Atomic Spectra

Atoms · CBSE

Theory

The difference between a continuous spectrum, an emission line spectrum and an absorption spectrum, and the conditions that produce each.
Why a line spectrum is a fingerprint of the element.
Writing Rydberg's formula and naming the series to which a given line belongs.
It is short: CBSE mostly folds it into 12.5 questions, so learn the definitions and the formula and spend your time on the series.

Condensed matter (solids and liquids) and dense gases at any temperature emit electromagnetic radiation containing a continuous distribution of wavelengths, with different intensities at different wavelengths. The radiation comes from the oscillations of atoms and molecules, each governed by its interaction with close neighbours, so no single wavelength is picked out.
Light from a rarefied gas — heated in a flame, or excited electrically as in a neon sign or a mercury vapour lamp — has only certain discrete wavelengths. The average spacing between the atoms is large, so each atom radiates on its own rather than as part of an interacting crowd, and the radiation is characteristic of the individual atom.
The spectrum then appears as a set of isolated bright lines on a dark background.

Definition: when an atomic gas or vapour at low pressure is excited, usually by passing an electric current through it, the emitted radiation contains only certain specific wavelengths. Viewed through a spectrometer this is a set of bright lines on a dark background — the emission line spectrum.
The set of lines is fixed for a given element and independent of how the gas was excited, so the emission spectrum serves as a fingerprint for identifying the gas. Sodium always gives its yellow doublet; hydrogen always gives the same set of lines with fixed relative positions.
This is the observational fact that suggested an intimate relationship between the internal structure of an atom and the radiation it emits — the fact Bohr's model was built to explain.

Definition: when white light (a continuous spectrum) is passed through a cool gas and the transmitted light is analysed with a spectrometer, dark lines appear in the otherwise continuous spectrum. This is the absorption spectrum of the gas.
Crucial observation: the dark lines fall at precisely the wavelengths that appear as bright lines in the emission spectrum of the same gas. The atom absorbs exactly what it emits.
Explanation, once Bohr's model is in hand: absorption raises an electron from a lower to a higher stationary state, and only a photon whose energy exactly matches an energy gap can do it. So the frequencies removed from the beam are exactly the frequencies the atom would emit on the reverse transitions.

Type	Source conditions	Appearance
Continuous	Hot solid, liquid or dense gas	Unbroken band of all wavelengths
Emission line	Rarefied gas excited electrically or thermally	Bright lines on a dark background
Absorption	White light passed through a cool rarefied gas	Dark lines on a continuous bright background, at the same wavelengths as the emission lines

$\dfrac{1}{\lambda}=R\left[\dfrac{1}{2^2}-\dfrac{1}{n^2}\right],\qquad n=3,4,5,\dots$

$\dfrac{1}{\lambda}=R\left[\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\right],\qquad n_2>n_1,\qquad R=1.097\times10^{7}\ \text{m}^{-1}$

$n_1$ is the lower level and fixes which series the line belongs to; $n_2$ is the upper level and fixes which line within the series. $R$ is the Rydberg constant, $1.0973\times10^7$ m $^{-1}$ .
The formula predicts $1/\lambda$ , the wave number. Always invert to get $\lambda$ .
The five series, for reference (the details are worked in 12.5): Lyman $n_1=1$ , ultraviolet; Balmer $n_1=2$ , visible; Paschen $n_1=3$ , infrared; Brackett $n_1=4$ , infrared; Pfund $n_1=5$ , far infrared.
The $H_\alpha$ line, wavelength $656.3$ nm, is the first line of the Balmer series, $n_2=3\to n_1=2$ — the standard 1-mark identification.
Nothing in classical physics explains why such a simple integer formula should work. That is exactly the puzzle Bohr's model solves in 12.4 and 12.5, where the same formula emerges with $R=\dfrac{me^4}{8\varepsilon_0^2h^3c}$ .

Continuous spectrum from dense matter; line spectrum from rarefied gases.
Emission lines and absorption lines of the same element coincide in wavelength.
$\dfrac{1}{\lambda}=R\left[\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\right]$ with $R=1.097\times10^7$ m $^{-1}$ .
Line spectra are element-specific and are used to identify unknown gases.