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Spectrum of Hydrogen

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Spectrum of Hydrogen SPECTRUM OF HYDROGEN When atoms in the gas phase are excited either electrically or by heating in a flame, they emit light of a characteristic color. If this light is dispersed (i.e. the various wavelengths are separated into different beams, by a prism or a diffraction grating), the spectrum of the sample can be studied. In the case of gaseous atoms, the spectrum consists of individual lines, which provide much information about the energy levels of the electron(s) in the atoms. Figures 7.12 and 14 in the Tro text provide examples of several atomic spectra. Hydrogen atoms give a particularly simple spectrum of lines occurring in the visible, ultraviolet and infrared regions. The wavelengths of these lines fit a mathematical pattern which was recognized by spectroscopists around 100 years ago. The Danish physicist Niels Bohr developed the first theory that was able to account for the wavelengths of these lines in 1913. According to Bohr’s model, the electron in the hydrogen atom follows a circular path or orbit, centered on the nucleus. Not all circular paths are permitted, only those of particular energies. Bohr developed a theory that gave an expression for the energy of an orbit as E = - 2.178 x 10-18 J n2 (eq 1) In this expression, n is a quantum number, and its allowed values are 1, 2, 3,.., ∞. Each value of n corresponds to a different orbit, or energy level. The orbit of smallest radius -18 has n = 1, has the lowest energy (-2.178 x 10 J), and hence is called the ground state of the atom. All other states are called excited states; unless the atom is somehow energized, the electron will not occupy one of these higher energy orbits. When n → ∞, the energy has its highest value, zero. This arbitrary zero of energy corresponds to the electron beginning at an infinite distance from the nucleus. In other words, the atom has been ionized. According to Bohr, when the electron changes from a higher energy (higher n) orbit to a lower one, the atom must lose energy which is emitted in the form of light. The energy of the photon is equal to the difference in energies between the initial and final states of the atom, and is related to the photon’s wavelength, λ, as follows: Ephoton = hc / λ = ΔEatom = Eupper − Elower (eq 2) Here h is Planck’s constant (6.626 x 10-34 Js) and c is the speed of light (2.998 x 108 m/s). The amount of energy in a photon given off when an atom makes a transition from one level to another is very small, on the order of 1 x 10-19 J. This is not surprising since atoms are tiny particles. To avoid such small numbers, we will work with one mole of 1 atoms. To do this we will multiply Equation 2 by Avogadro’s number. 23 23 23 ΔEatom = (6.02 x 10 )hc / λ = (6.02 x 10 )Eupper − (6.02 x 10 )Elower (eq 3) If the values of h and c are plugged in and the units converted to nanometers, we arrive at the following relationships: 5 ΔEatom = 1.19627 x 10 kJ/mole / λ (in nm units) = Efinal −Einitial (eq 4) 5 λ (in nm units) = 1.19627 x 10 kJ/mole / ΔEatom (in kJ/mol units) (eq 5) Equation 5 is useful in the interpretation of atomic spectra. Using equation 1 you can calculate, very accurately, the energy levels for hydrogen. Transitions between these levels give rise to the wavelengths in the atomic spectrum of hydrogen. These wavelengths are also known very accurately. Given both the wavelengths and the energy levels, it is possible to determine the actual levels associated with each wavelength. In this experiment your task will be to make determinations of this type for the observed wavelengths in the hydrogen atomic spectrum that are listed in Table 1. Table 1 Some Wavelengths (in nm) in the Spectrum of the Hydrogen Atom as Measured in a Vacuum Wavelength Assignment Wavelength Assignment Wavelength Assignment nhigh nlow nhigh nlow nhigh nlow 97.25 410.29 1005.2 102.57 434.17 1094.1 121.57 486.27 1282.2 389.02 656.47 1875.6 397.12 954.86 4052.3 Experimental Procedure: There are several ways we might analyze an atomic spectrum, given the energy levels of the atom involved. A simple and effective method is to calculate the wavelengths of some of the lines arising from transitions between some of the lower energy levels, and see if they match those that are observed. We shall use this method in our experiment. All the data are good to at least five significant figures, so by using electronic calculators you should be able to make very accurate determinations. A. Calculations of the Energy Levels of the Hydrogen Atom Given the expression for En in equation 1, it is possible to calculate the energy for each of the allowed levels of the H atom starting with n =1. Using your calculator, calculate the energy in kJ/mole of each of the 10 lowest levels of the H atom. Note that the energies are 2 all negative, so that the lowest energy will have the largest allowed negative value. Enter these values in the values of the energy levels, Table 2. On the energy level diagram provided, plot along the y-axis each of the six lowest energies, drawing a horizontal line at the allowed level and writing the value of the energy alongside the line near the y- axis. Write the quantum number associated with the level to the right of the line. B. Calculation of the Wavelengths of the Lines in the Hydrogen Spectrum The lines of the hydrogen spectrum all arise from jumps made by the atom from one energy level to another. The wavelengths in nm of these lines can be calculated by Equation 5, where ΔE is the difference in energy in kJ/mole between any two allowed levels. For example, to find the wavelength of the spectral line associated with a transition from the n = 2 level to the n = 1 level, calculate the difference, ΔE, between the energies of those two levels. Then substitute Δ E into Equation 5 to obtain the wavelength in nanometers. Using the procedure we have outlined, calculate the wavelengths in nm of all the lines we have indicated in Table 3. That is, calculate the wavelengths of all the lines that can arise from transitions between any two of the lowest levels of the H atom. Enter these values in Table 3. C. Assignment of Observed Lines in the Hydrogen Spectrum Calculate the wavelengths you have calculated with those listed in Table 1. If you have made those calculations properly, your wavelengths should match, within the error of your calculation, several of those that are observed. On the line opposite each wavelength in Table 1, write the quantum numbers of the upper and lower states for each line whose origin you can recognize by comparison of your calculated values with the observed values. On the energy level diagram, draw a vertical arrow pointing down (light is emitted ΔE < 0) between those pairs of levels that you associate with any of the observed wavelengths. By each arrow write the wavelength of the line originating from that transition. There are a few wavelengths in Table 1 that have not yet been calculated. Enter those wavelengths in Table 4. By assignments already made by an examination of the transitions you have marked on the diagram, deduce the quantum states that are likely to be associated with the as yet unassigned lines. This is most easily done by first calculating the value of ΔE, which is associated with a given wavelength. Then find two values of En whose difference is equal to ΔE. The quantum numbers for the two En states whose energy difference is ΔE will be the ones that are to be assigned to the given wavelength. When you have found nhigh and nlow for a wavelength, write them in Table 1 and Table 4; continue until all the lines in the table have been assigned. D. The Balmer Series This is the most famous series in the atomic spectrum of hydrogen. The lines in this series are the only ones in the spectrum that occur in the visible region. Your instructor has a hydrogen source tube and a spectroscope with which you should be able to observe some of the lines in the Balmer series. In the Data and Calculations section are some questions you should answer relating to this series. 3 Name ___________________________________ Data and Calculations: The Atomic Spectrum of Hydrogen A. The Energy Levels of the Hydrogen Atom Energies are to be calculated from equation 1 for the 10 lowest energy states. Table 2 Quantum Number, n Energy, En (kJ/mol) Quantum Number, n Energy, En (kJ/mol) 1 6 2 7 3 8 4 9 5 10 B. Calculation of Wavelengths in the Spectrum of the H Atom In the upper half of each box write ΔE, the difference in energy in kJ/mole between Enhigh and Enlow. In the lower half of the box, write λ in nm associated with that value of ΔE. Table 3 nhigh 6 5 4 3 2 nlow 1 2 3 4 5 C. Assignment of Wavelengths 1. As directed in the procedure, assign nhigh and nlow for each wavelength in table 1 which corresponds to a wavelength calculated in Table 3. 4 2.
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