Laboratory Exercise. Quantum Mechanics

Transcription

1 Laboratory Exercise Quantum Mechanics Exercise 1 Atomic Spectrum of Hydrogen INTRODUCTION You have no doubt been exposed many times to the Bohr model of the atom. You may have even learned of the connection between this model and bright line spectra emitted by excited gases. In this experiment, you will take a closer look at the relationship between the observed wavelengths in the hydrogen spectrum and the energies involved when electrons undergo transitions between energy levels. Objectives In this experiment, you will Use a spectrometer to determine the wavelengths of the emission lines in the visible spectrum of excited hydrogen gas. Determine the energies of the photons corresponding to each of these wavelengths. Use a modified version of Balmer s equation to relate the photons energies to specific transitions between energy levels. Use your data and the values for the electron transitions to determine a value for Rydberg s constant for hydrogen. Materials Vernier LabQuest or computer with Logger Pro Spectrophotometer Spectrum Power Supply and Hydrogen Tube VIS-NIR Optical Fiber straight-line filament incandescent bulb and socket spectrometer Page 1 of 9

2 Pre-Lab 1. Place the incandescent bulb in the socket, plug it in, and turn on the electrical power to the bulb. View the spectrum through the diffraction grating.. Place the hydrogen discharge tube in the power supply socket. Turn on the power and observe the bright line spectrum of hydrogen through the diffraction grating. What are the differences in the spectra from these two sources? The appearance of bright line spectra for excited gases created a seemingly insoluble problem for physicists in the late 19 th century. They could not find a simple way to explain why only certain wavelengths were emitted by excited gases. In 1885, Johann Balmer, a Swiss high school mathematics teacher, found an empirical equation; nm m m where m was an integer greater than, that related the wavelengths of the lines in the visible spectrum of hydrogen. The spectral lines that Balmer described are now called the Balmer series. Three years later, Johannes Rydberg, a master of spectroscopy, rearranged Balmer s equation and expressed it in a more general form; 1 R 1 H n 1 1 n where 1/ is the wavenumber (the reciprocal of the wavelength expressed in m), RH is Rydberg s constant (1.097 x 10 7 m -1 ) and n1 and n are integers such that n1 < n. This equation led to the discovery of similar sets of spectral lines in the ultraviolet (Lyman series) and infrared (Paschen series) in the early part of the 0 th century. Nevertheless, it was not until Niels Bohr proposed his model of the hydrogen atom in 1911 that a causal explanation for the existence of the bright line spectra emerged. Bohr assumed that the electron circled the nucleus in certain well-defined orbits corresponding to specific energy states (see Figure 1 at right). In his model of the hydrogen atom, the electron can exist only in one of these energy states. Ordinarily, the electron exists in its lowest energy condition (called the ground state). So long as the electron is in a particular energy state, the atom does not emit light energy. However, when a hydrogen atom is given enough energy (via an inelastic collision or photon absorption), the electron is bumped up from its ground state (n = 1) to an excited state (n > 1). When the electron drops back to a lower energy state, a photon is usually emitted. The lines in the hydrogen spectrum represent various transitions made by electrons from higher to lower energy states. Figure 1 Further detail on the connection between the Bohr model and bright line spectra can be found by watching the 5 minute video, Quantized Electrons, posted on the physics web site. In this experiment you will analyze the visible bright line spectrum for hydrogen and use a variation of the Rydberg equation to relate the energy of the photons associated with each bright line to the energy levels in the Bohr model of the atom. Page of 9

3 Procedure 1. Connect the spectrometer to a USB port on the computer and choose New from the File menu. Connect the optical fiber cable to the spectrophotometer.. Set up the data-collection parameters. a. Choose Change Units Spectrometer Intensity from the Experiment menu. The software will measure the intensity in relative units. b. Next, choose Set Up Sensors Spectrometer from the Experiment menu. Change the data-collection duration to 40 ms. 3. Turn on the power to the hydrogen spectrum tube and aim the end of the optical fiber at the middle of the spectrum tube. Use the bracket that holds the end of the optical fiber in position. Note: Hydrogen tubes have a limited lifetime, so do not leave the tube on when you are not taking data. 4. Start data collection. If the peak for the red line (H-) of the spectrum saturates (flat, wide peak at an intensity value of 1.0), move the tip of the optical fiber slightly farther away. If this peak is too small, shift the position of the tip so that more light from the discharge tube enters it. When the intensity of this peak reaches a value of at least 0.8, stop data collection. You need to see at least four distinct peaks to perform this analysis. Depending on your hydrogen tube, you may need to collect two runs: one with no peak height above 0.9, and another with the two strongest peaks saturated at 1.0 so that you can detect the smaller peaks. You may see peaks from contamination as the tubes age. Ask Mr. Ballog which peaks are due to the hydrogen and which are due to contamination. 5. Once you have identified four peaks, save your experiment file. Turn off the hydrogen discharge tube. Analysis 1. Use the Examine tool to find the maximum intensity for the red line you observed in the H-spectrum. Record the wavelength. Repeat this step for the remaining peaks.. Use the value for the speed of light to calculate the frequency of each of these bright lines. Keep in mind that your wavelengths were measured in nm. 3. Using Planck's equation E = hf, calculate the energy of the light emitted for each of the 34 observed lines. Planck s constant is h J s. You will often find electron energy levels expressed in electron volts (ev; 1 ev = 1.60 x J). Record your values in the table below: Color Wavelength (nm) Frequency Hz (1/s) Δ Energy (J x ) red 4. The photon energies you calculated in Step 3 result from differences in the allowed energy states of the electron in hydrogen atoms, E = Efinal Einitial. While they provide a clue to the allowed energy states, these values alone are insufficient to determine the actual energy of Page 3 of 9

4 the allowed energy states. We can, however, make use of the fact that E = hf = hc/λ to obtain a form of the Rydberg equation that relates the energy of the emitted light to the initial and final energy states. E J n f n i Your task is to determine values of nf and ni that that will give light energies that approximate the values that you calculated in Step 3.One way to approach this task is to divide your calculated values of E by the constant, J, then match this fraction by substituting various pairs of integers in the =, then the expression 1 n 1 f n i portion of the equation. For example if nf = 1, and ni 1 n 1 f n 1 i Record your values and best guesses for n f and n i in a table like the one below Remember that n f will be smaller than n i since the electron drops down when releasing energy. Hint: n f and n i for these lines are both less than 8, also, transitions that end at n f of either 1 or 3 will give values outside of the visible spectrum (Search for the Lyman and Paschen series to find out more about these transitions). Color red E (J) Ratio E/constant reasonable values for nf ni 6. Using the wavelengths you obtained in your analysis of the hydrogen spectrum and the values of nf and ni corresponding to the transitions producing these lines, determine an average value for the Rydberg constant for hydrogen. The Rydberg equation is in the Pre-Lab Investigation. How does your experimental value compare to the accepted value for this constant? Report your % error along with the rest of this exercise on your web site. Visit the Visual Quantum Mechanics web site, and run the simulation for hydrogen spectroscopy. Note that a hydrogen gas discharge tube is lit and a spectrum appears at the top. Step-by-step directions are provided. See how closely your simulation spectrum matches the observed spectrum for hydrogen. Page 4 of 9

5 Exercise Determining Plank s Constant INTRODUCTION Quantum Mechanics began with a catastrophe. It was called the ultra-violet catastrophe. The problem had to do with a thought experiment concerning how energy would be reradiated from a totally absorptive black body. A black body that would absorb all radiation would always reradiate wavelengths that corresponded to the temperature of the box, not the wavelengths it absorbed. The problem was that as the black body gathered more energy the radiated energy would reradiate an increasingly more energetic wavelengths well beyond ultraviolet and eventually be radiating more energy that it was absorbing. This was a clear violation of the Law of Conservation of Energy. It was catastrophic for classical mechanics. Max Plank to the rescue with the idea that energy must come in discrete packets rather than continuous. Plank called those packets quanta. His idea led to the observation that there must be some smallest amount of energy that makes up a quanta. His equation fit the data and became the birth of quantum mechanics. The h in the equation is now one of the basic constants in physics, Plank s constant. Plank s constant can be experimentally derived using LED s. The amount of energy needed to just light an LED to its threshold can be used to determine the lowest energy state of an electron which is what Plank s constant is based upon. An LED is constructed using semiconductor technology where the transmitted electron moves from one side of the semiconductor material to the other side. Electrons cannot move freely from one side to the other but must reach a certain threshold energy level. Once the threshold amount of energy is given to individual electrons they move from one side of the semiconductor to the other. An LED is constructed such that the electrons are not conducted further in a circuit so they must give up their energy; they do so as photons of light. The wavelength of light corresponds to the energy required to pass the threshold value of the semi-conductor. By measuring the threshold value of several different LED s you will experimentally confirm Plank s constant. Procedure 1. Set up a circuit as shown at right. Some components may be preassembled or you may have to engineer the entire setup. Mr. Ballog is still developing this exercise and it will evolve over time. Whatever form it takes the basic circuitry is the same. Page 5 of 9

6 . Using Logger Pro, connect the spectrophotometer to a USB port on the computer and choose New from the File menu. Connect the optical fiber cable to your spectrophotometer. 3. Set up the data-collection parameters. a. Choose Change Units Spectrometer Intensity from the Experiment menu. The software will measure the intensity in relative units. b. Next, choose Set Up Sensors Spectrometer from the Experiment menu. Change the data-collection duration to 40 ms. 4. Power the red LED and measure its peak wavelength. Be careful not to burn the LED out with too much current. Each LED should be protected by a 10K resistor and powered by no more than 4 volts. Measure the peak wavelength of each LED. Select save previous run each time you measure a new LED. Record your measurements. LED Wavelength ʎ (nm) 5. Set up the measuring apparatus to test the Voltage and Amperage of each LED. Begin testing with the potentiometer turned so no voltage or current is recorded. Slowly open the potentiometer. You will first see a voltage reading without current flowing. At this point the power provided to each electron is not enough to allow it to move through the LED semiconductor. Continue to slowly turn up the voltage. Stop when the ammeter begins to display a current reading. At this point the electrons have exceeded the threshold value of the LED semiconductor. The threshold value is the activation voltage of the semiconductor, Va. As mentioned earlier the energy of each electron is given up as a photon of light as the electron drops back to a lower energy level. The relationship between Va and the energy of the emitted photon (Ep) is given by the equation; V a = E p/e + φ/c (#1) Where e is the charge on an individual electron (1.60 x coulombs) and φ/c is a constant associated with each LED. φ/c determines the amount of energy lost internally as heat in the electron transfer. The constant φ/c is unknown for the LEDs but can be assumed to be equivalent for each of the LEDs. In order to determine Ep we will measure the current of each LED at several voltages above Va and then use linear regression to determine where the activation energy of the particular LED. This works because above the actual Va the increase in voltage/current is due to the internal resistance of the LED alone and not the energy required to emit a photon. The LED becomes brighter due to an increase in the Page 6 of 9

7 number of electrons (more photons) not because of higher energy levels of each emitting electron. By plotting back to the x axis on an amperage vs. voltage graph you can determine the Va of the particular LED. Record the meter readings as you slowly increase the voltage of each LED. Analysis 6. Use LoggerPro to plot amperage (y) vs. voltage (x) of all of the wavelength (color) LEDs. Perform a linear regression and determine the x intercept of the regression line. Record the values as Va for that particular LED. Include the graphs and data table on your web site. LED Va 6. Using LoggerPro plot the Va data for each LED. Perform a linear regression and use the data obtained to derive your experimental value for Plank s constant. Using Plank s equation we can rearrange the Va equation given above (#1) into a usable linear equation Va = hc/e(1/λ) + φ/c (#) By plotting the Va (y axis) vs. 1/λ (x axis) of the different LEDs and performing a linear regression the slope (hc/e) of the Va equation above (#) may be found. Plank s constant (h) can then be obtained from the slope (m) as; m=hc/e h= me/c Determine the percent error in your observations by comparing your derived value with the accepted value for h (6.63 x Js). Include all graphs, data tables, and calculations on your web site. Page 7 of 9

8 LED Color Voltage Amperage LED Color Voltage Amperage LED Color Voltage Amperage LED Color Voltage Amperage Page 8 of 9

9 LED Color Voltage Amperage LED Color Voltage Amperage Page 9 of 9