Hydrogen Spectrum by Grating Made Easy with Simple Examples

When hydrogen gas is excited in a discharge tube it does not glow white; it emits a few sharp, pure colours: a red line, a blue-green line, a violet line and more in the ultraviolet. This set of discrete wavelengths is the hydrogen spectrum, and it was the key evidence that atoms have quantised energy levels.

A grating spectrometer spreads these lines out and lets each wavelength be measured from the angle at which it diffracts. From the angles of the visible Balmer lines the wavelengths follow, and from the wavelengths the Rydberg constant, one of nature's fundamental numbers, can be calculated.

Key Concepts

• Line spectrum: emission at only certain discrete wavelengths, unlike the continuous rainbow of a hot solid.
• Balmer series: the visible lines of hydrogen, produced when electrons fall to the n = 2 level (H-alpha red, H-beta blue-green, H-gamma violet).
• Diffraction grating: a glass plate ruled with thousands of fine parallel lines per centimetre that splits light into spectra.
• Grating element (a + b): the distance between neighbouring rulings, equal to 1 / N where N is the number of lines per unit length.
• Order (n): the spectrum number, n = 1, 2, 3, appearing at larger angles for higher orders.

Formulas

Grating equation (normal incidence): n λ = (a + b) sin θ, so λ = (a + b) sin θ / n, with (a + b) = 1 / N. Variables: n the order, λ the wavelength in metres, θ the measured diffraction angle. Balmer formula: 1 / λ = R (1 / 2² − 1 / n²) for n = 3, 4, 5, …, where R = 1.097 × 10⁷ m⁻¹ is the Rydberg constant. Assumptions: the light strikes the grating at normal incidence, the grating lines are uniform, and only first-order angles are used for the cleanest measurement.

Worked Example

A grating ruled with N = 6000 lines per cm is used at normal incidence. The red hydrogen line appears in the first order at θ = 23.2°. Find its wavelength and check it against the Balmer formula for n = 3.

1. Grating element: (a + b) = 1 / 6000 cm = 1.667 × 10⁻⁴ cm = 1.667 × 10⁻⁶ m.
2. Grating equation: λ = 1.667 × 10⁻⁶ × sin(23.2°) / 1 = 1.667 × 10⁻⁶ × 0.3938.
3. Multiply: λ = 6.563 × 10⁻⁷ m = 656.3 nm, the famous H-alpha red line.
4. Balmer check: 1 / λ = 1.097 × 10⁷ × (1/4 − 1/9) = 1.097 × 10⁷ × 0.13889 = 1.5237 × 10⁶ m⁻¹.
5. Invert: λ = 1 / (1.5237 × 10⁶) = 6.563 × 10⁻⁷ m = 656.3 nm, agreeing exactly.

Common Misconceptions

• Hydrogen does not give a rainbow. A prism or grating shows isolated bright lines on a dark background, because only fixed energy jumps are allowed.
• Higher orders are not new lines. The second-order spectrum repeats the same wavelengths at larger angles; overlapping orders must be identified by their order, not mistaken for extra lines.
• The grating does not create the colours. It only separates what the hydrogen already emits; the wavelengths are properties of the atom.

Real-World Applications

• Spectroscopy identifies elements in stars: the Sun's and stars' compositions are read from their line spectra.
• Gas-discharge lamps and lasers are designed around known atomic lines.
• Forensic and environmental labs identify trace elements by their emission lines.
• The Rydberg constant measured this way anchors atomic physics and quantum theory.

The grating spectrometer turns the hydrogen atom's secret code into angles on a scale: measure θ, apply n λ = (a + b) sin θ, and each coloured line yields its wavelength. Those wavelengths then confirm the Balmer formula and the Rydberg constant behind them.