Spectrum of Hydrogen: MDCAT Chemistry notes
Spectrum of Hydrogen for MDCAT: line vs continuous spectra, Lyman, Balmer and Paschen series, Rydberg formula, Zeeman and Stark effects explained.
Continuous and line spectra
A spectrum is the pattern of wavelengths emitted or absorbed by a substance. A continuous spectrum contains all wavelengths merging into one another, as from white light or a hot solid. An atomic (line) spectrum consists of sharp, separate lines, each of a definite wavelength, separated by dark gaps.
To obtain the atomic spectrum of an element, the element or its compound is first volatilized (vaporised) in a flame or electric arc so that it exists as free, excited atoms. The emitted light is analysed by a spectrometer. Every element gives its own characteristic line spectrum, like a fingerprint, so line spectra are used to identify elements.
| Type | How produced | Appearance |
|---|---|---|
| Emission spectrum | Excited atoms give out light | Bright lines on a dark background |
| Absorption spectrum | White light passed through a vapour | Dark lines on a continuous bright background |
Origin of hydrogen lines (Bohr)
When hydrogen gas is excited, its electron jumps to a higher orbit. On falling back from a higher level $n_2$ to a lower level $n_1$ it emits a photon of energy $\Delta E = E_{n_2} - E_{n_1} = h\nu$. Only fixed energy differences are allowed, so only definite wavelengths appear.
Spectral series
| Series | $n_1$ (lower level) | $n_2$ | Region |
|---|---|---|---|
| Lyman | 1 | 2, 3, 4 ... | Ultraviolet |
| Balmer | 2 | 3, 4, 5 ... | Visible |
| Paschen | 3 | 4, 5, 6 ... | Infrared |
| Brackett | 4 | 5, 6, 7 ... | Infrared |
| Pfund | 5 | 6, 7, 8 ... | Infrared |
Key formulas
$$\bar{\nu} = \frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$
- $R_H = 1.09678 \times 10^{7}\ \mathrm{m^{-1}}$ (Rydberg constant).
- $\Delta E = h\nu = hc/\lambda$.
- Series limit: put $n_2 = \infty$, so $\bar{\nu} = R_H/n_1^2$.
Worked idea: the first Lyman line ($n_2 = 2$) has $\bar{\nu} = R_H(1 - \tfrac14) = \tfrac34 R_H$, the largest energy gap of low-lying lines, which is why the Lyman series lies in the ultraviolet.
Splitting of lines
- Zeeman effect: splitting of spectral lines in a magnetic field.
- Stark effect: splitting of spectral lines in an electric field.
Bohr's model could not explain these splittings or the fine structure of lines; this was one of its main defects.
Common MDCAT traps
- Magnetic field = Zeeman; electric field = Stark. Students often swap them.
- Aufbau and Pauli principles concern electron filling, not line splitting.
- The sample is volatilized, not frozen, and atoms give a line spectrum, not a continuous one.
- Only the Balmer series is visible; Lyman is UV, the rest are IR.
Quick revision
- Atomic spectra are line spectra; each element has its own.
- Lines arise from electron jumps between fixed energy levels.
- Lyman $n_1=1$, Balmer $n_1=2$, Paschen $n_1=3$.
- $1/\lambda = R_H(1/n_1^2 - 1/n_2^2)$.
- Zeeman: magnetic; Stark: electric.