EXPERIMENT 13 Balmer Series of Hydrogen Any atomic gas or element (heated to vapor form) can be made to radiate light when suitably "excited" by an electric discharge, spark, or flame. If this light is viewed through a prism or diffraction grating, the spectrum consists of a series of discrete lines whose wavelengths are characteristic of the element concerned. The wavelengths that occur are directly related to the structure of the atom; in fact, our modern theory of atomic structure developed as a result of attempts to explain observed spectra. The purpose of this experiment is to gain insight into the structure of the simplest atom by analyzing the wavelength of the Balmer series of hydrogen. The results are then correlated to Bohr's theory of the hydrogen atom. Theory In 1885 J.J. Balmer succeeded in fitting a mathematical equation to the visible wavelengths of hydrogen, namely 1 1 = R 4 1 n 2 n = 3,4,5... (1) where R was an experimentally determined constant, called the Rydberg constant. The values of n corresponded to successive lines of the so-called Balmer series as shown in figure (1), where the symbols H, H, H, etc. are the traditional names given to the lines. series limit n=8 n=7 n=6 n=5 n=4 n=3 H ξ H ε H δ H γ H β H α Ultraviolet Violet Blue Blue-Green red Figure 1 The Balmer series of spectral lines for hydrogen. The wavelengths λ are given in nanometers. Although Balmer's equation accurately predicted the wavelengths, no one understood why. It was not until 1913 that Neils Bohr applied quantum ideas to develop a theory, which successfully explained the origin of the lines and showed how Balmer's equation could be derived theoretically. Bohr postulated that the electron, orbiting around
13-2 the nucleus, could only do so in certain discrete orbits. The energy E of the atom when the electron is in a stationary orbit with quantum number n is given by E = me4 Z 2 8εo 2 h 2 n 2 n = 1,2,3... (2) where m is the mass of the electron, e is the electronic charge, h is Planck's constant, is the permittivity of free space, Z is the atomic number of the atom. The energies specified by equation (2) can be represented by an energy-level diagram as shown in figure (2) where E 1 is the ground state of the electron and, E 2, E 3, E 4, etc. indicated excited states. Bohr showed that the observed spectral lines were caused by electrons making transitions from one energy level to another. He stated that if an electron in an initial energy state E i drops to a final lower state E f, the energy difference E = E i E f is radiated as a photon whose energy is given by Einstein's equation E = hf. Substituting equation (2) into E i - E f = h(c / ), Bohr obtained the following expression for the wavelength of the emitted radiation: o 1 λ = Z 2 4 me 8ε o 2 h 3 c 1 n 2-1 f n 2 i (3) n= E = 0 n=5 n=4 n=3 E E 5 4 E 3 This theoretical expression corresponds exactly to the form of Balmer's empirical equation if Z = 1 and n f = 2. The numerical value of the quantity n=2 H H H H 123 Balmer series E 2 me 4 8ε o 2 h 3 c agrees very closely with the experimental value of R. However, even better agreement can be obtained if the electron mass m in equation (3) is replaced by the "reduced mass" m* given by n=1 ground state E 1 Figure 2 Energy level diagram for the Balmer series of hydrogen. m* = mm (m + M) (4)
13-3 where M is the mass of the nucleus. This correction is necessary to account for the fact that the nucleus is not stationary but that both the nucleus and electron revolve about their common center of mass. Using the reduced mass m*, equation (3) can be rewritten as 1 / = Z 2 R(1 / n f 2 1 / n i 2 ) where the appropriate expression for R becomes: R = M m +M R R =- me 4 8ε o 2 h 3 c (5) The quantity R is simply a notation representing the value of R when M/m. For the case of hydrogen, the ratio M/m is about 1836, so that the value of R Hydrogen is slightly smaller than R. Today, the best available value of R is 1.0973732 x 10 7 m -1. As a final note, Bohr predicted the existence of other series in the hydrogen spectrum by substituting different values of n f in equation (3). These series, which were all later confirmed experimentally, are: Lyman series (n f = 1), Paschen series (n f = 3), Brackett series (n f = 4), and Pfund series (n f = 5). Furthermore, Bohr's theory for the hydrogen atom also works for other single-electron systems such as He +, Li ++, etc. each of which have a slightly different Rydberg constant. The downfall of the Bohr theory came when it could not give the correct energies for more complex atoms and this is where the need for modern quantum mechanics arose. Grating Spectrometer spectrometer table φ R slit θ R φ 0 (center) light source collimator θ L grating mount eyepiece telescope Figure 3 A schematic diagram of a grating spectrometer used to measure wavelengths of the light source.
13-4 In this experiment, a spectrometer mounted with a diffraction grating is used to measure the wavelengths of light (see figure 3). The spectrometer consists of a collimator which allows parallel light rays to pass through the grating, a telescope to view the resultant spectrum, and a rotatable table. If care is taken to ensure that the light enters the grating with normal incidence, the wavelength of a spectral line is given by the familiar relation mλ = d sinθ m = 0,1,2,3,.. (6) where d is the spacing of the "slits" in the grating, m is the order of the diffracted wavelength, and is the angle through which the wavelength has been deviated. A schematic diagram showing the arrangement for the first three orders of the hydrogen spectrum is illustrated in figure (1) below. m=2 m=1 Left m=0 m=1 Right m=2 Figure 4 Representation of 3 successive orders for the hydrogen spectrum. The theory of the diffraction grating is available in most texts for those who are not familiar with it. Procedure: Part I: Spectrometer Alignment When using the spectrometer, wear your glasses or contacts if you normally do so. In order that the spectrometer is properly set for use with this experiment, the following adjustments are necessary: 1. Focus the eyepiece so that the cross-hairs are clearly visible. 2. Focus the telescope for parallel light by viewing a distant object and obtain a clear image with no parallax relative to the cross-hairs. Parallax occurs if the image appears to move when you gently bob your head from side to side while sighting into the telescope. Note: Once the telescope has been set for parallel light, it should not be re-adjusted during the rest of the experiment.
13-5 3. Turn on the hydrogen lamp for about 5 minutes, and place it as close as possible to the entrance slit of the collimator. Line up the collimator slit and telescope cross hairs. This is done by centering the cross hairs on the image of the slit in the telescope. Adjust the collimator so that a sharp image of the slit is seen when sighting through the telescope. This is the zeroth order diffraction line. The accuracy of the spectrometer is limited by the size of the collimator slit and by the angular sensitivity of the spectrometer table. A narrow slit is necessary for precise measurements but it may give too faint an image. A slightly wider slit can be used and then narrowed prior to the final reading. Part II: Grating Alignment 1. Carefully mount the grating in the holder and rotate the spectrometer table so that the slide is perpendicular to the collimator axis. 2. To verify that the grating is perpendicular to the collimator axis first, measure the position of the zeroth order diffraction line 0 (m=0). Next rotate the telescope to the left and locate the first order red spectral line. Depending on the diffraction grating this rotation will be 15 to 20 degrees from the central (m=0) bright line. Record the angular position of the line L. Next record the angular position of the first order red line to the right of the central line R. Finally check that the diffracted angles to the left and right are equal within 0.1. If they are not equal then rotate the grating until they are equal within 0.1. Once the grating is aligned, tighten the table-clamp allowing the telescope to rotate freely. 3. If the spectral lines appear too faint, first adjust the lamp position and then, if necessary, increase the slit-size slightly using the thumb screw at the end of the collimator. If you have difficulty seeing 2 violet lines in hydrogen, check the alignment of the light source so that the spectral lines are of maximum brightness. Also blocking stray light or covering the spectrometer arrangement with a dark cloth is helpful. Part III: Hydrogen Spectrum measurements 1. Record the position for the zeroth order diffraction line 0 (m=0). 2. Locate as many of the first order spectral lines as possible to the left of the central line. Center the cross hairs of the telescope on each line and record the position calculate the corresponding diffracted angle., then 3. Repeat the measurements using the first order spectral lines located to the right of the central line.
13-6 Analysis 1. Using equation (6), calculate the wavelength λ ± δλ for one set of spectral lines. 1 1 2. Graph vs n. 2 3. Find the value for R and its uncertainty from the slope of the resulting graph. 4. Repeat steps (1) to (3) above using the second set of data. Questions: 1. What is the shortest wavelength in the Balmer series? Is it visible? 2. What is happening to the electron relative to the nucleus when E = 0? E > 0? E < 0? Refer to figure (2). 3. Bohr's theory helped in the discovery of the isotope deuterium, 2 H, by predicting a very small difference between the wavelengths of ordinary hydrogen and deuterium. How far apart should the H α line of these two kinds of hydrogen be? 4. If light incident on the grating makes an angle with respect to the normal to the grating, show that equation (6) becomes d[sin(θ ϕ) + sin ϕ] = mλ. 5. Derive equation (4) for m*. (Hint: The total angular momentum Iω = m*r 2 ω equals the sum of the individual angular momenta, where r = r e + r n ; r e, r n are the distances of the electron and nucleus respectively, from the center of mass.)