In the Classroom
Do the Series in the Hydrogen Atom Spectrum Ever Overlap? David W. Ball Department of Chemistry, Cleveland State University, Cleveland, OH 44115;
[email protected] General chemistry and physical chemistry textbooks alike comment on the simplicity of the spectrum of the hydrogen atom. The equation proposed in 1890 by Johannes Rydberg for all of the lines in the hydrogen atom spectrum is famous: ν∼ =
1 1 1 = R − λ n 22 n12
(1)
where n1 and n2 are integers. The modern value of R, the Rydberg constant, is 109,737.31534(13) cm᎑1 and is one of the most accurately known physical constants (1, 2). This equation was later justified by Bohr in his theory of the hydrogen atom (3) and, ultimately more satisfactorily, by Schrödinger (4) and Heisenberg (5). Some references display the hydrogen atom spectrum in only energy-level form (6, 7). On those occasions when multiple series are displayed in line spectrum form (8–11), it is easy to notice that the series are discretely grouped. The infrared series (the Paschen series), the visible series (the Balmer series), and the ultraviolet series (the Lyman series) do not overlap. Is this always the case? A check of the literature found Brackett’s original experimental measurements of the hydrogen atom spectrum into the far infrared (12). Figure 1 is a reprint of Brackett’s experimental measurements of emitted light from a long hydrogen discharge tube, in which infrared light intensity was measured with a rock-salt prism and “an extremely sensitive vacuum thermo-junction”. The peaks labeled Pα through Pζ (right to left) are the previously announced Paschen series (9; n1 = 3 in eq 1). The two peaks labeled Nα and Nβ are two new lines, ultimately part of the Brackett series (n1 = 4 in eq 1). Nβ looks very close to Pα, suggesting the possibility that overlap may occur. This prompts the question: when do the series overlap (if ever)? That is, at what value of n does the series limit of the next series overlap with the first line of the previous series? This is rarely addressed, although it is illustrated in one physical chemistry text (13) and tables are available for reference (14). What would be nice would be some sort of proof. Although the hydrogen atom spectrum has been part of several papers in this Journal (11, 15–19), we are unaware that a proof has ever been demonstrated directly. Such a proof, while perhaps not scientifically significant, should be pedagogically interesting. The Solution
Figure 1. F. S. Brackett’s original figure showing light he detected from a long hydrogen discharge tube. The lines labeled “P” are from the Paschen series, while the two lines labeled “N” are newly-detected lines. The curved line superimposed on the spectrum is the calibration curve, relating the number of micrometer turns on the monochromator (which is what is plotted on the x axis) to the wavelength of infrared light (plotted on the y axis). (From Reference 8. Reproduced by permission of the American Astronomical Society.)
while the first line of the previous series is
Since we are in wavenumber units, we want to find the value of n for which the series limit of the next series is greater than the first line of the original series (if we were doing this in terms of wavelength, we would require the first line have higher wavelength than the series limit, but the algebra is simpler if we do it in terms of wavenumber): R
What we want to determine is at what value of n the first line of one series overlaps with the series limit of the next (that is, the n + 1) series. In wavenumber units, the series limit for the n + 1th series is
R
1
( n + 1)
> R
2
1 2
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( n + 1) •
1 1 − n2 ( n + 1) 2
What follows now is all algebra. We cancel the Rydberg constants, R, from both sides:
1
( n + 1)
1 1 − 2 n ( n + 1) 2
R
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2
>
1 1 − 2 n ( n + 1)2
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883
In the Classroom
Now, we multiply all terms by the product of the denominators, (n2)(n + 1)2: n 2 ( n + 1)
( n + 1) 2
2
n 2 ( n + 1)
>
2
n2
−
n 2 ( n + 1)
2
( n + 1) 2
We cancel out the terms in the numerator and denominator that are the same, leaving us with: 2
n 2 > ( n + 1) − n 2
Now we expand the (n + 1)2 term and simplify. We get n 2 > n 2 + 2n + 1 − n 2
Literature Cited
which ultimately becomes n
2
> 2n + 1
Bringing all terms over to one side of the equation, we have n 2 − 2n − 1 > 0 The value of n for which overlap of consecutive series occurs can be determined by inspection of this last equation. The smallest value for which the above expression is greater than 0 is for n = 3, indicating that the series limit for the n = 4 series (the Brackett series) occurs at a higher wavenumber than the first line of Paschen series. We can confirm this easily: the first line of the Paschen series occurs at
109737 cm −1
1 3
2
−
1 4
2
= 5334 cm −1
whereas the series limit of the Brackett series is 109737 cm −1
1 42
= 6858 cm −1
In fact, the line in the Brackett series that has n2 = 9 has a predicted wavenumber of 5504 cm᎑1, so every line in the Brackett series having n2 ≥ 9 will overlap with the Paschen series. If Brackett had been able to observe additional lines in his spectrum, he would have observed the first overlapping in the hydrogen atom spectrum. However, as is obvious from Figure 1, the dramatic decrease in intensity of the emission lines precluded his observation of any line greater than n2 = 6. A few words regarding serendipity are in order. The hydrogen atom spectrum was first investigated in the visible region of the electromagnetic spectrum (post-1860, after invention of the spectroscope by Kirchhoff and Bunsen), then the ultraviolet (by Lyman in 1905), then the infrared (by
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Paschen, in 1908). These choices were, no doubt, anthropologic in origin: it makes sense to first investigate the region of the spectrum that our own vision is sensitive to. As the analysis above just demonstrated, only these three series are mutually non-overlapping. Thus, the initial investigations into the hydrogen atom spectrum focused on those series whose positions in the electromagnetic spectrum provide the simplest, most regular spectra. How different the progress of physical science might have been if, for example, we first investigated atomic spectra in the microwave region, where overlap of spectral series would make the hydrogen atom spectrum much more complex.
1. Value taken from the International Union of Pure and Applied Chemistry Web site. http://www.iupac.org (accessed Mar 2006). 2. Ball, D. W. Physical Chemistry, 1st ed.; Brooks-Cole: Pacific Grove, CA, 2003; p 250. 3. Bohr, N. Phil. Mag. 1913, 26, 1– 5 (part I), 576–602 (part II). 4. Schrödinger, E. Ann. der Phys. 1925, 81, 109. 5. Heisenberg, W. Z. Physik 1925, 33, 879. 6. Hill, J. W.; Petrucci, R. H.; McCreary, T. W.; Perry, S. S. General Chemistry, 4th ed.; Prentice-Hall: Upper Saddle River, NJ, 2005; p 279. 7. Ebbing, D. D.; Gammon, S. D. General Chemistry, 8th ed.; Houghton Mifflin: New York, 2005; p 275. 8. Kotz, J. C.; Treichel, P. M.; Weaver, G. C. Chemistry & Chemical Reactivity, 6th ed.; Brooks/Cole Publishing: Pacific Grove, CA, 2006; p 311. 9. Olmsted, J.; Williams, G. M. Chemistry, 3rd ed.; John Wiley and Sons: New York, 2002; p 253. 10. Silberberg, M. S. Chemistry: The Molecular Nature of Matter and Change, 4th ed.; McGraw-Hill: Boston, 2006; p 265. 11. Ramachandran, B. R.; Halpern, A. M. J. Chem. Educ. 1999, 76, 1266. 12. Brackett, F. S. Astrophys. J. 1922, 56, 154. 13. Atkins, P. W.; de Paula, J. Physical Chemistry, 7th ed.; W. H. Freeman and Company: New York, 2002; p 366. 14. Hollenberg, J. L. J. Chem Educ. 1966, 43, 216. 15. Kauzmann, W. Quantum Chemistry; Academic Press: New York, 1957; p 282. Some versions of the CRC Handbook also have tables of hydrogen atom spectral lines, but not the most recent editions. 16. Companion, A. L.; Schug, K. J. Chem Educ. 1966, 43, 591. 17. Douglas, J.; von Nagy Felsobuki, E. I. J. Chem Educ. 1987, 64, 552. 18. Reiss, E. J. Chem Educ. 1988, 65, 517. 19. Khundkar, L. R. J. Chem Educ. 1996, 73, 1055.
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