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Light Spectra

Skills to develop

• Explain the terms white light, line, emission, & absorption spectra
• Interpret the regularity of hydrogen spectra
• Light Spectra

The intensity of light as a function of wavelength is a spectrum. To see a spectrum, a beam of electromagnetic radiation is spread according to wavelength or colour so that the intensity as a function of wavelength is represented. A visible spectrum has many colors, and a rainbow is a typical spectrum.

Visible light is only in a small range of the total electromagnetic radiation spectrum and a partial spectrum may be in regions that are not visible. Classified according to region of radiation, we have infrared-, ultraviolet-, X-ray-, and gamma-ray- spectra.

Kinds of Spectra

Light beams (radiations) from (hot) solids may be spread (by a prism) into a continuous display of color, and such beams give continuous spectra or white spectra.

Light beams from a gaseous sample usually show colored lines called line spectra or discontinued spectra. These continuous and line spectra are called emission spectra. The spectra of H, Hg, and Ne are line spectra.

When a white light beam passes through a medium, light of some colors will be absorbed. As a result, the spectrum shows dark (absent) lines. Such a spectrum is called an absorption spectrum.

The dark lines in the absorption spectrum of a gas correspond exactly to some of the bright lines in the emision spectrum of that gas.

Spectra of the H Atom

Since the invention of the spectrometer to analyze light, the light source from hydrogen gas has been intensely studied. Lines were observed in the infrared, visible, and ultraviolet regions. The wavelengths or frequencies of these lines had been known for a long time.

The frequency (Hz) wave number (number per meter) and wavelength (nm) of some of the lines in the visible region are given below:

Frequency
in 1e14 Hz
Wavelength
nm
Wave number
mm-1
Energy
in e-19 J
4.57656.315233.028
6.17486.120564.088
6.91434.023034.578
7.32410.224374.845
7.56397.025185.006
7.71388.925715.110
7.82383.526065.181

Many people tried to find the rules that govern their variation, and in the process many mathematical techniques have been employed. Among these attempts, there has been suggestions that the wavenumbers vary according to

 RH --- - K, n2

where RH is the Rydberg constant, n an integer, and K some constant.

At the time when Bohr worked on this problem, Rutherford had shown that atoms consisted of small dense nuclei surrounded by very light electrons. Thus, Bohr thought the atom might be a minature solar system with electrons revolving around (like the planet) the nucleus (the Sun). He also applied Planck's quantum concept and implied that when the angular momentum of the revolving electron is of a certain value, the electron orbital is stable. He derived a formula that showed the energy of the photons of these lines vary according to

 ``` 1 1 E = - RH ( --- - --- ) n2 4 ```
and RH = 2.179E-18 J, is the Rydberg constant.

Depending on the units used for RH, the energy E of this formula can be wavenumbers or frequencies.

Furthermore, the formula also applies to lines of the hydrogen spectrum in the infrared and ultraviolet regions if the 4 is replaced by the square of some other integer, n'. Thus, the formula has a general form of

 ``` 1 1 E = - R ( --- - --- ) n2 n'2 ```
Depending on the value of n', we have the following series: n' = 1, Lyman series (ultraviolet)
n' = 2, Balmer series (visible) Wavelength vary from 400 to 700 nm
n' = 3, Ritz-Paschen series (short wave infrared)
n' = 4, Pfund series (long wave infrared)

Thus, this formula agrees with all observed lines in the hydrogen spectrum. The above formula has been employed to calculate the spectra of the hydrogen atom.

Confidence Building Questions

• The spread of a light beam into colors so the variation of intensity is observed is called what?
p - photon
s - spectrum
w - wavenumber

Discussion:
The term spectrum can have other meanings, but this is a good start. A spectrum sometimes also display the intensity of the various colors. In this case, the spectrum is presented as a plot.

• What type of spectrum does one get from a incandescent lamp?
a - absorption spectrum
c - continuous spectrum
l - line spectrum

Discussion:
Light emitted from a solid (wire) is white, and it has a continuous spectrum. White radiation or a continuous spectrum is emitted by a hot solid.

• What type of spectrum does one get from a fluorescent lamp?
a - absorption spectrum
b - broken spectrum
c - continuous spectrum
l - line spectrum

Discussion:
Light from any gas consists of lines. A plasma (hot gas) is used in fluorescent lamp as light source.

• Which series of the hydrogen spectrum is in the visible region?
1 Lyman series
2 Balmer series
3 Ritz-Paschen series
4 Pfund series

Discussion:
Lyman series is in the UV region, Ritz-Paschen and Pfund series are in the IR region. The n' (or nf in some texts) has a value of 2 for the Balmer series.

• What is approximately the low limit of wavelength (nm) in the visible spectrum?

Discussion:
The visible spectrum range from about 400 to 700 nm. The visible region is really narrow. A wavelength of 400 nm corresponds to a frequency of what?