HYDROGEN SPECTRUM. Figure 1 shows the energy level scheme for the hydrogen atom as calculated from equation. Figure 1 Figure 2

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1 15 Jul 04 Hydrogen.1 HYDROGEN SPECTRUM In this experiment the wavelengths of the visible emission lines of hydrogen (Balmer series) will be measured and compared to the values predicted by Bohr s quantum theory. Theory: According to the quantum theory of Niels Bohr (1913), the hydrogen atom can only exist in discrete energy states whose energies are given by: E n 4 me = 8ε h 1 n o (1) where e and m are the electron charge and mass, respectively; ε o is the dielectric permittivity of free space; h is Planck s constant; and n is the quantum number (n = 1, 2, 3,...). (1). Figure 1 shows the energy level scheme for the hydrogen atom as calculated from equation Figure 1 Figure 2 Under normal conditions, the hydrogen atom remains in its ground state (n = 1). It is possible to populate the various excited states (n > 1) by heating, electric discharge, irradiation, etc. The following decay from an excited state (n i ) to a lower-lying excited state or the ground state (n f ) is

2 15 Jul 04 Hydrogen.2 accompanied by the emission of electromagnetic radiation (a photon). The frequency of this radiation is determined by Bohr s second postulate: E = hν where E is the energy difference between the two hydrogen atom states. Using equation (1), for a transition from the initial energy state n i to the final energy state n f : E = hν = E i E f 4 me hν = 8ε h 1 n 4 me 8ε h 1 n o i o f 4 me 1 1 hν = ε h n n o f i Now ν = c/λ, so 4 1 me 1 1 = λ ε h c n n o f i = R λ 2 2 n n where R = m 1 (the Rydberg constant). f i All transitions which end at the same energy level, called final state (n f ), form a series. The various series are named in honour of their discoverers. n f = 1 is the Lyman series, n f = 2 is the Balmer series, n f = 3 is the Paschen series, n f = 4 is the Bracket series, and n f = 5 is the Pfundt series (see Figure 2). The first four transitions of the Balmer series of hydrogen can be observed as a line emission spectrum in the visible region. They are traditionally named H α (red, n i = 3), H β (blue-green, n i = 4), H γ (violet, n i = 5), and H δ (violet, n i = 6). The H δ line is of very low intensity and so the lines originating in higher energy states are of even lower intensity. Apparatus: The apparatus used to observe the spectrum consists of a spectrometer and a diffraction grating. A diffraction grating is a piece of transparent material with very fine, closely-spaced rulings or grooves cut or moulded into it.

3 15 Jul 04 Hydrogen.3 The grating is extremely fragile and must not be touched. When a monochromatic beam of light is incident on the grating, an interference pattern is formed as the beam passes through each of the grooves. The pattern consists of sharp bright lines whose angular positions are determined by mλ = d sinθ provided the grating is perpendicular to the incident light. m is the order of the line (m = 0, ±1, ±2,...); λ is the wavelength; d is the spacing between the rulings on the grating; and θ is the deviation angle of the transmitted beam. If the incident beam is composed of light of different wavelengths, these wavelengths will be separated into distinct lines. Figure 3 shows the spectrum formed by light of different wavelengths passing through a grating. Figure 3 The hydrogen source is an electric discharge tube filled with water vapour. The discharge is confined in a capillary tube 1 mm in diameter and 50 mm long, thus providing a high intensity line source. The tube is powered by 5000 V so use extreme caution when near the power supply and tube. Figure 4 shows the arrangement of the grating spectrometer.

4 15 Jul 04 Hydrogen.4 Figure 4 Note the locknuts and fine adjustment screws on the spectrometer. Note that the vernier scale yields measurements in decimal degrees to an accuracy of 0.1. A mercury light source is provided for calibration of the grating spectrometer. Procedure and Experiment: 1. Determine the controls which allow locking and unlocking of the grating table, locking and unlocking of the spectrometer table, and fine and coarse positioning of the telescope. 2. Practise reading the vernier scale until you are confident of your measurements. 3. Place the collimator slit of the spectrometer about 0.5 cm from the mercury light source. Remember, do not touch the diffraction grating. 4. For the grating equation, mλ = d sinθ, to be applicable, the grating must be perpendicular to the incident light. a) Set the spectrometer to view the slit directly as shown in Figure 5a. b) Unlock the spectrometer table and set it to read 0, then lock it. c) Unlock the telescope and swing it clockwise through 90 ; lock it. d) Unlock the grating table and rotate the grating so as to view the reflected image of the slit. This places the grating at 45 to the incident light at shown in Figure 5b. Lock the grating table. e) Unlock the spectrometer table and swing it through 135 so that the grating is perpendicular to the incident light and on the side furthest from the slit. See Figure 5c. f) Lock the spectrometer table and unlock the telescope. You are now ready to make measurements.

5 15 Jul 04 Hydrogen.5 Figure 5 5. The grating constant d is determined by using light of known wavelength, measuring the angle of diffraction, and using mλ = d sinθ. 6. In first order, measure the angular locations, φ 1 and φ 2, of the green line of mercury on each side of the incident direction. See Figure 6. θ = φ φ Figure 6

6 15 Jul 04 Hydrogen.6 Measure the locations of the violet 1, blue 3, and one of the yellow lines of mercury as well. Using the wavelength values given in the following table, calculate a d value for each line, then average your results to obtain the grating constant value that will be used for the remainder of the experiment. Mercury Spectrum Colour Wavelength (nm) Violet Violet Blue Blue Blue Blue-Green Blue-Green Green Green-Yellow Yellow Yellow Orange Orange Red Red Relative Intensity 7. Replace the mercury source with the hydrogen source. Measure the angular positions on each side of the incident direction, and calculate the wavelengths, in as many orders as possible for the H α, H β, H γ, and H δ hydrogen lines. Other lines may appear due to impurities in the source but these will be easily distinguishable from the hydrogen lines. The H α (red), H β (bluegreen), and H γ (violet) lines should be measurable in at least two orders, while the H δ (faint violet) will likely only be visible in first order. To facilitate observation of the spectral lines, turn off the room and hall lights and drape the black cloth over the spectrometer. 8. Compare the average experimental wavelength values for the H α, H β, H γ, and H δ lines with those calculated theoretically from the Bohr equation.