In the Laboratory
The Ionization Energy of Helium M. J. Kaufman* and C. G. Trowbridge Department of Chemistry, Emory University, Atlanta, GA 30322
Introduction and Theory
Experimental Procedures
The first allowed absorption transition of helium (to 1s2p 1P) lies at 171,129 cm᎑1 (58.4 nm) in the vacuum UV and cannot easily be observed in an undergraduate laboratory. However, if students are given the frequency of this transition, they can combine it with measurements on the visible emission spectrum of helium to estimate the ionization energy of this two-electron atom. The extrapolation to the ionization limit takes advantage of the fact that some of the observed transitions can be analyzed quite accurately by the Bohr equation, which treats the excited levels as one-electron systems. Since two-electron excitations of helium lie above the ionization level, only singly excited levels appear in its spectrum. Similarly to hydrogen, the visible emissions of helium terminate on the n = 2 level. The helium spectrum is more complicated, however, since in a two-electron system there is splitting between s, p, d, … configurations, due to differing penetration of the outer electron into the 1s orbital. In addition, further complications are introduced by the presence of both singlet and triplet levels, as can be seen in an energylevel diagram for the helium atom (1). Nevertheless, some transitions are well described by one-electron theory (i.e., the Bohr equation), in which, for a multi-electron system, the concept of an effective nuclear charge, Zeff, is incorporated.
The experiment is performed in the undergraduate physical chemistry laboratory, using a Gaertner hand-tuned spectrograph with visual observation of emission lines. Source lamps and power supplies are purchased from Edmund Scientific Co. Measurements are best made with minimal room lighting to avoid contaminating the readings with emissions from Hg lines emitted by fluorescent lamps. The spectrograph has good precision, but poor accuracy, so it is calibrated with observations of the atomic hydrogen spectrum. The wavenumbers of the Balmer hydrogen emissions are calculated with the Bohr equation, and a linear calibration of the spectrograph is prepared. We use Mathcad to generate the calibration and derive its statistics. If the hydrogen 6 → 2 transition at 410.2 nm is observed, the four observation points can also be used to prepare a quadratic calibration whose variance is compared with the linear calibration. The chosen calibration is then used to convert the observed wavelengths of the helium lines (usually 9 are observed) to more accurate values.
ωn =
εn – ε2 2 = R He Z eff 1 – 1 2 2 hc 2 n
(1)
where ω n is the wavenumber of the transition, εn and ε2 are the energies of the upper and lower levels of the transition, n is the principal quantum number of the excited electron in the upper state, h is Planck’s constant, and c is the velocity of light. The Rydberg constant for helium is
µ R He = R H µHe = 109,722 cm᎑1 H and RH is that for hydrogen, with µHe and µH being the reduced masses of these atoms. The transitions expected to best follow the one-electron theory are those for which, in both the upper and lower state, the excited electron does not appreciably penetrate into the electron cloud of the 1s electron. For an n = 2 electron, this will be for the excited electron in the 2p orbital, which has a node at the origin. The upper states of the allowed 1snd → 1s2p transitions are extremely one-electron-like, because of their Rydberg nature and because the wave function and its first derivative vanish at the origin for a d wave function. Singlet transitions should be more one-electron-like than triplet transitions, since triplet-state wave functions must be spatially antisymmetric (2). *Email:
[email protected].
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Results and Discussion Students are told to analyze only lines in the spectrum that seem to be well described by one-electron theory. The key to finding these lines is that the ratios of their wavenumbers are as predicted by eq 1. The proper transitions can be selected by forming a matrix of all possible ratios of the observed He line wavenumbers, ω i . (The ij th element of this matrix is ω i /ω j ). The sought-for transitions should lie in a single row or column of this matrix. In Table 1, one student’s results are given as well as the literature (3) values for the wavenumbers of the 1snd 1D → 1s2p 1P transitions. The ratios of the wavenumbers of the transitions to the wavenumber of the first of the series (n = 3) are also shown. While there is a systematic difference between the transition wavenumbers measured by the student and those reported in the literature, the ratios to the n = 3 transition are within experimental error in the two sets. These ratios are close enough to those predicted by eq 1 for the oneelectron-like transitions to be identified. The ratios agree with the one-electron theory to within 1%. (The n = 6 transition will probably not be reported by students with visual detection of the emissions.) Table 1. Wavenumbers of 1sn d 1D → 1s2p 1P Transitions of He
n
104 cm ᎑1
ratio to n = 3
Student
Ref 3
Ref 3
3
1.498
1.4970
–
–
4
2.033
2.0312
1.357
1.350
5
2.281
2.2783
1.522
1.512
6
–
2.4126
1.612
1.600
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
From eq 1
In the Laboratory
Direct application of eq 1 to the transition wavenumbers gives Zeff slightly less than 1.0 for all transitions (Zeff = 0.993 ± 0.002 for student and 0.994 ± 0.002 for literature). This unexpected result arises from relatively more electron repulsion in 1s2p than in any of the D levels. A better way to analyze the data is by linear regression of the transition wavenumbers against [(1/22) – (1/n2)], which gives a slope of 109,900 cm᎑1 and an intercept of ᎑295 cm᎑1 for the literature data. From the former, Zeff for the nd levels is found to be 1.0008; that is, almost complete shielding of the nd electron by the 1s electron. The intercept is the upshift, due to electron repulsion, of the 1s2p level from its value calculated from the one-electron formula for the nd levels. The energy of the 2p level is more greatly affected by the multi-electron nature of the system than the energies of the d levels. The first ionization energy of He can be estimated by considering the following singlet transitions. To the energy of the first allowed absorption (1s2 → 1s2p), 171,129 cm᎑1, is added the energy of the 1s2p → 1s3d transition, 14,970 cm᎑1. The d levels are then extrapolated to n = ∞, using the oneelectron formula analogous to eq 1, giving
109,722 1.0008
2
1 = 12,210 cm᎑1 2 3
The sum, 198,309 cm᎑1, is in excellent agreement with the literature value, 198,310.74 ± 0.02 cm ᎑1. More accurate analysis of the data in terms of quantum defects (4, 5) is not warranted by the quality of student data and is not as revealing of basic quantum and spectroscopic principles as is the present discussion. It should be emphasized to the students that analysis of spectra of atoms with more than two electrons by the simple Bohr equation does not yield the accuracy here obtained for the two-electron helium system. Literature Cited 1. Herzberg, G. Atomic Spectra and Atomic Structure; Dover: New York, 1944; p 65. 2. Snow, R. L.; Bills, J. L. J. Chem. Educ. 1974, 51, 585–586. 3. Moore, C. E. Atomic Energy Levels, Vol. I; NBS Circular 467; National Bureau of Standards: Washington, DC, 1949. 4. Seaton, M. J. Proc. Phys. Soc. (London) 1966, 87, 337–339. 5. Kuhn, H. G. Atomic Spectra; Academic: New York, 1961; p 133.
JChemEd.chem.wisc.edu • Vol. 76 No. 1 January 1999 • Journal of Chemical Education
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