cation of the thin lens equation and the magnification equation to single lenses
systems will be investigated. In a later lab (The Telescope), these properties will ...
Chapter 6 Spectroscopy 6.1
Purpose
In the experiment atomic spectra will be investigated. The spectra of three know materials will be observed. The composition of an unknown material will be determined.
6.2
Introduction
Spectroscopy, or the study of the spectrum of light, is probably the most important tool in all of modern astrophysics. By analyzing the spectrum of astronomical objects, Earthbound science has been able to determine the temperature and chemical makeup of the stars, measure their velocities relative to our own, and even confirm that the Universe is expanding. In fact, it is the information extracted from the spectra of astronomical objects that gives the strongest and most incontrovertible evidence that the physical laws discovered on Earth also apply throughout the entire Universe. Without the use of spectral analysis, it is difficult to imagine that we could understand very much at all about the distant Universe. In this lab, you will make measurements of known spectra emitted by several hot dilute gases and use this information to calibrate a simple spectroscope. Then, this information will be used to identify the chemical composition of several unknown gases, some of which exist in the atmospheres of stars. As you work the lab, try to remember that what you are doing is no different in principle from what professional astronomers do when they measure the spectrum of a faint object using a 10-meter telescope and discover that the object is a cloud of hydrogen gas lying at the edge of the visible Universe. 45
6.2.1
Types of Spectra
The spectrum of light emitted (or absorbed) by an object may be broken down into three major types. The first is a continuum spectrum. This is a familiar spectrum—the type emitted by a hot metal such as the filament of a light bulb—and appears continuous like a rainbow when analyzed with a spectrograph. The second type is an emission spectrum, which is emitted by a hot dilute gas. This spectrum has sharp lines at specific wavelengths that correspond to discrete changes in the energies of the electrons bound to the atoms in the gas. The third type is an absorption spectrum. This spectrum can be thought of as the complement to the emission spectrum and consists of dark lines superposed on a continuum spectrum. They are produced when a continuum spectrum travels through a cool, dilute gas. For the same gas, the absorption and emission wavelengths are in correspondence. The emission lines correspond to the wavelengths at which the electrons lose energy and emit a photon of light. The absorption lines correspond to those wavelengths at which an electron absorbs a photon from the incident spectrum and gains energy.
6.2.2
Why Spectra Exist
In 1913, Niels Bohr proposed his Bohr model of the Hydrogen atom in which a negatively charged electron orbits the positive nucleus. The amount of energy that the electron has determines its orbit—lower energy results in an orbit closer to the nucleus, and higher energy results in an orbit farther from the nucleus. The lowest orbit is called the ground state. In a stable atom, electrons “want” to live in the ground state, as they want to have the lowest energy possible. Unlike planets orbiting a star, electrons can only reside in certain discrete energy levels, due to the quantum nature of the electron—energy only comes in packages. These allowed energy levels are unique to each element. For Hydrogen, the energy of an electron in the nth energy state1 is: En = −
13.6 eV n2
(6.1)
The unit of eV stands for electron-volt is a non-SI unit for energy. The SI unit of energy is the Joule (J ). We can convert from one to the other by using the conversion factor: 1 eV = 1.6 × 10−19 J. When a precise amount of energy is added to the atom, the electron jumps to the next energy level. But, it doesn’t want to be here, so it has to get rid of the extra energy. It does this by emitting a particle of light, called a photon. The energy of the photon, E = hf , is the 1
n = 1 corresponds to the ground state. n = 2 is the first excited state, etc.
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difference in energy between the energy levels. h is called Planck’s constant, 6.626 × 10−34 Joule-seconds. Figure 6.1 depicts what’s happening.
Figure 6.1: An electron in the Hydrogen atom jumps from the first excited state to the ground state; a photon is emitted. The energy difference between two energy levels, ni and nf is: ∆E = Ei − Ef 13.6 eV = − − n2i = (−13.6 eV)
13.6 eV − n2f ! 1 1 − 2 . n2i nf
!
Since ∆E = hf , we can determine the frequency of light that’s emitted when an electron falls from energy level ni to level nf : ! 1 1 − 2 hf = (−13.6 eV) n2i nf Dividing both sides by hc: f 1 −13.6 eV hf = = = hc c λ hc
1 1 − 2 2 ni nf
!
+13.6 eV = hc
1 1 − 2 2 nf ni
!
It can be shown that (13.6 eV)/hc = 1.097 × 10−2 nm−1 ≡ R, Rydberg’s constant. Thus, we’re left with: 1 =R λ
1 1 − 2 2 nf ni
!
(6.2)
The Bohr model fails to predict spectral lines when we introduce nuclei with more protons in the nucleus and thus more electrons around the nucleus. But the idea remains basically 47
the same: the electron energy levels of each element and isotope are unique! Thus, the wavelengths of emitted light are also unique, and each type of atom has its own distinct spectral fingerprint. The positions, widths, and intensities of spectroscopic lines are unique to each chemical element and reveal a wealth of information about the object that’s doing the emitting or absorbing. In this lab, you will be investigating emission spectra. Astronomically, emission spectra are very important in determining the pressure, density, and chemical composition at the surface of a star. Astronomers use spectra to classify stars. Emission spectra are also essential for measuring the red shift of a distant galaxy, in order to calculate its distance.
6.2.3
How the Spectroscope works
The tool that you’ll use to see spectra is the spectroscope. Light from the sample enters the spectroscope through a small slit in the scope (see Figure 6.2). The light reflects off of the collimator, which (as its name suggests) acts to collect the light onto the diffraction grating. When light is incident onto a diffraction grating, it is reflected in such a way as to break the light up into its component colors, which appear on the screen. We can then match each color up with its wavelength, λ. The effect is similar to the way a prism separates the colors in white light into a rainbow spectrum, although the physics is quite different.
Figure 6.2: The Spectroscope As you may know, when we see white light, we are actually seeing the superposition, or adding together, of the entire color spectrum. Now, each color in the spectrum has a different amount of energy, proportional to its frequency (E = hf ). If we think of a diffraction grating as a staircase, and light as a stream of tiny super-balls, then when the light hits the grating, the colors with higher energy will bounce “higher” than the colors with lower energy. Aside: Specifically, the sine of the angle of reflection, θ, is proportional to the wavelength, λ ∝ sin (θ). Also, λ is related to f as: λ · f = c, where c is the speed of light. 48
But you’ve already seen phenomena like this. Take an ordinary CD and hold the bottom side under a light source. You’ll see a really neat rainbow. In this lab, we will be examining optical emission lines - light in the frequency range in which our eyes are sensitive. Light of wavelength 700 nm appears red and light of wavelength 400 nm appears violet. The position of the line in the spectroscope depends on the line’s wavelength and hence its color. Note that some of the spectroscopes are calibrated in angstroms ( A). which is related to nanometers (nm) by 1 ˚ A= 10−10 m = 0.10 nm. Figure 6.3 shows the signature yellow spectral lines (minus the color) of the element Sodium. This is what you should see when you look through the spectroscope:
Figure 6.3: Spectrum of Sodium
6.3
Procedure
Special Cautions: • The light sources used in this lab use high voltage. Do not touch the tubes or the connections. We will first use the spectroscope by measuring the positions of emission lines of known wavelengths. Then we will examine the lines of an unlabeled sample and determine its spectral wavelengths. From the spectrum of the unknown sample, we will identify the sample from it spectrum.
6.3.1
The Experiment
1. Using your spectroscope, measure and record the spectral lines (λmeasured ) for each of the three known samples (Hydrogen, Sodium and Mercury). Record the measured value for each spectral line wavelength (λmeasured ) in a column of data Table 6.1 with the approximate color of each line. Some of the samples may have only a few spectral lines while others may have several. 2. Using the wall chart, record the actual (λexact ) wavelength for each of the lines for each samples in the data table. 3. After you have recorded the actual wavelengths (λexact ) for each of the known samples, record the difference (∆) between the measured (λmeasured ) and actual wavelengths(λexact ) 49
for each line of each sample. If several of the differences are large (greater than 10 nm) ask your TA to advise you on ways you might correct your data. 4. The spectrometer is a fairly complex instrument. To get a ’rough’ idea of the error in your measurements, take all of your difference (∆), square them, add them together, divide by the number of different ∆s and then take the square root. Show your calculation below and record the value:
’Rough’ error in your measurements
Hydrogen (H) Approx. Color λmeasured λexact ∆ (λmeasured - λexact ) Sodium (Na) Approx. Color λmeasured λexact ∆ (λmeasured - λexact ) Mercury (Hg) Approx. Color λmeasured λexact ∆ (λmeasured - λexact )
line 1
line 2
line 3
line 4
line 5
line 6
Table 6.1: Data table for measurements of H, Na and Hg
6.3.2
Determining the Identity of the Unknown Sample
Now we will look at an unknown sample. 1. Record the measured measured wavelengths (λmeasured ) for the unknown sample. Record them in the unknown sample data Table 6.2 along with their approximate color. 2. Compare the spectral line wavelengths to the wall chart. Determine the substance of the unknown sample and record the the name of the element below. Record the actual wavelengths (λexact ) of the unknown sample after you identified it. 50
Unknown Approx. Color λmeasured λexact Table 6.2: Data table for the Unknown sample What element is the unknown sample?
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6.3.3
Questions
1. What is the range of wavelengths for visible light?
2. How are frequency and wavelength related?
3. You should have found three spectral lines for Hydrogen. Using Equation 6.2, can you determine which electron transitions resulted in these lines? ( let nf = 2 and solve for ni )
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4. If energy were not quantized, and electrons were allowed to be at any radius away from the nucleus, how would an element’s spectrum be affected?
6.4
Conclusion
Write a conclusion about what you have learned. Include all relevant numbers you have measured with errors. Sources of error should also be included.
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