Absolutely Small - Michael D. Fayer [53]
FIGURE 9.2. The visible portion of the solar spectrum. The continuous range of colors is the black body spectrum. The dark lines or bands are colors that do not reach Earth, so they appear as colors missing from the solar spectrum. The wavelengths of the lines and the spectrum are given in nm, nanometers, which are billionths (10-9) meters.
The same wavelengths that are seen as dark lines in the solar spectrum can also be seen as distinct colors from an arc lamp filled with hydrogen gas. A hydrogen arc lamp or discharge lamp is a sealed glass cylinder filled with hydrogen gas with electrodes at either end. When a sufficiently high voltage is connected to the lamp, positive connected to one electrode and negative to the other electrode, electricity arcs through the lamp like a small continuous lightning bolt. The colors (wavelengths) in the visible coming out of the lamp are the same as wavelengths of the black lines shown in Figure 9.2.
The Hydrogen Line Spectrum
The first attempt at understanding the line spectrum of hydrogen in the visible region was made in 1885 by the Swiss schoolteacher and mathematician, Johann Balmer (1825-1898). Balmer noted that the frequencies of the lines, f, in the visible part of the spectrum were related by the formula
The symbol ∝ means proportional to, so there is a multiplicative constant that is discussed below. In this equation, n is an integer greater than 2, that is, 3, 4, 5, etc. The spectral lines in the visible are called the Balmer series.
Later, lines were discovered in the ultraviolet and the infrared. These are called the Lyman series and the Paschen series, respectively, after their discoverers Theodore Lyman (1874-1954), a U.S. physicist and spectroscopist, and Louis Karl Heinrich Friedrich Paschen (1865-1947), a German physicist. In 1888, the Swedish physicist and spectroscopist, Johannes Rydberg (1854-1919) presented a formula that described all of the spectral lines seen in emission from a hydrogen arc lamp or in the absorption spectrum of solar or stellar light. The Rydberg formula for the frequency of the hydrogen atom spectral lines is
n1 is an integer beginning at 1. n2 is another integer that must be greater than n1. n1 = 1 gives the Lyman series. n1 = 2 gives the Balmer series. n1 = 3 gives the Paschen series. The constant, RH, is called the Rydberg constant for the hydrogen atom. It has the value, RH = 109,677 cm-1. Here the constant is given in wave numbers (cm-1). When this value is used in the Rydberg formula, the frequency of a spectral line determined by the integers n1 and n2 is in wave numbers. To get it in Hz, the result is multiplied by the speed of light in cm/s, that is, 3 × 1010 cm/s. To find the wavelength of a spectral line, take the inverse of the frequency in wave numbers, that is, take 1 and divide it by the frequency in wave numbers. For example, if n1 = 2 and n2 = 3, then
the frequency in wave numbers. The inverse of this number is 6.56 × 10-5 cm = 656 × 10-9 m. 10-9 m is a nanometer, so the wavelength is 656 nm. This is the red line in the Balmer series shown in Figure 9.2.
In connection with Figure 8.7, we already discussed discreet optical transitions between quantized energy levels for the particle in a box. Figure 8.7 shows transitions between the particle in a box states for n = 1 going to n = 2 and n = 1 going to n = 3. So it should come as no surprise that the optical transitions of the hydrogen atom could involve discreet frequencies that depend on integers. However, in 1888, at the time of the Rydberg formula, it was still 12 years before the first use of the idea of quantized energy levels by Planck to explain black body radiation, and 37 years before true quantum theory