R LEP 5.1.07 Balmer series / Determination of Rydberg’s constant Related topics Diffraction image of a diffraction grating; visible spectral range; single electron atom; Bohr’s atomic model; Lyman, Paschen, Brackett and Pfund Series; energy level; Planck’s constant; binding energy. Principle and task The spectral lines of hydrogen and mercury are examined by means of a diffraction grating. The known spectral lines of Hg are used to determine the grating constant. The wave lengths of the visible lines of the Balmer series of H are measured. Equipment Spectrum tube, hydrogen Spectrum tube, mercury Holders for spectral tubes, 1 pair Cover tube for spectral tubes Connecting cord, 50 KV, 1000 mm Object holder, 535 cm Diffraction grating, 600 lines/mm High voltage supply unit, 0-10 kV Insulating support Tripod base -PASSBarrel base -PASSSupport rod -PASS-, square, l 400 mm Right angle clamp -PASSStand tube Meter scale, demo, l = 1000 mm 06665.00 06664.00 06674.00 06675.00 07367.00 08041.00 08546.00 13670.93 06020.00 02002.55 02006.55 02026.55 02040.55 02060.00 03001.00 1 1 1 1 2 1 1 1 2 1 1 1 3 1 1 Cursors, 1 pair Measuring tape, l = 2 m 02201.00 09936.00 1 1 Problems 1. Determination of the diffraction grating constant by means of the Hg spectrum. 2. Determination of the visible lines of the Balmer series in the H spectrum, of Rydberg’s constant and of the energy levels. Set-up and procedure The experimental set-up is shown in Fig. 1. Hydrogen or mercury spectral tubes connected to the high voltage power supply unit are used as a source of radiation. The power supply is adjusted to about 5 kV. The scale is attached directly behind the spectral tube in order to minimize parallax errors. The diffraction grating should be set up at about 50 cm and at the same height as the spectral tube. The grating must be aligned so as to be parallel to the scale. The luminous capillary tube is observed through the grating. The room is darkened to the point where it is still possible to read the scale. The distance 2 l between spectral lines of the same color in the right and left first order spectra are read without moving one’s head. The distance d between the scale and the grating is also measured. Three lines are clearly visible in the Hg spectrum. The grating constant g is determined by means of the wavelengths given in Table 1. Rydberg’s constant, and thus the energy levels in hydrogen, are determined from the measured wavelengths by means of Balmer’s formula. Fig. 1: Experimental set-up to determine the spectral lines of the hydrogen atom. PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany 25107 1 R LEP 5.1.07 Balmer series / Determination of Rydberg’s constant Fig. 2: Diffraction at the grating. 2l screen spectral tube 2. Hydrogen spectrum Due to collision ionization, H2 is converted to atomic hydrogen in the spectral tube. Electrons from the H atoms are exited to higher energy levels through collisions with electrons. When they return to lower energy levels, the atoms emit light of frequency f given by the energy difference of the concerned states: DE = h · f (3) where h is Planck’s constant. Applying Bohr’s atomic model, the energy En of a permitted electron orbit is given by: d sin a = a l En = – !êêêê d2 + l2 1 e4me 1 8 «20 h2 n2 n = 1,2,3… (4) where «0 = 8.8542 · 10-34 As/Vm is the electric field constant, e = 1.6021 · 10-19 C is the electronic charge and me = 9.1091 · 10-31 kg is the mass of the electron at rest. The emitted light can therefore have the following frequencies: grating fnm = 1 n, m = 1,2,3… (5) If the wave number N = l-1 is used instead of the frequency f, substituting c = l · f one obtains: N = Ryth 1 1n 2 – eye where Ryth = Theory and evaluation 1. Diffraction grating If light of wavelength l impinges on a grating with constant g, it is diffracted. Intensity peaks occur when the angle of diffraction a fulfills the following condition: n · l = g · sin a ; n = 0, 1, 2, ... n=` 0 – 0.9 BrackettSeries – 1.5 H« PaschenSeries Hd energy level In the examples given in Table 1, the average obtained for the three measurements of the grating constant is g = 1.672 µm. 1 e4me = 1.097 · 107 m-1 8 «20 h3 c eV (1) (2) Ed 2 + l 2 (6) Here Ryth is Rydberg’s constant, which follows from Bohr’s atomic model. Light is collected by the eye on the retina, therefore the light source is seen in the color of the observed spectral line on the scale in the prolongation of the light beams. For the diffraction of the nth order, the following relation is deduced from the geometrical structure (Fig. 2): l 2 1 m2 n=4 n=3 Hg Hb Ha – 3.4 BalmerSeries n=2 ionization energy 13.6 eV a n·l=g· 2 1 e4me 1 1 – 8 «20 h3 n2 m2 Tab. 1: Determination of the grating constant from the wavelengths of the Hg spectrum 2 Color l / nm 2 l / mm g / µm yellow green blue 578.0 546.1 434.8 330 311 244 1.680 1.672 1.661 25107 – 13.6 LymanSeries n=1 Fig. 3: Energy level diagram of the H atom. PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany R LEP 5.1.07 Balmer series / Determination of Rydberg’s constant n = 1 : Lyman series Spectral range: n = 2 : Balmer series Spectral range: n = 3 : Paschen series Spectral range: n = 4 : Bracket series Spectral range: n = 5 : Pfund series Spectral range: Notices ultraviolet ultraviolet till red infrared infrared infrared Fig. 3 shows the energy level diagram and the spectral series of the H atom. For m R `, one obtains the limits of the series; the associated energy is thus the ionization energy (or the binding energy) for an electron in the nth permitted orbit. The binding energy can be calculated by means of the equation: En = –Ryth · h · c - Next to the atomic hydrogen spectrum, the molecular H2 band spectrum may be observed if the room is sufficiently darkened. The numerous lines, which are very close to each other, are due to the oscillations of the molecule. - The Hd line is situated on the border of the visible spectral range and is too weak to be observed by simple methods. - The treatment of more complex atoms requires quantum mechanics. In this case, the energies of the states are determined by the eigenvalues of the hamiltonian of the atom. For atoms similar to hydrogen, calculations yield the same results as Bohr’s atomic model. 1 n2 where c = 2.99795 · 108 m/s and h = 6.6256 · 10-34 J s = 4.13567 · 10-15 eV s. The ground state is found to be 13.6 eV. Tab. 2: Examples of measurements for the H spectrum (Balmer series) Distance d = 500 mm Line 2l lexp llit Ha Hb Hg Hd 384 mm 275 mm 243 mm – 656 nm 489 nm 436 nm – 656.28 486.13 434.05 410.17 Ryexp nm nm nm nm 1.097 · 107 m-1 1.093 · 107 m-1 1.092 · 107 m-1 – average: Ryexp = 1.094 · 107 m-1 PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany 25107 3