30.2

Line Spectra

We have seen in Sections 13.3 and 29.2 that all objects emit electromagnetic waves, and we will see in Section 30.3 how this radiation arises. For a solid object, such as the hot filament of a light bulb, these waves have a continuous range of wavelengths, some of which are in the visible region of the spectrum. The continuous range of wavelengths is characteristic of the entire collection of atoms that make up the solid. In contrast, individual atoms, free of the strong interactions that are present in a solid, emit only certain specific wavelengths rather than a continuous range. These wavelengths are characteristic of the atom and provide important clues about its structure. To study the behavior of individual atoms, low-pressure gases are used in which the atoms are relatively far apart.
A low-pressure gas in a sealed tube can be made to emit electromagnetic waves by applying a sufficiently large potential difference between two electrodes located within the tube. With a grating spectroscope like that in Figure 27-36, the individual wavelengths emitted by the gas can be separated and identified as a series of bright fringes. The series of fringes is called a line spectrum because each bright fringe appears as a thin rectangle (a “line”) resulting from the large number of parallel, closely spaced slits in the grating of the spectroscope. Figure 30-3 shows the visible parts of the line spectra for neon and mercury. The specific visible wavelengths emitted by neon and mercury give neon signs and mercury vapor street lamps their characteristic colors.

Figure 30-3 The line spectra for neon and mercury, along with the continuous spectrum of the sun. The dark lines in the sun’s spectrum are called Fraunhofer lines, three of which are marked by arrows. (Courtesy Bausch & Lomb)

The physics of neon signs and mercury vapor street lamps.

The simplest line spectrum is that of atomic hydrogen, and much effort has been devoted to understanding the pattern of wavelengths that it contains. Figure 30-4 illustrates in schematic form some of the groups or series of lines in the spectrum of atomic hydrogen. The group of lines in the visible region is known as the Balmer series, in recognition of Johann J. Balmer (1825–1898), a Swiss schoolteacher who found an empirical equation that gave the values for the observed wavelengths. This equation is given next, along with similar equations that apply to the Lyman series at shorter wavelengths and the Paschen series at longer wavelengths, which are also shown in the drawing:

| (30.1)

| (30.2)

| (30.3)

In these equations, the constant term R has the value of and is called the Rydberg constant. An essential feature of each group of lines is that there are long and short wavelength limits, with the lines being increasingly crowded toward the short wavelength limit. Figure 30-4 also gives these limits for each series, and Example 2 determines them for the Balmer series.

Figure 30-4 Line spectrum of atomic hydrogen. Only the Balmer series lies in the visible region of the electromagnetic spectrum.

| Example 2 | | | The Balmer Series

Find (a) the longest and (b) the shortest wavelengths of the Balmer series. Reasoning Each wavelength in the series corresponds to one value for the integer n in Equation 30.2. Longer wavelengths are associated with smaller values of n. The longest wavelength occurs when n has its smallest value of n = 3. The shortest wavelength arises when n has a very large value, so that 1/n2 is essentially zero.
Solution

a. | With n = 3, Equation 30.2 reveals that for the longest wavelength

b. | With 1/n2 = 0, Equation 30.2 reveals that for the shortest wavelength

Equations 30.1, 30.2 and 30.3 are useful because they reproduce the wavelengths that hydrogen atoms radiate. However, these equations are empirical and provide no insight as to why certain wavelengths are radiated and others are not. It was the great Danish physicist, Niels Bohr (1885–1962), who provided the first model of the atom that predicted the discrete wavelengths emitted by atomic hydrogen. Bohr’s model started us on the way toward understanding how the structure of the atom restricts the radiated wavelengths to certain values. In 1922 Bohr received the Nobel Prize in physics for his accomplishment.

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