10.1 Introduction: The Bohr Hydrogen Atom

Transcription

1 Chapter 10 Experiment 8: The Spectral Nature of Light 10.1 Introduction: The Bohr Hydrogen Atom Historical Aside The 19 th century was a golden age of science. We could explain everything we observed. Near the turn of the century that began to change. The arrival of the microscope allowed us to see random motion of wheat grains. Light emitted from hot objects had spectrum different than expected; indeed the expected spectrum was unreasonable. Light emitted or absorbed by atomic gases had sharp lines instead of continuity. In this lab we will resolve some of the reasons for the confusion surrounding the state of science in nineteen hundred. These explanations involve accepting that light has a particle or quantum nature as well as the wave nature that our interference experiments have verified. These explanations involve an atomic nature for materials instead of the continuous, two charged fluid model suggested by Benjamin Franklin. These explanations even involve stable quantum states in atoms where the accelerating electron does not radiate EM waves as expected. An atom has a diameter of about m and is made up of a small (about m), heavy, and positively charged nucleus, surrounded by orbiting electrons. The nucleus consists of protons (positive electric charge) and neutrons (no electric charge) bound together by nuclear forces. In a neutral atom there are as many negatively charged electrons as there are protons. Each electron occupies one of the many possible well defined atomic orbit states. Each particular orbit state has an associated total energy (kinetic + potential). Electrons jumping from one orbit with less energy to one with more (binding) energy will release discrete quantities of energy in the form of light. The spectrum of quantized frequencies observed in the light emitted by excited atoms is a direct measurement of the allowed transitions of 147

2 -e me CHAPTER 10: EXPERIMENT 8 r e M Figure 10.1: Schematic force diagram for the nucleus and electron in a hydrogen atom. electrons among different orbits in the atom. The emission spectrum of a given atom is a unique signature of this atom. Even different isotopes in an element family have different spectra. Although the existence of discrete line spectra emitted by atoms had been known for a long time, it was Niels Bohr who first calculated the spectrum of the hydrogen atom. This is the simplest atomic system and consists of one proton as nucleus and one orbiting electron. The electron describes a circular orbit in the electric field of the proton; the force acting on the electron is e2 4πε 0 r = F v 2 2 e = m e a r = m e r. (10.1) From Equation (10.1) the kinetic energy of the electron is KE = 1 2 m e v 2 = e2 8πε 0 r. (10.2) The electrostatic potential due to the proton s positive charge at a distance r is V = e 4πε 0 r (10.3) and the potential energy, U e = ( e)v of the electron is e2 U e = 4πε 0 r = 2KE (10.4) 148

3 The total energy of the electron-proton system (of the atom) is Energy (ev) Series Limit Lyman Series Hydrogen Energy Levels Series Limit Balmer Series Paschen Series Series Limit Series Limit Brackett Series Figure 10.2: This figure shows an energy level diagram for hydrogen along with some of the possible transitions. An infinite number of energy levels are crowded between n = 6 and the escape energy n =. from Equation (10.2) and Equation (10.7) to find E = KE + U e = KE 2KE = e2 8πε 0 r (10.5) Bohr quantized the value of r by postulating that the electronic orbit must be a standing wave with wavelength λ, consequently the length of the circle describing the electron s trajectory must be the product of an integer and λ, 2πr n = nλ (10.6) where n = 1, 2, 3,... The French physicist Louis de Broglie postulated that one can assign a wavelength λ to any object having a momentum so that for the electron p = m e v; λ = h = h p m ev, where h is Planck s constant. Combining this with Equation (10.6) we see that 2π r n = nλ = nh m e v. (10.7) We eliminate the electron s velocity e 2 8πε 0 r = 1 2 m e v 2 = 1 ( ) 2 nh 2 m e = (nh)2 (10.8) 2π m e r 8π 2 m e r 2 and ( ) 1 e 2 r = π m e. (10.9) nh ε 0 Checkpoint What did Bohr postulate in calculating the energy spectrum of the hydrogen atom? 149

4 Table 10.1: Details about the photon interactions with hydrogen atoms for several lines. Most humans can see the first 3-4 Balmer lines. Series Name n l lower level n u upper level Wavelength (nm) Lyman Balmer Paschen Checkpoint What does quantization mean? Name five quantities that are quantized in nature. Now we substitute this into Equation (10.5) to determine the atom s energy to be E n = e2 8πε 0 r n = e2 8πε 0 ( ) e 2 π m e nh ε 0 = m e 2 ( e 2 where the ionization energy for hydrogen is found to be E I = m e 2 ( e 2 2 hε 0 2 hε 0 ) 2 1 n = E 1 2 I (10.10) n 2 ) 2 = ev. (10.11) The radius r n of the orbiting electron and the binding energy E n are quantized; they can assume only discrete values! The quantity n is called the principal quantum number. The 150

5 Figure 10.3: The spectrum of white light as viewed using a grating instrument like the one in this experiment. The different orders identified by the order number m, are shown separated vertically for clarity. In actuality they would overlap. radius r n increases as n 2. (See Equation (10.9).) The energy values are negative because we are calculating a binding energy; (one must supply energy in order to achieve an isolated electron and an isolated proton at rest). Figure 10.2 shows the energy of stationary states and their associated quantum numbers. The upper level marked n =, corresponds to a state in which the electron is completely removed from the atom; the atom is ionized and the ionization energy is required to achieve this when n = 1. Checkpoint Why does the binding energy of an atomic electron have a negative value? 10.2 The Apparatus An atom radiates when, after being excited, it makes a transition from one state with the principle quantum number n, and consequently with energy E n, to a state m with lower energy E m. The quantum energy carried by the photon is hν = E γ = E u E l = E I ( 1 n 2 l 1 n 2 u ). (10.12) 151

6 Figure 10.4: A sketch of our grating spectrometer. Our gas is ionized to provide the source of light. Lens A collimates the light into a beam the size of the grating. Lens B focuses the light onto the cross-hair (see inset) and the eyepiece allows us to see the spectral lines. The Vernier angular scale allows us to determine the wavelength of the line at the cross-hair. The wave speed, the wavelength, and the frequency of oscillation are related by c = λν, so we can find the wavelength of the emitted photon to be 1 λ = ν c = E ( γ 1 hc = Ry n 2 l 1 n 2 u ) (10.13) where we have defined the Rydberg to be Ry = E I hc = nm 1. Equation (10.13) is Rydberg s law and was the first relation that successfully described hydrogen s emission and absorption spectrum. In our experiment we will wish to verify Rydberg s law and to measure the Rydberg constant using our measured wavelengths, R y = ( λ 1 n 2 l 1 1 n 2 u ). (10.14) Historical Aside Rydberg discovered this law empirically simply by examining the emitted wavelengths some time before Bohr discovered this theoretical basis. 152

7 Hydrogen has thousands of spectral lines from the ultra-violet to high-frequency radio waves, but the only four which are visible to humans are the lowest energy Balmer series photons having n l = 2 in the de-excited atom and n u = 3, 4, 5, 6. The n u = 6 transition appears very dim and many students cannot see it; therefore we will only view the first three. We must keep in mind that Bohr s model above applies only to hydrogen and to hydrogenlike atoms. To understand more complex atoms we must incorporate the revolutionary theory of quantum mechanics. Even quantum mechanics applied to these complex systems cannot be solved exactly, but we have developed approximate solutions that agree quite well with experiment. Additionally, quantum mechanics can explain the subtle variations in hydrogen s spectrum that are visible in the data you will gather today. The Grating Spectrometer As already discussed in the fourth laboratory exercise, gratings are used to analyze the frequency spectrum of light. Light of a given wavelength, λ, will be deflected by a grating with a line spacing constant d, by an angle θ, according to the relation ±mλ = d sin θ with m = 0, 1,... (10.15) where ±m is the order number. Figure 10.3 shows how a grating spectrometer diffracts visible light in groups of intensity distributions as a function of θ, and each of these groups corresponds to a Figure 10.5: A sketch of the apparatus (side view) illustrating how to view the spectrum of a hot tungsten filament. different order m. Previously, our experience with diffraction gratings was limited to illumination by monochromatic lasers and we saw the spots into which the laser was diffracted. Previously we saw spots because we were using only one color (or wavelength, λ) of light. Today, we are illuminating the gratings with many colors simultaneously and each color will excite every order of the gratings. Because of this it is important that we not allow ourselves to become overwhelmed by what the spectrometer does with our light. Adjacent distributions do overlap at the larger orders; third order violet occurs for smaller θ than second order red. Figure 10.4 shows the spectrometer used for this experiment. It consists of a circular table with a diffraction grating at its center and an angular Vernier scale along its circumference. Two pipes are mounted on the table to keep out room light and these pipes intersect in the region of the grating. The mounting of the first pipe is permanent. It is designed to carry parallel light from an external source to shine upon the grating. The second pipe is mounted in such a way as to be able to pivot about the center of the circular table. This rotation makes it possible to separate the light, collecting only what has been diffracted through some chosen angel, θ. If this pipe is set to an angle such as 153

8 θ = 10, then whatever light the grating diffracts through an angle of 10 will travel down the length of the pipe. It will be focused by the adjustable lens (B) and the eyepiece. A pair of cross-hairs identify the center of the pipe and, together with the vernier scale located on the equipment, allow the accurate measurement of the angle, θ. Checkpoint Draw the path of two parallel light rays diverging from the spectrometer slit and entering the observer s eye. Checkpoint What is the physical significance of the order m of the spectrum produced by a grating? Figure 10.6: Focusing system of the eye. Most of the refraction occurs at the first surface, where the cornea C separates a liquid with index of refraction n 1 from the air. The lens L, which has a refractive index varying from 1.42 in the center to 1.37 at the edge, is focused by tension at the edges. The main volume contains a jelly like vitreous humour (n = 1.34). The image is formed on the retina. Illumination levels are controlled by the aperture of the iris. The grating constant d is engraved on the frame holding the grating. Slit S, in Figure 10.4 defines the light source; the slit is parallel to the lines of the grating so the diffracted light will form vertical lines instead of spots. Lens A is separated from the slit by a distance equal to its focal length, f, and consequently it projects a beam of parallel light on to the grating. Such a collimated beam utilizes most of the grating so that the spectra are quite sharp. Lens B focuses the light diffracted by the grating into a focal plane which contains the cross-hair. Both positions of lens B and the cross-hair can be adjusted in order to focus the image better. First focus the cross-hair and then focus the slit line. The optics of the human eye is shown schematically in Figure The optimum focal distance of our eyes is about 25 cm. In order to observe an image at a closer distance, a lens called an eyepiece may be used as indicated in Figure The spectrometer has an eyepiece with focal length of 2.5 cm. Each student will have to adjust the position of the of lens B to see a sharp image the spectral fringe. The placement of the cross-hairs should then be adjusted so that the two are in focus. A small light bulb controlled by a push-button switch allows the cross- 154

9 hairs to be illuminated. To determine the axis (θ = 0) of the spectrometer, place the table light in front of the spectrometer slit, S, remove the diffraction grating and look through the pivoting arm of the spectrometer directly at the light source. Adjust the position of the slit, of lens B, and of the cross-hair in order to see a sharp, white image (order m = 0). When a sharp vertical image of the slit is centered on the cross-hair, you can read the angle corresponding to the axis of the spectrometer; use the Vernier scale as shown in the appendix. All measurements of θ that you will make must be with respect to the spectrometer axis so subtract this θ 0 from all remaining angle measurements. Ga3 can do this for you if you simply incorporate the subtraction into the column formula. Don t forget to record the units and experimental error for your measurements. Figure 10.7: Simple lens used as a magnifier. An object at O close to the eye can be focused by the eye to appear to be much larger and at a more distant point O. Figure 10.8: A sketch of the apparatus (side view) illustrating how to view the spectrum of an ionized gas. The Light Sources In this laboratory you will observe the spectrum of one hot filament and four gases: H, He, Ne, and Hg. The light source consists of a glass tube which contains one of these gases at a pressure of about cm Hg. A 60 Hz voltage of a few kilovolts is applied across the gas in the tube via electrodes at each end. The non-uniform electric field between the two electrodes will excite the gas in the tube. An electron will remain in an excited state for a time on the order of 10 8 s before jumping down to a state with higher binding energy. The excited state, with a lifetime τ 10 8 s, has then decayed into a lower energy state. In order to prolong the usefulness of these tubes do not keep the light source on any longer than necessary. Checkpoint How might you excite an atom and how long does it stay excited? 155

10 The Spectrum of White Light The electromagnetic spectrum emitted by the hot filament of the table lamp is a continuous spectrum whose relative intensity depends upon its temperature. The shape of the spectrum of a black body is shown in Figure It was Planck who first calculated the shape of this spectrum. Planck introduced a constant h, now known as Planck s constant, to reproduce mathematically the observed distribution of this spectrum. Observe the visible part of the electromagnetic spectrum emitted by your incandescent lamp. Look at the spectrum associated with the order numbers m = 0, 1, and 2. Observe the different colors associated with different frequencies or wavelengths Watts/(m2 nm) Planck Law T=5000 K Rayleigh-Jeans Law T=3000 K Planck Law T=4000 K Planck Law, T=3000 K Wavelength (microns) Figure 10.9: Blackbody Radiation Spectrum. Planck s energy density of blackbody radiation at various temperatures as a function of wavelength. Note that the wavelength at which the curve is a maximum decreases with temperature. Measure the spectral sensitivity of your eye by observing the highest frequency your eye can discern. Use the order m = 1 to make these measurements. Measure the angle θ with respect to the spectrometer axis (don t forget to subtract off θ 0 ) and use Equation (10.15) to calculate the corresponding λ. Don t forget your units and errors. Now measure the lowest frequency your eye can see using the same method. Note the characteristics of the filament s spectrum in your Data. Are there holes in the spectrum or is it continuous? Humans can see 400 nm < λ < 700 nm so be sure your measured wavelengths are reasonable before continuing. Once you understand how to use the spectrometer, double-check your 156

11 knowledge by repeating the exercise on the other side of zero (for m = 1) The Hydrogen Spectrum Start Vernier Software s Ga3 graphical analysis program. A suitable Ga3 configuration file can be downloaded from the lab s website at The wavelength is calculated as you enter the order m and the diffracted angle θ using Equation (10.15) λ = d sin θ; since the grating specifies lines/inch, this becomes m 25.4 * 10^6 * sin( Angle - θ 0 ) / (lines * Order ) in Ga3 s calculation equation. Another calculated column Rydberg with units 1/nm and formula 1 / ( Wavelength * (1/4-1/ Upper ^2)) is your measured Rydberg constant and is plotted on the y-axis. Equation (10.14) has been used as this column s equation. Use the controls at the bottom left to tell Ga3 about your θ 0 and lines/inch. Observe and record as many spectral lines as you can see in the spectrum of hydrogen, using a hydrogen gas tube. Make sure that the light source is properly aligned with the spectrometer slit. At 0 the red and blue will merge into a bright sort of tan at proper alignment (the same color as the directly viewed hydrogen tube). Hold the table as you move the eyepiece or you will mess up your careful alignment and need to repeat it before continuing. Enter the angle into Ga3 (θ), the diffraction order (m), and the upper electron shell s principal quantum number (red n u = 3, cyan n u = 4, violet n u = 5). Compare the wavelength of the lines you observe with the value shown in Table Which series of lines do you observe? Table 10.1 also gives the principal quantum numbers involved in the observed transitions. Hints for measuring the wavelengths of spectral lines: 1. Align and center the light source to make the lines as bright as possible. 2. Optimize the focus for each measurement using lens B. Make sure the diffraction grating is aligned perpendicular to the incident, collimated beam (a peg on the slide s bottom should fit into a groove on the mount.) 3. Measure each color using all visible orders on both sides of the optical axis. Each order yields a measured value of the Rydberg constant. Note correlations between Rydberg constants gathered using one color and contrast these with constants gathered using other colors. 157

12 4. Use some intuition in matching the measured wavelengths to the chart of known lines. Even with careful measurement procedures your measured wavelengths may easily vary as much as a couple dozen nanometers from those in the chart. Also the chart only lists the brighter lines; other dim lines may also be present. Rescale your y-axis to be (0, nm -1 ) and decide whether the result is a constant plus experimental errors. If all of the points were taken from the same distribution, then it is reasonable to compute their statistics as described in Section 2.6. Ga3 can compute the statistics for you; do you remember how? Specify your best measurement of the Rydberg constant and its tolerance. Checkpoint Which one of the hydrogen series describes light in the visible spectrum? Checkpoint What is the difference between the electromagnetic spectrum produced by an incandescent object and the spectrum generated by excited atoms? The Spectrum of He, Ne, and Hg The He gas tube is labeled while the tubes with Ne and Hg are only marked with colored tape. Record the wavelength of 5 to 10 spectral lines of each one of these elements. Measure these wavelengths only once each rather than once for each visible order and note their units and errors. Notice that the spectrum of these gases is much more complicated, with many more lines than the one of hydrogen. This is due to the larger number of electrons orbiting the nuclei of these elements, the large number of electron-electron interactions, and consequently the larger number of possible transitions corresponding to spectral lines. Table 10.2 gives the wavelengths and the relative intensity of the line spectra of these elements. By comparing your observed values with those on the table you will be able to tell which gas is enclosed in a tube marked with a given color. Chemists find the spectroscopic uniqueness of atoms and molecules quite useful in identifying unknown substances Analysis Discuss your graph of measured Rydberg constants. What trends do you see in the graph? Bohr s model and quantum mechanics predicts that all of the Rydberg constants are the same; is it conceivable that your data agrees with this within your errors? Use the strategy in Section to compare your measurement of Ry to that measured by others on page

13 Table 10.2: Several known lines for hydrogen (H), helium (He), neon (Ne), and mercury (Hg). The units are nm and items marked with asterisks (*) are very dim. These may be used to identify these substances in chemicals when they are burned in a flame. Hydrogen Helium Neon Mercury Does your Rydberg constant agree with everyone else s? Disagree? How well do you know? What do you observe about the environment of the hydrogen atoms that might affect the electrons energies? What does this all mean versus Bohr s model? Now discuss the three more complex spectra. Take a few minutes and try to fit helium s spectrum to Bohr s model (Equation (10.10)). Do all of the lines fit or do we need a better theory? Does it seem like we might have better luck with neon or mercury? 10.4 Conclusions Is blackbody radiation fundamentally the same or different than gas discharge light? Is Bohr s model valid? Under what circumstances? Did you measure anything that you and/or others might find of value in the future? Remember to communicate using complete sentences and to define all symbols used in equations. Can you think of any applications for what we have observed? What improvements might you suggest? 159

14 Reference Sections of this write up were taken from: Bruno Gobi, Physics Laboratory: Third Quarter, Northwestern University 160

15 10.5 APPENDIX: The Vernier Principle A vernier consists of a movable scale next to a stationary scale. The movable scale is marked off in divisions that are slightly smaller than the divisions on the main scale. If you line up the zero mark of the vernier with the zero mark of the fixed scale (see Figure 10.10), you will notice that not all of the other divisions are aligned. However, if you examine the two scales closely, you should observe that the n th division of the movable scale exactly coincides with the n 1 division of the fixed scale. This implies that n 1 divisions of the main scale has the same length as n divisions on the vernier scale; so each vernier division is n 1 or 1 1 n n times the length of a main scale division. Determining the quantity 1 is an important first n step in making any measurement with a vernier; 1 is called the least count. n The least count gives an indication of the precision of the particular measuring device. For example, if we can read the main scale of our device to the nearest degree, and the device has a least count Figure 10.10: An illustration of the Vernier scale. The position of the movable 0 indicates the coarse reading on the fixed scale. One additional digit of precision can be obtained accurately from the Movable Scale digit that aligns best with a Fixed Scale number. of 1/4 (i.e. 4 vernier divisions is equal in length to 3 main scale divisions) then we can make measurements with an instrumentally limited error of ± 1/4 of a degree. Examine Figure What is n? What is the least count of this Vernier scale? If each division of the main scale corresponds to the one degree, what is the precision of the measurements we can make with this device? A well-designed instrument will have accuracy approximately equal to precision; the ability to provide faithful measurements (accuracy) should be about the same as the amount of information conveyed in a reading (precision). An instrument with too few displayed digits defaults to having round-off be its greatest experimental error. Contrarily, having too many displayed digits wastes money on useless precision and implies an accuracy which it is incapable of providing. Keeping this in mind, we should expect that the experimental error associated with an instrument is usually about the same as one or two least significant digits of its carefully read display. In the case of our spectrometers, we should expect 0.1 errors in our angle measurements with reasonable care and attention to detail. To read a measurement from a vernier such as the one in Figure 10.11, follow these steps: 1. Determine the least count of your device. 2. Look opposite the zero mark of the movable scale and read off what division of the fixed scale is at the zero mark. If the zero mark does not exactly align and is situated between two divisions, then read off the value of the lower of these two divisions. In Figure 10.11, we read 46 degrees. 161

16 3. Examine each mark along the movable scale and determine which division (from zero to n) is aligned exactly with a division of the main scale. Read off this number from the movable scale. In Figure 10.11, we can see that 3 on the movable scale is aligned with a division on the main scale. So we must add 3 least counts of 3/10 of a degree to the number read from the main scale earlier, 46. Thus our final answer is 46.3 degrees. If the zero mark (and simultaneously the n mark) of the movable scale had exactly coincided with a division of the main scale, as in Figure 10.10, then we would add zero least counts and arrive at an answer of 0.0 degrees. Figure 10.11: An example tutorial in reading the Vernier scale. The movable 0 is between the fixed 46 47, so the reading is 46.x. The x is determined by noting that the movable 3 matches the Fixed Scale numbers best (49 in this case); this makes x = 3 and our reading is We might also note that the movable 0 is indeed quite close to 46.3 to double-check our result. Historical Aside All of the above is brought to you courtesy of Pierre Vernier ( ), a French engineer who spent most of his life building fortifications, but who designed the Vernier scale in about