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Atomic Spectroscopy: The H-Atom When atoms are excited, either in an electric discharge or with heat, they tend to give off light. The light is emitted only at certain wavelengths that are characteristic of the atoms in the sample. These wavelengths constitute what is called the atomic spectrum of the excited element and reveal much of the detailed information we have regarding the electronic structure of the atom. Atomic spectra are interpreted in terms of quantum theory. According to this theory, atoms can exist only in certain states, each of which has an associated fixed amount of energy that is just equal to the difference between the energies of the initial and final states. This energy may be absorbed or emitted in the form of light. The emission spectrum of an atom is obtained when atoms fall from higher to lower energy levels. Since there are many such levels, the atomic spectra of most elements are very complex. Light is absorbed or emitted by atoms in the form of photons, each of which has a specific amount of energy, E. This energy is related to the wavelength of light by the equation: he (1) where his Planck's constant, 6.62618 x 10 joule seconds, c is the speed of light, 2.997925 10 meters per second, and is the wavelength in meters. The energy,Ephoton is in joules, and is the energy given off by one atom as it jumps from a higher to a lower energy level. Since total energy is conserved, the change in energy of the atom, datom, must equal the energy of the photon emitted: Ephoton = Aatom = Ephoton where Aatom is equal to the energy in the upper level minus the energy in the lower one. Combining equations 1 and 2, we obtain the relation between the change in energy of the atom and the wavelength of light associated with that change: ΔΕ Ξ Aatom = Eupper Flower = £photon == (3) The amount of energy in a photon given off when an atom makes a transition from one level to another is very small, of the order 1 x 10 joules. This is not surprising since after all, atoms are very small particles. To avoid such small numbers, we will work with one mole of atoms, much as we do in dealing with energies involved in chemical reactions. To do this, we need only multiply Equation 3 by Avogadro's number, N. NAEatom = AE = NEupper - NElower Eupper-Elower = Substituting the values for N, h, and c, and expressing the wavelength in nanometers rather than meters, (1 m = 1 × 10° nanometers). Obtain an equation relating energy change in kilojoules per mole of atoms to the wavelength of photons associated with such change? ane Let (6.02205 5x 1023) (6.62618 x 10-js) (2.997925 x 10) (1 x 10'nm) mol 1m λ (in nm) (2) 1 kj 1000/
AE Eupper-Elower = 119627×10 male 4 (in mm) À (in nm) = mat Equation 4 is useful in the interpretation of atomic spectra. Say, for example, we study the atomic spectrum of sodium and find that the wavelength of the strong yellow line is 589.16 nm (see Figure 1). This line is known to result from a transition between two of the three lowest levels in the atom. The energies of these levels are shown in the figure. To make the determination of the levels which give rise to the 559.16 nm line, we note that there are three possible transitions shown by the downward arrows in the figure. We find the wavelengths associated with those transitions by first calculating AE (EE) for each transition. If we know AE, we can calculate & using Equation 4. Clearly, the II-I transition is the source of the yellow line in the spectrum. Energy in kJ/mole -187.931 -292.802 11-1 -495.849- lonization occurs- 11-1 → or En E-E-187.931-(-292.802) 104.871 kJ Am-- 1.19627/10/ AE (in = 1.19627 x 10 104.871 -1140.71 m E-E-187.931-(-495.849)-307.918 kJ 119627 x 10 307.918 1.19627 x 10 203047 Figure 1. Calculation of the wavelengths of spectral lines from energy levels of the sodium atom An <-388.50 m E₂-E,- -292.802-(-495.849)-203.047 J <-589.16 nm The simplest of all of the atomic spectra is that of the hydrogen atom. In 1856, Balmer showed that the lines in the spectrum of the hydrogen atom had wavelengths that could be expressed by a rather simple equation. Bohr, in 1913, explained the spectrum on a theoretical basis with his famous model of the hydrogen atom. According to Bohr's theory, the energies allowed to a hydrogen atom are given by the equation: (4) (5) where B is a constant predicted by the theory and n is an integer, 1, 2, 3,.., called a quantum number. It has been found that all of the lines in the atomic spectrum of hydrogen can be associated with energy levels in the atom, which are predicted with great accuracy by Bohr's equation. When we write Equation 5 in terms of a mole of H atoms and substitute the numerical value for B. we obtain: -1312.04 k n = 1,2,3, n² mot (6) Using Equation 6, you can calculate the energy levels for hydrogen. Thansitions between these levels gives rise to the wavelengths in the atomic spectrum of hydrogen. These wavelengths are also known very accurately.
ven both energy levels and the wavelengths, 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 97.25 389.02 434.17 397.12 486.27 410.29 656.47 102.57 121.57 954.86 1005.2 1094.1 1282.2 1875.6 4052.3 Experimental Procedure There are several ways we might analyze an atomic spectrum given the energy levels of the atoms 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. 1. Calculations of the Energy Levels of the Hydrogen Atom. Given the expression for Es in Equation 6, it is possible to calculate the energy for each of the allowed levels of the H atom starting with = 1. Calculate the energy in of each of the ten lowest levels of the Hatom (-1 ton-10). Note that all energies are KJ mat negative, so that the lowest energy will have the largest allowed negative value. a. Enter these values in your report sheet in Table 2. b. On the energy level diagram provided (Figure 2), plot each of the six lowest energy levels by drawing a horizontal line at the allowed level and writing the value of the energy alongside the line, near the y- axis. 2. Calculation of the Wavelength of the Lines in the Hydrogen Spectrum. The lines in the hydrogen spectrum all arise from jumps made by the atom from one energy level to another. The wavelengths in mm of the lines can be calculated by Equation 4, where AE is the difference in energy (in) between any two allowed levels. For example, from the n=2 level to the n=1 level, calculate the difference, AE, between the energies of those two levels. Then substitute 4E into Equation 4 to obtain this wavelength in nanometers. Using the procedure outlined above, calculate the wavelength (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 six lowest levels of the H atom. Enter these values in Table 3. 3. Assignment of Observed Lines in the Hydrogen Spectrum. Compare the wavelengths you have calculated in Table 3 with those in Table 1. If you have made your 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 4, write the quantum numbers of the upper and lower states for each line whose origin you can
ognize by comparison of your calculated values with the observed values. On the energy level diagram (Figure 2), draw a vertical arrow pointing downward (light is emitted, 4E<0) between those pairs of levels that you associate with any of the observed wavelengths in Table 4. By cach arrow, write the wavelength of the line originating from that transition There are a few wavelengths in Table 4 that have not yet been calculated. Enter those wavelengths in Table 5. By assignments already made and by 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 perhaps most easily done by first calculating the value of AE. which is associated with a given wavelength. Then find two values of E. whose difference is equal to JE. The quantum numbers for the two Es states whose energy difference is JE will be the ones that are to be assigned to the given wavelength. When you have found and for a wavelength, write them in Table 4 and Table 5; continue until all the lines in the table have been assigned. 4. 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 and near ultraviolet regions (-250-750 nm). Answer the questions in the report sheet relating to this series.
Figure 2. Energy Level Diagram Energy in kJ/mole -100 -200 -300 -400 -500 -600 -700 -800 -900 -1000 -1100 -1200 -1300 -1400 -1500