named series of spectral lines: $n=1$ gives the Lyman series (ultraviolet region); $n=2$ gives the Balmer series (visible region - the only series bright enough to be seen directly with the eye); $n=3$ gives the Paschen series (near infrared); $n=4$ gives the Brackett series (mid infrared); and $n=5$ gives the Pfund series (far infrared). Within any one series, successive lines (as $m$ increases) crowd progressively closer together, approaching a limiting wavelength - the series limit - as $m\to\infty$.

This same formula also lets the total number of distinct spectral lines an excited hydrogen sample can emit be counted: if an electron has been excited up to level $n$, it can subsequently make a transition down to any lower level, and the total number of distinct transitions (and hence distinct emitted wavelengths) possible works out to $n(n-1)/2$.

Despite this considerable success in matching the observed hydrogen spectrum precisely, the underlying Bohr model that predicts these formulas has real limitations: it works only for hydrogen and other single-electron (hydrogen-like) ions, not for multi-electron atoms; it cannot account for the faint "fine structure" lines that accompany each main spectral line under close examination; it does not explain why different spectral lines have different intensities; and it does not fully describe how electrons are actually distributed within an atom - gaps that were only resolved later by full quantum mechanics.

"Hydrogen spectrum series formula Lyman Balmer Paschen" and "hydrogen spectral series important questions" are among the most searched topics in the Atoms chapter of the NCERT/CBSE Class 12 Physics curriculum, since this Rydberg-formula-based series appears almost every year in JEE Main, JEE Advanced and NEET. The formula for counting the total number of spectral lines, $n(n-1)/2$, is a favourite standalone JEE Main question in its own right.