Explain the spectral series of hydrogen atom. (Diagram not necessary)

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Q. Explain the spectral series of hydrogen atom. (Diagram not necessary)

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Concept understanding — Hydrogen Spectral Series

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,

\frac{1}{\lambda}=R\left(\frac{1}{n^2}-\frac{1}{m^2}\right),\qquad m>n\ \text{positive integers}

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 $n(n-1)/2$ . …

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The spectral lines of atomic hydrogen fall into distinct series depending on which lower energy level the electron falls to; each series is described by the Rydberg formula.

According to the Bohr model, when an electron in a hydrogen atom jumps from a higher orbit $n_2$ to a lower orbit $n_1$ , a photon of definite wavelength is emitted, with wave number given by the Rydberg formula: $\frac{1}{\lambda} = R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right), \quad n_2 > n_1$ where $R$ is the Rydberg constant ( $1.097\times10^7\ \text{m}^{-1}$ ).
Lyman series: $n_1 = 1$ , $n_2 = 2,3,4,\dots$ — lies in the ultraviolet region.
Balmer series: $n_1 = 2$ , $n_2 = 3,4,5,\dots$ — lies in the visible region (the only series visible to the naked eye).
Paschen series: $n_1 = 3$ , $n_2 = 4,5,6,\dots$ — lies in the infrared region.
Brackett series: $n_1 = 4$ , $n_2 = 5,6,7,\dots$ — lies in the infrared region.
Pfund series: $n_1 = 5$ , $n_2 = 6,7,8,\dots$ — lies in the far-infrared region. …

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Q.Explain the spectral series of hydrogen atom. (Diagram not necessary)

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