Atomic Spectra. d sin θ = mλ (1)

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1 Atomic Spectra Objectives: To measure the wavelengths of visible light emitted by atomic hydrogen and verify that the measured wavelengths obey the empirical Rydberg formula. To observe emission spectra and make careful measurements on one to identify an unknown molecule. Apparatus: diffraction grating, hand-held and precision grating spectrometers, spectral tubes, and power supply. Background: Light interference and diffraction gratings: When light passes through a large opening such as a window (wide compared to the wavelength of the light) it casts a bright "shadow" on the wall. But when the width of the opening gets much smaller (comparable to the wavelength) then the "shadow" becomes fuzzy without sharp edges. This phenomenon is called diffraction and is due to the wave nature of light; waves (of any nature) are able to bend around the edges of an object into the geometric shadow region. More specifically diffraction arises from the interference of waves from a source arriving at a point after traveling over different paths. If the difference in path length for the waves is an exact multiple number of wavelengths, then the waves are said to be inphase and will add to give a bright spot. This is called constructive interference. If the path difference is an odd multiple half wavelengths, then the waves are out-of-phase and destructively interfere, giving no light. Figure 1 shows light from a distant source impinging on two closely spaced narrow slits (spacing d). The light passing through the slits is diffracted as shown. The slits act as two light sources radiating in-phase. The two waves interfere when they overlap on the distant screen. The interference maxima occur when the path difference from the two sources S 1 and S 2 to a point P on the screen is an integer number of wavelengths (Fig. 2.) Mathematically, maxima occur when d sin θ = mλ (1) where m = 0, ±1, ±2,... and θ is defined in Fig. 2. We see from Fig. 1 that this theory predicts that the image of the slits on the screen will be a series of almost equally spaced, intense but fairly broad maxima. But now suppose that instead of two slits we had a large number of identical slits space d apart. What will the pattern on the screen look like? There will still be a series of maxima at the locations given by Eq. (1). The reason for this is easy to see. Consider m = 1. At the value of θ given by Eq. (1) the path difference for light from adjacent slits (separation d) will be λ, while for slits separated by 2d, 3d, etc. it will be 2λ, 3λ, etc. Thus the waves from the various sources will all be in phase and a maximum will occur. However, these maxima will be very sharp instead of broad as shown in Fig. 1. The reason for this is also easy to see. Suppose θ increases slightly from the value given by Eq. (1) so that the path difference for adjacent slits is 1.05 λ, then for slits separated by 2d, 3d,..., 10d will be 1.1λ, 1.15λ, λ. The waves will quickly get out of phase and no longer add up to give a maximum making the maximum much narrower. Spectra 1

2 Figure 1: Interference of Light Waves Passing Through Parallel Slits. A glass slide (or film) with a large number of parallel slits is called a diffraction grating. Because the maxima are so sharp for diffraction gratings, it is easy to accurately measure the angle θ and the wavelength can then be accurately calculated using Eq. (1). Thus diffraction gratings are frequently used to measure the wavelength of light. Figure 2: Approximation for Interference Maxima Atomic Spectra. We now want to investigate the inner structure of an atom. This is difficult to do; the unraveling of the structure of the atom was one of the major scientific achievements of the last century. The reason this is a difficult problem, of course, is that the atom is so small. We cannot take it apart like a watch and examine its parts and see how they fit together. Spectroscopy is the special tool we need to examine atomic structure. We can deduce the structure of an atom by exciting the atom and observing the Spectra 2

3 emission spectrum as it returns to its initial state. This may be compared to studying the structure of a watch by throwing it against the wall and studying the pieces that come out. The way we learn about the structure of an atom is by exciting it from its lowest energy state -- the ground state -- to an excited state, and then studying how the atom returns to the ground state by the emission of light. The wavelength of the emitted light is determined by the energy difference between the ground and excited states. Thus the emitted light occurs with very specific, sharp wavelengths that are characteristic of the atom; the emission spectrum of hydrogen is very different from that of say argon. Emission spectroscopy is a frequently used tool to identify elements. To use emission spectroscopy, you do not need a model of the atom; all you do is compare the pattern of emission lines for the unknown with that of the known elements. Figure 3: The Balmer Series for Hydrogen The simplest atom to study is the hydrogen atom, which consists of only one electron and one proton. Early spectroscopists didn't have a model for hydrogen (much less for more complex atoms) to use to interpret the spectra they saw, so they used an empirical formula that fit the data: = (2) λ n0 n where λ is measured in nm (=10-9 m) and the number is called the Rydberg constant. Equation (2) is called the Rydberg equation after the physicist who first showed it fit the data. n o and n are called the principal quantum numbers of the atom and are positive integers. Series of spectral lines, spectral series, are observed that are related by all having the same value of n o. Many of these series lie in the ultraviolet or infrared region and are more difficult to observe. We will study one series of visible lines, the Balmer series (named after Balmer who in 1885 first found an empirical formula that fit the data) which correspond to n o = 2. For this series n can have values n = 3, 4, 5,.... Four of these lines (wavelength given in nm) lie in the visible, as shown in Figs. 3. It wasn? until 1913 that Niels Bohr developed a model (using the new concepts of quantum mechanics) that was able to give a theoretical explanation of Eq. (2). In the Bohr model the hydrogen electron can orbit the positively charged nucleus only in specific orbits characterized by the principle quantum number n. Normally the electron moves in the n = 1 orbit closest to the nucleus where the electric force is strongest and the electron is most tightly bound to the atom. This is the lowest energy state or ground state of the atom. When the atom is excited, perhaps by collision with a free electron, the orbiting electron is raised to a higher, less tightly bound orbit; the atom is in an excited Spectra 3

4 state. The atom quickly returns to its ground state by emitting the extra energy as light. However, the orbiting electron does not have to drop immediately into the n = 1 orbit; it can drop into any lower orbit. The Balmer series, as shown in Fig. 4, is produced when electrons drop into the n = 2 orbit on the way to the ground state. Figure 4. Bohr Model of the Hydrogen Atom: the Balmer Series Even though Bohr was able to derive Eq. (2) and theoretically calculate the exact value of the Rydberg constant, his theory was unable to explain more complicated atoms and is now regarded as fundamentally wrong. This illustrates an important point about scientific theories -- theories can never be proven right, but they can be proven wrong! The best that one can say about a theory is that it explains all known data. Theories are useful as a way of categorizing and explaining experimental data and in giving guidance through predictions on what type of experiments would be important to do. But there is always the expectation that future more careful experiments, or experiments extending to unexplored regions may show that any existing theory is wrong (i.e. incomplete or makes predictions that disagree with experiment). Spectral (discharge) tubes: To generate emission spectra in the lab you will use several spectral tubes. These are glass tubes that are evacuated and filled with a small quantity of the element you want to study. When a very high (~5000 V) voltage is applied to electrodes at either end of the tube, a current flows through the tube and the free electrons are accelerated to high enough energy that they excite the gas to a variety of excited states. The emission spectra result when the excited atoms drop back into lower energy states and eventually the ground state. The fluorescent light bulb is a commonly used example of a discharge tube. The fluorescent bulb contains a drop of mercury that vaporizes and is excited by the electron current. The mercury emits a few visible lines, but also much ultraviolet. To convert the UV into visible light and also to create a less harsh "warmer" light, the inside of the tube is coated with a powder that absorbs the UV and reemits it at longer wavelengths. This process is termed fluorescence. By carefully choosing an appropriate mix of elements in the powder, the tube's output can be tailored to fairly closely resemble that of an incandescent bulb or the sun. Spectra 4

5 Figure 5: The Diffraction Grating Procedure: You will use a diffraction grating spectrometer, Figs. 5 and 6, to measure some of the wavelengths of the visible members of the hydrogen spectrum Balmer series and compare the results with the Rydberg equation, Eq. (2). The relationship between the wavelength λ, the measured deviation angle θ produced by the grating, and the spacing d between adjacent grating lines is given by Eq. (1). This implies that you may see a diffracted beam at more than one angle for a given wavelength or color (multiple orders of m), as long as sin λ < 1. (Note: be sure to keep clear the difference between the integers m, used in the grating equation, and the principal quantum number, n). Figure 6: The Optical Spectrometer The essential elements of the spectrometer are: * The slit entrance, which is pointed at the spectral lamp. The slit can be rotated and its width is adjustable: The narrower the slit, the sharper the resolution of the spectrometer. * A fixed diffraction grating. Most diffraction gratings for this experiment have 6000 lines per cm ==> 6 x 10 5 per meter. This implies that the slit separation distance is d = 1/(6x10 5 ) m = x 10-6 m = 1667 nm. * A telescope tube with an eyepiece fitted with a cross hair for pointing. This assembly rotates about the grating center. The angle with respect to the slit barrel can be read to 0.1 o with a Vernier scale. [The Sargent-Welch spectrometers are calibrated in wavelength, but for a different grating spacing than usually used. Therefore, the wavelength scale is not valid. You must use the measured angle θ to calculate wavelength from Eq. (1). ] Adjusting the Spectrometer: the hydrogen spectrum Spectra 5

6 a. First, pull out the telescope from the viewing end. Look through and adjust the eyepiece to focus on the cross-hairs. The eyepiece slides in and out of its draw-tube. Put the telescope assembly back into the spectrometer. Be careful not to disturb the focused eyepiece. b. Insert the hydrogen discharge tube into spring loaded tube holder. Do not touch the center part of the tube. By pushing down on a red button, turn on the power supply that applies a high voltage to the tube containing atomic hydrogen gas. Release the button when not in use. BE CAREFUL ABOUT ELECTRIC SHOCKS. c. Align the entire spectrometer in a perfect straight line. Look through the eyepiece and put the slit in the center of the field. Set θ = 0 o on the base plate. This is the m = 0 maximum which occurs at θ = 0 for all wavelengths; thus, you see a mix of all original colors. You may need to swivel the slit to get it upright and parallel with the spectral tube. To adjust the slit width, turn the screw on the front of the barrel. d. Increase θ by swinging the telescope to the left, and locate the angular position of the first four (red, green-blue, and two (?) violet) lines in the visible spectrum of hydrogen. Ignore the fainter lines, which are from contamination in the tube (water and air). You might start each line with the slit fairly wide open for easy location. e. If the cross hairs cannot be observed, center the slit image in the telescope eyepiece and average a few readings. Atomic Spectra of Other Elements: The spectrum of atomic hydrogen, an atom with just one electron, is simple, and the energy levels are given by the simple relation given in Eq. (2). Other elements with many orbiting electrons have more complicated atomic spectra; their electrons interact in complex ways. Nevertheless, the spectral line pattern is unique for each element. You will be given spectral tubes of "unknown" elements, helium (He), neon (Ne), or mercury (Hg), which you must identify. Measure the wavelengths of the lines in the emission spectrum of each. Identify each element by referring to the list of wavelengths of prominent spectral lines in the table below. Use your wavelength measurements, not the colors. Note that some lines will be brighter than others. Be careful when changing spectral tubes: they become hot. Do not hold the central region, hold them at the larger ends. Spectra 6

7 SPECTRAL LINE WAVELENGTHS The wavelengths are in nanometers. The colors are approximate and should not be used to identify the elements Neon 489, 496 (blue-green), 534 (green), 585 (yellow) 622, 633, 638, 640, (orange-red) 651, 660, 693 (red) Helium 447 (blue) 471 (turquoise) 492 (blue-green) 501 (green) 587 (yellow) 667(red) Mercury 404 (violet) 435 (blue) 491 (green) 546 (green) 576 (yellow) 579 (yellow) Spectra 7

8 Interference of Light and Optical Spectra Name Section Preliminary Questions (Not required to hand in): 1a. A hydrogen atom is excited into the n = 3 state. It quickly returns to its ground state, but not necessarily directly; it can stop momentarily at any of the intermediate states. List all possible photons that might be emitted while the atom de-excites and use the Rydberg formula to calculate the wavelengths of the longest and shortest wavelength photons. Show your work. 1b. Which of these photons will fall in the visible range, if any? 2a. A laser of unknown wavelength strikes a diffraction grating with 600 lines per mm. The grating produces a first order diffraction spot at θ = 20 o. What is the wavelength of the laser? 2b. How many diffraction spots will the laser produce? Include all orders, both positive and negative. [Hint: what is the largest possible value for sin θ?] Spectra 8

9 Interference of Light and Optical Spectra Name Section Partners Activity 1: The Balmer Spectrum of Hydrogen. Insert a hydrogen spectral tube into the power supply. Using the precision optical spectrometer, measure the angles θ at which first order lines are observed. Calculate λ (experimental) from Eq. (1):. [Use m = 1; the spectrometer grating has 600 slits per mm.] Identify the principal quantum numbers for the lines you observe. Calculate the ratios of the experimental λ and the wavelength calculated from the Rydberg equation, Eq. (2). Turn off the hydrogen lamp when not taking data. This will extend the life of these expensive lamps. VISIBLE HYDROGEN SPECTRUM Line color θ(deg) λ(exp) (nm) Transition: n n 0 λ(rydberg)(nm) λ(exp)/λ(rydberg) For one of the lines you and each of your partners should repeat the measurement twice. θ(deg) λ(exp) (nm) θ(deg) λ(exp) (nm) Average λ Standard Deviation nm = % Rydberg λ Discrepancy nm = % Discuss the meaning of these results. Can you conclude that your measurements have proved Bohr? theory of the atom? Explain. Spectra 9

10 Activity 2: Identification of an unknown element. Carefully measure the wavelengths of the emission lines from one unknown gas. Use the table in the procedure section to identify the gasses. Unknown # Element identification Line Color θ(deg) λ(exp) (nm) λ(accepted) (nm) λ(exp)/λ(th) Spectra 10