Physics 223 Name: Exercise 5: The electromagnetic spectrum and spectroscopy Objectives: Experience an example of a discovery exercise Predict and confirm the relationship between measured quantities Using different instruments to make the same measurements, and assessing each instrument s performance Equipment: Hydrogen lamp (caution: high voltage) Learning Technologies Project STAR Spectrometer ($42) Vernier Spectrovis spectrophotometer with Spectrovis Optical Fiber ($470) Vernier spectrophotometer (Ocean Optics-produced) ($1700) Laptop with LoggerPro software Introduction and relevant equations: In 1913, Niels Bohr published a paper titled On the Constitution of Atoms and Molecules in Philosophical Magazine. In the paper, he describes a model of the atom in which electrons orbited the central positively charged nucleus at certain prescribed distances, much like a model of the solar system. Even though this simple model was greatly modified over the next decade, the Bohr model set up the notion of quantization within the atom the existence of discrete energy levels that electrons in atoms were allowed to occupy. At once, this idea explained the existence of the discrete emission peaks of hot gases (as you will be observing in this lab) and other phenomena, previously unexplained by classical mechanics. Bohr reinterpreted Max Planck s original quantization of energy (E = h n), labeling each of the allowed energy levels with integers. The principal quantum number, n, is therefore labeled n 1, n 2, n 3 and so forth, with n 1 being the lowest allowed energy of an electron around an atom. The idea of quantized energies actually precedes Planck: the hydrogen emission spectrum was interesting to late 19 th century physicists precisely because it could not be explained by classical mechanics and Maxwell s electromagnetic wave equations. At a conference in 1888, Johannes Rydberg presented a formula that showed the proportionality of the energy of emissions from a heated hydrogen gas sample to the difference in the reciprocal of integers: E = R 1 n ' ( 1 n ( ( where E is the energy difference between electron orbits, R is a constant (now called the Rydberg constant) and n 1 and n 2 are different principal quantum numbers with n 1 < n 2. In analyzing the hydrogen lamp spectrum, Johann Balmer had stumbled on this relationship for the visible wavelength emissions of hydrogen in 1885; these correspond to n 1 = 2. As technology improved, Theodore Lyman was to discover the ultraviolet wavelength emissions of hydrogen in 1906 (n 1 = 1) and Friedrich Paschen did the same for the infrared wavelength emissions (n 1 = 3). 1
You will measure the wavelengths of emission of a hydrogen lamp with three different spectrometers. From those measurements, you will calculate R, and give an estimate of its uncertainty. 1. Given the Rydberg formula, and E = hn and c = nl, derive an expression that connects the wavelength of a hydrogen atom emission with the difference in the reciprocal of quantum numbers squared. 2. Describe what data need to be plotted on a graph, such that the slope equals R. Watch out for that factor hc. Keep in mind that, for the part of the electromagnetic spectrum you are studying, n 1 = 2. Procedure (work in teams of two or three): Obtain a laptop, the Vernier Spectrovis spectrophotometer with Spectrovis Optical Fiber (in a tub), the Project STAR Spectrometer (the blue triangular plastic thing) and a hydrogen lamp. Plug in the hydrogen lamp, and make sure it works. Due to the limited lifespan of the hydrogen lamp bulb, please turn off the hydrogen lamp when not in use. If starting with the Project STAR Spectrometer, hold it horizontally with the Project STAR Spectrometer label on top. Look through the eyepiece (narrow end of the triangle) and aim the slit (right side of the view through the eyepiece) at the hydrogen lamp. There should be four different color vertical lines visible on the left side of the view. Read the wavelength of each line (in nanometers); record these in the data section. Get an estimate of the uncertainty in the wavelength (in other words, what ± can you read the line wavelength to?). If starting with the Vernier Spectrovis spectrophotometer with Spectrovis Optical Fiber, plug the USB cable attached to the spectrophotometer into the laptop and open the LoggerPro software. 2
There should be a brightly-colored window (arranged from red to violet) that is the background to an absorption vs. wavelength graph. Insert the Optical Fiber into the Spectrovis spectrophotometer, and make sure that the white triangles printed on the side of the box on the fiber and the cavity in the spectrophotometer line up. Then, on the LoggerPro menu, go to Experiment Change Units Spectrometer 1 Intensity, which will change the y-axis to the proper units. Point the fiber tip at the hydrogen lamp, turn on the lamp, click Collect, then view the spectrum. You can stop the data collection at any point when you have a reasonable graph that shows emission peaks. You can use the mouse to determine the wavelength of each peak; record these in the data section. Get an estimate of the uncertainty in the wavelength (in other words, what ± can you read the peak wavelength to?). When you have a chance, obtain the Vernier Ocean Optics spectrophotometer from me (there s only one, so the teams will take turns). Plug the USB cable attached to the spectrophotometer into the laptop and use the LoggerPro software. There should be a brightly-colored window (arranged from red to violet) that is the background to an absorption vs. wavelength graph. Then, on the LoggerPro menu, go to Experiment Change Units Spectrometer 1 Intensity, which will change the y-axis to the proper units. Point the fiber tip at the hydrogen lamp, turn on the lamp, click Collect, then view the spectrum. You can stop the data collection at any point when you have a reasonable graph that shows emission peaks. You can use the mouse to determine the wavelength of each peak; record these in the data section. Get an estimate of the uncertainty in the wavelength (in other words, what ± can you read the peak wavelength to?). Data Table of Measured Wavelengths 1 (nm) 2 (nm) 3 (nm) 4 (nm) Uncertainty (nm) Project STAR Spectrometer Vernier Spectrovis Vernier Ocean Optics Analysis On a separate piece of graph paper, plot the graph suggested by your answer to question 2. The axes will have odd labels and odder units, but recall that the slope is the goal. Each spectrophotometer should have its own graph (or its own line, anyway, overlaid on the same set of axes). Determine R for each spectrophotometer, and record in the table below. 3
Determine the uncertainty in R for each spectrophotometer, and record it in the table below. While not necessarily needing a full GUM approach, elements of the GUM protocol may be helpful to you here. Don t forget to match precision. Table of Rydberg Constants TVSpec spectrometer Vernier Spectrovis Vernier Ocean Optics R u(r) Show a sample calculation of u(r): Results, discussion and conclusion 3. Wikipedia gives the Rydberg constant as 1.097303 10 7 m 1. Calculate the percent error for each of the three spectrophotometers. 4
4. Which spectrophotometer did the best job of (a) seeing all of the hydrogen visible wavelength emission lines clearly and (b) giving the best value of the Rydberg constant? The idealized hydrogen spectrum is available on the wall chart at the front of the room. Note the word best includes the ease of the spectrometer s setup and use, the percent error comparison, and the size of the uncertainty! 5. Describe at least one strength and one weakness for each spectrophotometer (and its corresponding software), which helped you decide how to answer the question above. You may also take into account the cost of the hardware (given in the equipment section) as a plus or a minus. 5