PC1144 Physics IV Atomic Spectra 1 Objectives Investigate how well the visible light wavelengths of hydrogen predicted by the Bohr theory agree with experimental values. Determine an experimental value for the Rydberg constant from a fit of the measured values of hydrogen wavelengths to the form of the Balmer equation. Identify two unknown elements by examining their visible optical spectra. 2 Equipment List Spectrometer, diffraction grating in holder Hydrogen gas discharge tube, mercury discharge tube Two discharge tubes for unknown elements Power supply for the discharge tubes 3 Theory The spectrum from a hot gas of an element consists of discrete wavelengths that are characteristics of the element. In 1885, in an attempt to understand these spectra, Johann Balmer published an empirical relationship that described the visible spectrum of hydrogen. Although Balmer published the relationship in a somewhat different form, the modern equivalent is ( 1 1 λ = R H 2 1 ) 2 n 2 n = 3, 4, 5, 6,... (1) where R H = 1.097 10 7 m 1 is a constant called the Rydberg constant, λ stands for the wavelength and n is an integer that takes on successive values greater than 2. In 1913, Neils Bohr was able to derive the Balmer relationship by making a series of revolutionary postulates. The Bohr theory was historically of great importance in the developments that eventually led to modern quantum theory. In his attempts to explain the spectrum of hydrogen, Bohr was influenced by several recently developed theories. He incorporated concepts from the quantum theory of Max Planck, from the photon description of light by Albert Einstein and from the nuclear theory of the atom suggested by Ernest Rutherford s α-particle scattering from gold. Page 1 of 5
Atomic Spectra Page 2 of 5 The central ideas of Bohr s theory are contained in a series of four postulates that are stated below: 1. The electron moves in a circular orbits of radius r n around the nucleus under the influence of the Coulomb force between the negative electron and positive nucleus. 2. The electron of mass m can only have velocity v n and orbits r n that satisfy the relationship mr n v n = nh/2π where h = 6.626 10 34 Js and n = 1, 2, 3, 4,...,. 3. In an allowed orbit the electron does not radiate energy. The atom is stable in these orbits and this is called a stationary state. This postulate was a radical departure from classical physics. Classical electromagnetic theory predicts that an electron moving in a circle is accelerated and must radiate electromagnetic energy continuously. 4. The atom radiates energy only when an electron makes a transition from one allowed orbit to another allowed orbit. If E i and E f stand for the energies of the initial and final stationary states, the energy radiated by the atom is in the form of a photon of energy hf = E i E f where f is the frequency of the photon. With these postulates, it is possible to derive an expression for the energy of the stationary states. They are given by E n = ( me 4 8ɛ 2 0ch 3 ) 1 n 2 with n = 1, 2, 3, 4,..., (2) This expression for allowed energies can be used to obtain values for 1/λ predicted by the Bohr theory. The transitions that produce photons that correspond to the first four visible Balmer wavelengths are those from the states n = 3, 4, 5, 6 down to the n = 2 state. They are 1 λ = me4 8ɛ 2 0ch 3 ( 1 2 2 1 n 2 ) with n = 3, 4, 5, and 6 (3) Bohr showed that the value of the constant me 4 /8ɛ 2 0ch 3 was in excellent agreement with the value of the Rydberg constant in Balmer s formula. This is striking confirmation of the validity of the Bohr theory of hydrogen. The four wavelengths of the visible hydrogen spectrum that are easily seen and measured are also in excellent agreement with the first four wavelengths predicted by equation (3). In this experiment, the wavelengths will be measured with a diffraction grating spectrometer. Images of the slit for different wavelengths will appear in the first order at angles θ given by λ = d sin θ (4) Initially, the grating spacing d will be assumed to be unknown and the wavelength of mercury will be considered as known. Measurements of the angles at which the mercury spectrum occur can then be used to determine d. Using that value of d, measurement of the angles at which the hydrogen wavelengths occur will allow the determination of those wavelengths.
Atomic Spectra Page 3 of 5 4 Laboratory Work Part A: Mercury Spectrum In this part of the experiment, you will setup a diffraction grating spectrometer and determine experimentally the grating spacing d from the diffraction images of mercury spectrum (known wavelengths). Figure 1: Experimental arrangement for the diffraction grating spectrometer. A-1. The setup of the diffraction grating spectrometer is schematically shown in Figure 1. Remove the diffraction grating from the spectrometer. A-2. Adjust the eyepiece of the telescope, which is basically a magnifier, by sliding it in and out until the cross-wires are in focus. A-3. Position the spectrometer so that the telescope can be pointed at some distant object (e.g. building outside the window). A-4. View the distant object through the telescope and turn the focussing knob until the image is sharp. The telescope is then focussed for parallel light. A-5. Turn on the power supply for the mercury discharge tube and position the mercury discharge tube as close to the slit at the end of the collimator. Note: DO NOT touch the high voltage electrodes while the power supply is on. provides a voltage of 5000 V. A-6. Turn the telescope and collimator until they are in line with each other and the mercury A-7. Narrow down the slit and adjust the collimator until the image of the slit as seen through the telescope is sharply focused on the cross-wires. The image of the slit can be made vertical by rotating the slit. The collimator is then set to produce parallel light from the slit. It
Atomic Spectra Page 4 of 5 A-8. Align the cross hair on the telescope to the collimator when the telescope and collimator are aligned in straight line. Record the angular position of the telescope as θ 0 in Data Table 1. This will then be the reference angular position for the rest of the measurements. A-9. Place the diffraction grating it its holder on the turntable such that the plane of the grating is almost (as best as you can) perpendicular to the incoming rays. A-10. Rotate the telescope to the left or right until images of the spectral lines for mercury are located. The wavelengths of the mercury spectrum with the relative intensities in parentheses are: violet 4.047 10 7 m (1800), blue 4.358 10 7 m (4000), blue-green 4.916 10 7 m (80), green 5.461 10 7 m (1100), yellow 5.770 10 7 m (240), yellow 5.790 10 7 m (380). Rotate the telescope to the other side to be sure that all the lines can be located. This is just a preliminary check to be sure that all the lines are visible. It may not be possible to resolve the two yellow lines. If not, just assume one line at 5.780 10 7 m. A-11. Measure the angular positions of the brightest line of the mercury spectrum on each side (denote the two measured angles as θ 1 and θ 2 ). If θ 1 θ 0 and θ 2 θ 0 differ significantly, then rotate the diffraction grating such that their values are as close to each other as possible. In this way, the diffraction grating is considered to be perpendicular to the incoming rays. Note: It is extremely important that the grating is never moved after it is originally located. A-12. Measure and record the two angle positions for each of the diffraction lines of mercury spectrum in Data Table 1.
Atomic Spectra Page 5 of 5 Part B: Hydrogen Spectrum In this part of the experiment, four wavelengths of the visible hydrogen spectrum will first be determined using the diffraction grating spectrometer. An experimental value for the Rydberg s will then be determined from the linear least squares fit of the measured data. B-1. Without moving the diffraction grating, turn off the power supply for the mercury B-2. Turn on the power supply for the hydrogen discharge tube and place the hydrogen discharge tube as close to the slit as possible. Note: Again, be very careful not to touch the high voltage electrodes while making these adjustments in the position of the supply. B-3. Rotate the telescope back to the reference angular position θ 0 and carefully adjust the position of the hydrogen discharge tube until a sharp image of the slit is seen directly through the grating. Everything should be in focus from the mercury measurements and it should just be necessary to place the hydrogen discharge tube in the correct position to the give the brightest image. At all time be extremely careful not to move the grating. B-4. Carefully measure with as much precision as possible the angles at which the first four wavelengths of the visible hydrogen spectrum occur on both sides of the reference angular position. Record the two angles for each of the wavelengths in Data Table 2. Part C: Unknown Spectrums In this part of the experiment, diffraction patterns of two unknown sources will be obtained and its identify will then be identified. C-1. Without moving the diffraction grating, turn off the power supply for the hydrogen C-2. Turn on the power supply for the discharge tube labeled A (or B) and place the discharge tube as close to the slit as possible. C-3. Rotate the telescope back to the reference angular position θ 0 and carefully adjust the position of the discharge tube until a sharp image of the slit is seen directly through the grating. C-4. Carefully measure with as much precision as possible the angles at which those visible lines of the spectrum occur on both sides of the reference angular position. Repeat the measurement for at least FOUR visible lines of the spectrum. Record the colour as well as two angles for each of the wavelengths in Data Table 3 (or 4). C-5. Repeat the above steps for the discharge tube labeled B (or A). Last updated: Sunday 22 nd February, 2009 11:10pm (KHCM)