Structure of Atom · Lesson 5 of 12
Atomic Spectra and the Hydrogen Pattern
“Read emission and absorption spectra as evidence of discrete atomic energy differences.”
• Distinguish continuous spectra, line emission spectra, and absorption spectra. • Explain how a characteristic spectrum identifies an element. • Use the Balmer and Rydberg expressions to describe hydrogen spectral series. • Connect the hydrogen pattern with the need for discrete electronic energies.
Separating the colours in light
A prism changes the direction of light because light changes speed when it enters a different medium. Its refractive effect depends on wavelength. In an ordinary prism, shorter visible wavelengths deviate more strongly than longer ones. White light therefore spreads into a band from red to violet.
A continuous visible spectrum contains an unbroken range of wavelengths, with one colour blending into the next. The rainbow is a familiar example of wavelength separation, although a raindrop’s optical path also involves internal reflection. An atomic line spectrum looks strikingly different: it contains separated lines with dark gaps.
The distinction is not merely one of brightness. A dim continuous spectrum still contains a continuous wavelength range, while a bright line spectrum still selects particular wavelengths. To understand the difference, we must consider what happens when matter gains and loses energy.
Emission and absorption tell a paired story
Energy supplied by heating, electrical discharge, or radiation can put an atom, molecule, or ion into an excited state. We focus here on atomic spectra. When the atom returns to a lower-energy state, it emits radiation. Passing that light through a spectroscopic arrangement reveals the wavelengths emitted. A dilute excited gas produces characteristic bright lines.
For an absorption spectrum, broad-spectrum light passes through a sample before being separated into wavelengths. The sample removes certain wavelengths, leaving dark lines or reduced intensity in the transmitted spectrum. The absorption wavelengths correspond to transitions available from the states actually occupied in the sample; under suitable conditions they match corresponding emission lines.
Each line represents a definite energy difference, because photon energy is . An atom cannot choose an arbitrary photon frequency during a particular transition. This is the central evidence connecting line spectra with quantised electronic energies.
A spectral fingerprint
Different elements have different electronic structures and therefore different characteristic sets of spectral lines. Comparing an unknown sample’s lines with a known reference can identify an element. Matching a pattern of several lines is stronger evidence than matching one colour by eye.
Spectroscopy helped discover elements such as rubidium, caesium, thallium, indium, gallium, and scandium. Helium was recognised in the Sun’s spectrum before being identified on Earth. These applications show how radiation can reveal the identity of matter even when direct sampling is difficult.
A macroscopic sample contains enormous numbers of atoms. Individual atoms need not all be in the same excited state. Radiation from many available transitions across the collection produces the set of observed lines. One atom does not occupy every excited state simultaneously.
Problem
An unknown gas has three measured emission lines matching three lines of element A. Only one line matches element B. Which identification has stronger support?
- 1.A line corresponds to an allowed energy difference. An accidental match of one wavelength is possible.
- 2.Matching several characteristic lines supports element A more strongly.
- 3.A full identification should check additional lines and possible mixtures. The reasoning depends on the pattern, not simply the apparent colour of the gas.
The regularity in hydrogen’s lines
An electrical discharge through hydrogen produces excited hydrogen atoms. Their spectrum is simpler than those of heavier atoms, making regularities easier to identify. Balmer found that the visible lines followed a mathematical relationship when expressed as wavenumbers.
For the Balmer series, one integer is fixed at 2 and the other takes values 3, 4, 5, and higher. The empirical formula describes the observations before an explanatory atomic model has been supplied. We should distinguish “a rule fitting the data” from “a physical reason for the rule”.
Rydberg generalised the pattern to other hydrogen series. The smaller integer labels the lower state, while the larger integer labels the upper state. The physical meaning of these integer-labelled energy states will emerge from Bohr’s model. In the empirical wavelength formula below, both numbers are positive and the larger is always placed second, giving a positive wavenumber.
| Series | Lower integer | Upper integers | Spectral region |
|---|---|---|---|
| Lyman | Ultraviolet | ||
| Balmer | Visible members; limit in ultraviolet | ||
| Paschen | Infrared | ||
| Brackett | Infrared | ||
| Pfund | Infrared |
The familiar Balmer lines are visible, but the series does not remain wholly within visible light: its high members converge towards an ultraviolet limit near 365 nm. This qualification prevents a simplified series table from becoming a misleading absolute rule.
Problem
Find the wavelength for the hydrogen series member with and .
- 1..
- 2..
- 3.This is a visible red line. The exact measured value depends on the constants and medium convention used.
Problem
Which series has lower integer 3, and what happens to wavelength as the upper integer grows?
- 1.The fixed lower integer 3 identifies the Paschen series.
- 2.As increases, decreases. Therefore the positive wavenumber increases and wavelength decreases.
- 3.At the limit , . The lines converge towards a shortest series wavelength, rather than spreading indefinitely.
A line spectrum is not evidence that an electron emits continuously while moving around a fixed orbit. It is evidence of specific energy differences. Also, Rydberg constants written in inverse centimetres must not be mixed with wavelengths left in metres.
Quiz
A bright set of isolated wavelengths from an excited dilute gas is a:
An absorption line appears dark because:
Which fixed lower integer identifies the Paschen series?
Within a hydrogen series, increasing the upper integer makes successive lines:
Why do different elements have characteristic spectra?
Practice Problems
- Explain the different experimental arrangements for obtaining emission and absorption spectra.
- Use the Rydberg expression to calculate the wavelength for , .
- Find the longest wavelength in the Balmer series and explain why the first member supplies it.
- Explain why a sample containing many excited atoms can emit several lines at once.
- Compare the wavelength limits of the Lyman and Balmer series.
- Describe how spectroscopy could identify an element in a distant star.
Key Takeaways
Continuous spectra contain an uninterrupted wavelength range; atomic line spectra select particular wavelengths. Emission and absorption reveal allowed electronic energy differences. A characteristic set of lines acts as an elemental fingerprint. Hydrogen series follow the Rydberg pattern and converge towards series limits.