Determining the Critical Potentials for Helium: The Franck-Hertz Experiment

Transcription

1 Determining the Critical Potentials for Helium: The Franck-Hertz Experiment Trent H. Stein, Michele L. Stover, and David A. Dixon Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, AL, Introduction: Sketch of Franck-Hertz Apparatus The model for the atom developed over a number of years on the basis of key experiments and insights. In 1897, J. J. Thomson showed that cathode rays are what we now call electrons and measured the charge to mass ratio of the electron by using crossed electric and magnetic fields. He showed that the mass of the electron was small, more than 1000 times smaller than the hydrogen atom. For this work, Thomson received the Nobel Prize in Physics in E. Rutherford shot alpha particles at a gold foil and showed that only a few back-scattered. This led to the nuclear model of the atom in Rutherford had already won the Nobel Prize in Chemistry in 1908 for his studies of the disintegration of the elements and the chemistry of radioactive processes. Building off of this work, Niels Bohr introduced his model of the hydrogen atom in 1913 with the energy of the states of the atom quantized. He used classical mechanics and electrostatics with the key idea of quantizing the angular momentum. He predicted that the electrons in atoms can only exist in certain bound states (energy levels). In 1914, J. Franck and G. Hertz confirmed the Bohr model for atoms that electrons only occupy discrete quantized energy levels and made the first non-optical measurement of the quantum nature of atoms. The experiment involved sending a beam of electrons though mercury vapor and observing the loss of kinetic energy when an electron struck a mercury atom and excited it 1

2 from its lowest energy state to a higher one. This occurred at 4.9 ev and all electrons with at least this amount of energy would lose only 4.9 ev showing the quantum nature of atoms. There were already hints of this in the solar spectrum and in the emission of light from atoms heated up in a Bunsen burner but this was the first proof of this. In a second paper in May 1914, Franck and Hertz then showed that the light emitted from a collision of the electrons with mercury atoms was exactly at wavelengths corresponding to 4.9 ev which showed the relationship between wavelength and energy as well as that between absorption and emission from excited atomic states. Remember that they did not have today s light sources so they used an electron whose energy they could precisely control as the excitation source. Franck and Hertz were awarded the Nobel Prize in Physics in 1925 for their discovery of the laws governing the impact of an electron upon an atom. Bohr s derivation Potential Energy (P.E.) of two charges = q 1 q 2 /4πε 0 r with ε 0 = x C 2 J -1 So for a nucleus with charge Z interacting with 1 electron, (r = electron-proton distance, e = charge on the electron, h = Planck s constant, F = force, L = angular momentum) P.E. = -Ze 2 /4πε 0 r Total E = ½(m e v 2 ) - Ze 2 /4πε 0 r F = m e a and F(coulomb) = dv/dr = Ze 2 /4πε 0 r 2 For uniform centrifugal motion: a = v 2 /r so we can write F = Ze 2 /4πε 0 r 2 = m e v 2 /r (1) Bohr hypothesizes that only discrete levels are present for an electron orbiting the nucleus so quantize angular momentum so L = m e vr = nh/2π with n a positive integer (fit n intervals into 2π = 360º, a circle) n = 1, 2, 3,.. L = m e vr = nh/2π (2) This condition was later reinterpreted by de Broglie to imply that the electron existed in a standing wave pattern with n full wavelengths along the orbit. We now have 2 equations in 2 unknowns: V and r Solve (2) for v = nh/2πm e r and substitute into (1) Ze 2 /4πε 0 r 2 = m e n 2 h 2 /r(2πm e r) 2 = n 2 h 2 /r(2π) 2 m e r 3 r n = n 2 h 2 4πε 0 /4π 2 Ze 2 m e = n 2 h 2 ε 0 /πze 2 m e = (n 2 /Z)a 0 2

3 With a 0 = Bohr radius = h 2 ε 0 /πe 2 m e = Å Then v n = Ze 2 /2ε 0 nh Substitute back into total E expression to get: E n = -Z 2 e 4 m e /8n 2 h 2 ε 0 2 = x J (Z 2 /n 2 ) = ev (Z 2 /n 2 ) or we can write E n = -Z 2 e 2 /2a 0 n 2 where we express the energy in units of Rydbergs with 1 R H = x J = ev = 0.5 a.u. (atomic units) What is the experiment? One implication of Bohr s assumptions is that the energy can only take on certain discrete values. In particular, if the electron s lowest possible energy is E o, then the other available energies for the H atom are: Eo Eo Eo Eo,,,, (3) are the available energy levels for the electron. The value for E o in the Bohr model is 13.6eV for the H atom as noted above. It is difficult to measure the energy of an electron orbiting an atom directly. One can eject the electron if a light source with enough energy is available but detecting the photoelectrons is hard and one has to have the atoms in the gas phase. One can measure the absorption of light but many atoms do not readily absorb in the visible. One has to get the atoms in the gas phase and have an intense light source plus be able to measure the light absorption. One can measure the energy emitted in the form of light, when an electron drops from a higher energy (higher n) state to one with lower energy (lower n). Conservation of energy says that the amount of energy emitted in such a transition is simply E emitted E atom = E f E i (4) 1 E o 2 n 1 2 f n, (5) i where n i and n f are integers corresponding to the electron s initial and final states, respectively. Because of hydrogen s simple atomic structure (a nucleus plus one electron), Bohr s model applies directly. 3

4 The helium transitions you will be investigating are different from the hydrogen spectrum in at least two important respects: 1. Helium is a two-electron atom. Each electron must therefore interact not only with the nucleus, but also with the other electron. This renders the energy spectrum of helium much more complicated than that of hydrogen. The energy levels cannot be calculated accurately without a complete quantum-mechanical treatment, which is quite difficult to do even though there are only two electrons. (One has to accurately predict the interaction of two electron probability densities as the electrons are not really point charges.) 2. You will be inducing transitions in the helium with incident electrons, rather than looking at spontaneous transitions that emit photons as with hydrogen. This means that the selection rule ( l = ±1) that applied to the spontaneous transitions may no longer apply because that rule is a consequence of the fact that photons have 1 unit of spin angular momentum and angular momentum is conserved. An energy level diagram for helium is shown in Figure 1. The singlet states are on the left and the triplet states are on the right. Note that "singlet" states are those in which the two helium electrons have opposite spin, while "triplet" states are those whose electrons have the same spin. 1s5 3 s 1s4 3 s 1s3 3 s 1s5 3 p 1s4 3 p 1s3 3 p 1s5 3 d 1s4 3 d 1s3 3 d 1s2 3 p 1s2 3 s Figure 1. Helium energy level diagram showing the electron configuration. Spectroscopic term and energy above the ground state of the first few energy levels of helium. The horizontal dashed line indicates the ionization potential. 4

5 Experiment: Basic idea: In a tube that has been evacuated and then filled with helium, free electrons are accelerated by a voltage V A to form a divergent beam passing through a space at a constant potential. To prevent the walls of the tube from becoming charged, the inner surface is coated with a conducting material and connected to the anode A (see Figure 2). In the tube, there is a ring-shaped collector electrode R, through which the divergent beam can pass without touching it, even though the ring is at a slightly higher potential. A small current I R, with a value in the order of picoamperes (10-12 amps), is measured at the collector ring, and is found to depend on the accelerating voltage V A. It shows characteristic maxima, which are caused by the fact that the electrons can undergo inelastic collisions with helium atoms during their passage through the tube and excite the He atom into electronically excites states. The kinetic energy E of an electron is as follows: E = e V A where e is the elementary electron charge. If this energy corresponds exactly to a critical potential of the helium atom (an excited state), all of the kinetic energy may be transferred to the helium atom. In this instance the electron can then be attracted and collected by the collector ring, thus contributing to an increased collector current I R. As the accelerating voltage is increased, successively higher levels of the helium atom can be excited, until finally the kinetic energy of the electron is enough to ionize the helium atom. As the accelerating voltage is increased further, the collector current shows a steady increase. Figure 2. Schematic diagram of critical potential tube 5

6 Safety Instructions: The Critical Potential tube is a hot cathode tube. Treat them carefully. Do not subject the tube or leads to any tension or mechanical stresses. If voltage or current is too high, or the cathode is at the wrong temperature, it can lead to the tube becoming destroyed. Do not exceed the stated operating parameters. Equiment: Critical Potential Tube (helium) Battery Unit (with AA battery) Grounded Shield Tube Holder Control Unit (with charger) DC Power Supply 7 Experiment Leads (connectors) Multimeter LabQuest 2 (with charger) 2 Differential Voltage Probes USB connector DC Power Supply Grounded Shield Critical Potential Tube Tube Holder Control Unit LabQuest Battery Unit Multimeter Differential Voltage Probes Figure 3. Franck-Hertz Apparatus 6

7 Procedure Step A: Using the LabQuest 1) The Franck-Hertz apparatus should have already been set up by your TA. It should look like Figure 3. 2) In this experiment, you will only be adjusting the voltage and current on the power supply and the minimum and maximum accelerating voltage on the control unit. 3) You will NOT be adjusting any connectors, chargers, or the tube itself. Follow the instructions carefully to avoid damage to the equipment. The pins on the tube are very fragile and the tube is under vacuum. 4) Turn on the LabQuest. The screen should display 2 different potentials (red and blue). 5) To the right of the screen, you will see a small grey box with mode, rate, and duration (see figure to the right). Use the stylus (located on the back of the LabQuest) to tap on the box. 6) Set the Mode to Time based, the Rate to samples/s, and the Duration to 0.1 s. (NOTE: The interval setting sets itself after the rate is set). Then tap OK. 7) You should now be back at the screen with the two potentials. Make sure that your rate and duration have changed. (NOTE: If they did not, repeat step 6.) 8) In the top right corner, tap on the box with the graph. 9) To collect a set of data, press the play button or tap on the green arrow at the bottom left of the screen. 10) After you collect your data, save it by tapping the file cabinet in the upper right corner. 11) To switch between runs, tap on the button directly to the left of the file cabinet that says run #. Then tap on the particular run you would like to see. Procedure Step B: Setting the Accelerating Voltage (V A ) 1) Twist the dial on the multimeter clockwise to 200m in the V DC section. The display should turn on and read 0.00 mv (NOTE: There is a 200 (V) setting and a 200m (V) setting.) 2) Since the leads on the multimeter have fixed pin tips, you will have to hold them in place to make sure they are getting good metal-to-metal connection to take measurements. 7

8 3) To adjust the maximum V A, connect the COM input (black lead) on the multimeter to the ground ( ) for the output and the Voltage input (red lead) on the multimeter to #3 for the output on the control unit (see Figure 4). COM V (Max) V (Min) knob 4) Make sure that the leads are touching the metal sides of the probe holes and slowly turn the knob #3 clockwise or counter clockwise to increase or decrease the maximum V A respectively (see figure 4). Figure 4. Control Unit knob 5) Adjust the maximum V A to be somewhere between 20 and 30 mv. 6) To adjust the minimum V A, leave the COM input on the multimeter in the ground ( ) for the output and move the Voltage input on the multimeter from #3 to #4 for the output on the control unit (see Figure 4). 7) Hold the leads in place and slowly turn the knob #4 to adjust the minimum V A to be somewhere between 10 and 20 mv (see figure 4). Note: Do not set the minimum and maximum V A s equal to each other. Procedure Step C: Data Collection 1) On the power supply, make sure that all 4 of the knobs (labeled current and voltage) are turned off (to their counter-clockwise limit). Do not force the knobs, they will stop turning at this limit. Then, press the power button on the power supply. 2) The top two knobs are the coarse (on the right) and fine (on the left) knobs for current whereas the bottom two knobs are the coarse and fine knobs for voltage (NOTE: the fine and coarse knobs are used for small and large adjustments respectively.) 3) You should see two values displayed on the power supply. Both should be approximately zero. Voltage is on the left and current is on the right. 4) On your power supply, in the upper right corner of the screen, you will see a red CC. This stands for constant current. 5) Slowly turn the coarse knob for the current clockwise until the red CC disappears and a green CV appears at the bottom of the screen. This means you are in constant voltage. 6) Next, slowly turn the coarse knob for the voltage clockwise until the green CV disappears and the red CC reappears. (NOTE: If your voltage gets to 4 V and the red light has not 8

9 turned on, STOP increasing the voltage. Raise your hand, and your TA will come check your apparatus.) 7) Repeat steps 5 and 6 until your voltage is ~ 3 to 4.5 Volts and current is ~ 1 to 1.3 Amps. 8) At this point, your bulb should be lit up (see Figure 3). (NOTE: Do not move the bulb or the stand to do this.) 9) On your LabQuest, tap play and collect your first data set which corresponds to the first ionization energy. Your data should look similar to Figure 5. (NOTE: Unlike Figure 5, your time is set to 0.1 s (which is twice as long as Figure 5), so you will either see pieces of or a complete duplicate set of curves. Ignore these.) 10) Does your data look like a smooth single line (Figure 5) or does it jagged and distorted? Figure 5. 1 st ionization 11) Slightly adjust your minimum and maximum V A on the control unit and your current and voltage on the power supply within the ranges provided in this step. a. Minimum V A : mv b. Maximum V A : mv 12) Save your first run on the LabQuest. Then press play again. 13) Continue to make adjustments until you get data that looks like Figure 5. (NOTE: You do not have to save all of the bad runs). 14) Save the best run on the LabQuest. Record which run is your best run on the next page under the 1 st ionization data section. Also record the minimum and maximum V A s from the control unit along with the voltage and current from the power supply. (NOTE: Don t forget units) c. Voltage: V d. Current: Amps 15) It is also possible to get two complete sets of curves in a single acceleration that correspond to the 1 st and 2 nd ionization energies (see figure 6). Figure 6. 1 st and 2 nd ionization 16) To achieve this result, adjust the minimum and maximum V A on the control unit and the current and voltage on the power supply within the ranges provided in this step. a. Minimum V A : mv b. Maximum V A : mv 9 c. Voltage: V d. Current: Amps

10 17) Tap play on the LabQuest. Your data should look similar to Figure 6. (NOTE: Remember you are running for twice as long as the figure, so you will see a duplicate.) 18) Slightly adjust your minimum and maximum V A on the control unit and your current and voltage on the power supply until you get a smooth single line that looks like figure 6 (see step 16 for the ranges). NOTE: You do not have to save all of the bad runs 19) Save the best run on the LabQuest and record which run is your best on this page under the 1 st and 2 nd ionization data section. Also record the minimum and maximum V A s from the control unit along with the voltage and current from the power supply. (NOTE: Don t forget the units) 20) On the LabQuest, tap File (in the upper left-hand corner of the screen) and then Save. Choose a name for your data collection and tap Save/OK. (NOTE: This will save all of the runs that you have collected regardless of which one is currently on the screen). 21) Unplug the two differential voltage probes from the LabQuest. 22) Slowly adjust the voltage and current down on the power supply until the knobs are turned off (to their counter-clockwise limit). Then press the power button. 23) At this point, you are done with your apparatus. Inform your TA and move on to the Data Analysis Section. Data: 1 st ionization (step 14) Best Run Number Minimum V A Voltage Maximum V A Current 1 st and 2 nd ionization (step 19) Best Run Number Minimum V A Voltage Maximum V A Current 10

11 Data Analysis: 1. Use the USB cord provided to connect the LabQuest to the computer in the lab. If the computer begins to install software/drivers/etc., wait until the installation is complete before moving on to the next step. 2. Open Logger Pro. Click on File, go the LabQuest Browser, and click Open. 3. Click the file name you choose in step 20 in the previous section and click Open. (NOTE: A prompt may appear. If it does, click continue without data collection.) 4. To the right of the screen, you should see your graphs superimposed on one another. To the left, you should see a table with all of your data. Your data will be separated into runs that contain 3 columns each (see figure below). 5. Use the scroll at the bottom of the screen to see your best run for your 1 st ionization data set on the previous page (NOTE: You may have to expand the table window to see all 3 columns for the run at once). 6. Click. The entire column should be selected. While holding down the Shift key, click the other two column titles ( and ). At this point, all of the data for the run should be selected. 7. Copy and paste the data into an Excel spread sheet. Insert a line above the data and add back your column titles as they did not transfer over with the data. 11

12 Potential (V) Potential (V) 8. Follow steps 5 7 to copy the data from your best run for the 1 st and 2 nd ionization data set on the previous page into the spreadsheet. (NOTE: Use a blank column or a line to separate your two runs on your Excel sheet) 9. Save your spreadsheet and a copy of it to you and your lab partner. 10. From this point forward, the rest of the lab report can be completed outside of lab. If you need help using Microsoft Excel, ask your TA before you leave lab. This lab requires you use Excel to do multiple graphs and calculations. Attach your four graphs to this report a) Give each of your graphs a title, label your axes (including units), and label your lines if you have more than one on a single graph b) Graphs should be scatter plots with smooth lines and markers (see figure to right) 11. Plot a graph of Time (x-axis) vs. Potential (y-axis) for each of your two runs. NOTE: Each graph should have 2 lines as you have two different potentials (see figures below). Also you may need to manually adjust the ranges on the axes to better see the data. 2 1 st ionization 3 1 st and 2 nd ionization Time (s) Time (s) 12. At this point, you are done with your 1 st and 2 nd ionization data set (step 19). 13. Copy and paste all 3 columns of your data from the 1 st ionization data set (step 14) onto a second tab in your Excel spreadsheet. 14. Delete the potential column that does not have the curves. 15. Additionally delete any partial data you may have. You only need one set of ionization curves (see figure on next page). 12

13 I R (pa) I R (pa) 16. Use the equation to calculate the collector current in picoamps (pa). (NOTE: You will need to do this for all of your voltages. You should have around 400.) 17. Plot a graph of Time (x-axis) vs. I R (y-axis). (see figure to the right) 1600 Time vs I R 18. Identify the time of the tallest peak. You can do this by hovering the mouse over the peak. Record this time in your spreadsheet as t 1. (see figure to the right) NOTE: Make sure you record the time of the peak and NOT the current of the peak Identify the time of the ionization Time (s) threshold, the point where the line begins to increase before it drops to zero. Record this time in your spreadsheet as t 2. (see figure above) NOTE: You may want to make your graph larger and/or adjust the values of the x-axis to make this easier to see t 1 t Use the equation to calculate the energy in ev at each of your times, t (you should have about 400 of them). (NOTE: t 1 and t 2 are constants you determined in steps 18 and 19) 21. Plot a graph of I R (y-axis) vs. Energy (x-axis). (see figure to the right) 22. Use the chart on page 4 to identify the energy levels of each peak. 23. Label each peak. (NOTE: Use the terms to the left of the energies in the chart, minus the 1s, as your labels. Example: 4p or 3 3 s) I R vs E E (ev) 13

14 Questions: 1) How do these measurements support the ideas of quantum mechanics? 2) What are some possible sources of error in this experiment? 3) How does this experiment compare with spectroscopy of the hydrogen atom? 14