## Emission Spectrum of the Hydrogen Atom

#### Learning Objective

• Calculate emission spectra for Hydrogen using the Rydberg formula

#### Key Points

• The wavelengths in a spectral series are given by the Rydberg formula.
• The observed spectral lines are due to electrons moving between energy levels in an atom.
• Further series are unnamed but follow exactly the same pattern as dictated by the Rydberg equation.

#### Terms

• spectrumA range of colors representing light (electromagnetic radiation) of contiguous frequencies; hence electromagnetic spectrum, visible spectrum, ultraviolet spectrum, etc.
• emissionIn a spectral sense, what occurs when an electron transitions between a higher energy level and a lower one, resulting in the release of a photon of predictable energy.

The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted by an atom’s electrons when they are returned to a lower energy state. Each element’s emission spectrum is unique, and therefore spectroscopy can be used to identify elements present in matter of unknown composition. Similarly, the emission spectra of molecules can be used in chemical analysis of substances.

The emission spectrum of atomic hydrogen is divided into a number of spectral series, with wavelengths given by the Rydberg formula: $\frac { 1 }{ \lambda_{vac} } =RZ^2( \frac { 1 }{ {n_1 }^{ 2 } } -\frac { 1 }{ { n_2 }^{ 2 } })$,

where R is the Rydberg constant (approximately 1.09737 x 107 m-1), $\lambda_{vac}$ is the wavelength of the light emitted in vacuum, Z is the atomic number, and n1 and n2 are integers representing the energy levels involved such that n1 < n2. All observed spectral lines are due to electrons moving between energy levels in the atom. The spectral series are important in astronomy for detecting the presence of hydrogen and calculating red shifts. Further series for hydrogen as well as other elements were discovered as spectroscopy techniques developed.

The spectral lines are grouped into series according to n′. Lines are named sequentially starting from the longest wavelength/lowest frequency of the series using Greek letters within each series. For example, the 2 → 1 line is called Lyman-alpha (Ly-α), while the 7 → 3 line is called Paschen-delta (Pa-δ). Some hydrogen spectral lines fall outside these series, such as the 21 cm line (these correspond to much rarer atomic events such as hyperfine transitions). The fine structure also results in single spectral lines appearing as two or more closely grouped thinner lines due to relativistic corrections. Typically, one can only observe these series from pure hydrogen samples in a lab. Many of the lines are very faint and additional lines can be caused by other elements (such as helium if using sunlight or nitrogen in the air). Lines outside of the visible spectrum typically cannot be seen in observations of sunlight, as the atmosphere absorbs most infrared and ultraviolet wavelengths through the action of water vapor and ozone molecules respectively.