Atomic Spectra

Continuous spectrum and line spectrum

• A heated metallic object emits radiation of all wavelengths; passed through a prism this gives a continuous spectrum.
• Hydrogen gas heated to a high temperature inside a glass tube behaves differently: the emitted radiation contains only a few selected wavelengths, and through a prism it gives a line spectrum.
• In the visible range, hydrogen emits at $410$, $434$, $486$ and $656\ \text{nm}$, and nothing at wavelengths in between. The lines seen in such a spectrum are called emission lines.
• Diagram in words: a dark horizontal band representing the spectrum with four bright vertical lines on it — violet at $410\ \text{nm}$, blue-violet at $434$, blue-green at $486$ and red at $656$ — separated by darkness.

Series of lines

• Hydrogen also emits at other wavelengths in the ultraviolet, in the infrared and at still longer wavelengths.
• The lines fall into groups called series, named after the scientists who studied them. In order of increasing wavelength: Lyman, Balmer, Paschen, Brackett, Pfund.
• Within any one series, the separation between successive lines decreases as we go towards shorter wavelengths, and the lines crowd up to a limiting value called the series limit.
• Diagram in words for Fig. 15.3: three horizontal wavelength scales one below the other, marked UV, visible and IR; on each, a group of vertical lines that get closer together towards the short-wavelength end, with the limiting wavelength of each group shown by a dotted line.

The empirical relation

$$\dfrac{1}{\lambda} = R\left(\dfrac{1}{n^2}-\dfrac{1}{m^2}\right)$$

• $\lambda$ is the wavelength of a line, $R$ is a constant, and $n$ and $m$ are integers. $n = 1, 2, 3, \ldots$ for the Lyman, Balmer, Paschen, … series respectively, and for a given series $m$ takes all integral values greater than $n$.
• The wavelength decreases as $m$ increases, and the difference in wavelength between successive lines of a series also decreases with increasing $m$.
• In the limit $m \to \infty$ the wavelength reaches the series limit $\dfrac{n^2}{R}$.

Standard series data

• The visible series in the hydrogen spectrum is the Balmer series — a one-mark question the board sets on its own.

Two further points

• Atoms of other elements also emit line spectra, and the set of wavelengths emitted by each element is unique, so an element can be identified from the wavelengths of its spectral lines.
• Rutherford's model could not explain atomic spectra at all; explaining them is what Bohr's model achieves.