In the Laboratory
The Balmer Spectrum of Hydrogen: An Old Experiment with a New Twist
W
B. R. Ramachandran and Arthur M. Halpern* Department of Chemistry, Indiana State University, Terre Haute, IN 47809; *
[email protected] The investigation of the Balmer spectrum of hydrogen is a classic experiment that is often included in the undergraduate physical chemistry laboratory curriculum. It introduces students to the fundamental experimental and theoretical aspects of emission spectroscopy. It also ties in well with the coverage of quantum chemistry and provides compelling evidence for the acceptance of the quantum mechanical treatment of the H atom. In this experiment students acquire the emission spectrum produced by a discharge tube in the 350–660 nm region and determine the Rydberg constant by fitting the data to the Rydberg equation. A typical procedure is described in Shoemaker et al. (1). A recent paper in this Journal (2) deals with an interesting modification of the experiment in which the mass ratio of 2H and 1H atoms is determined from the spacing between their α lines as observed from the emission from a commercially available deuterium discharge tube (which contains a small amount of residual hydrogen). When students analyze their data in these experiments, however, they must make two assumptions: that the emission lines belong to the Balmer series, and that the feature observed near 656 nm is, in fact, the α line (i.e., the longest-wavelength component in the series). We describe here a straightforward approach in which students can discover for themselves that their data indeed correspond to the Balmer spectrum (not to the Lyman, Paschen, Brackett, or Pfund series) and that the 656-nm feature is the α line. This approach, especially when used in addition to the deuterium isotope effect described in reference 2, makes for a conceptually simple, rigorous, and compelling spectroscopy laboratory experiment that is very suitable for the undergraduate physical chemistry course. Furthermore, the analytical methods that are used expose students to realistic and challenging regression analysis procedures. Background In 1885, J. J. Balmer empirically analyzed the set of lines produced by a hydrogen discharge in the visible–near-UV region according to the expression
λ=k
n2 n –4 2
(1)
where λ is the wavelength of a given line, k is a constant, and n is an integer index ≥3 that accounts for each line. The Balmer relationship can be transformed into the more familiar Rydberg equation: 7
10 = ν = RH 1 – 1 2 2 λ 2 n
1266
(2)
where λ is the wavelength in nm, ∼ν is the frequency of the lines in wavenumbers, and R H is the Rydberg constant for the H atom (also in wavenumbers). Presumably Balmer was aware of the involvement of squared integers such as in eq 2, but some 30 years elapsed before eq 2 was firmly linked to a theoretical basis by N. Bohr. Today, students are familiar with the results of a quantum mechanical treatment of the H atom, the consequences of energy quantization (i.e., energy levels), and the application to the hydrogen emission spectra through the more general equation
1 =ν=R 1 – 1 H λ n ,2 nu2
(3)
where nu and n, denote the quantum numbers of the upper and lower states, respectively. Once students are told that a certain emission series arises from transitions involving a given value of n, and a range of values for nu (where nu > n,), they can predict that there will be several clusters or series of lines, depending on the particular value of n,. This situation is shown schematically in Figure 1, which illustrates the various line series mentioned above. But how are students to know that the spectrum they acquire is the Balmer series, that is, that n, = 2? Furthermore, although they can recognize from their data that, on the short wavelength end, the series they obtain converges in the near UV (ca. 370 nm), how do they know that on the long wavelength end there isn’t another line that lies at a longer wavelength than the feature they observe at 656 nm? Experimental Procedure Hydrogen- and deuterium-filled discharge tubes and the power supply were acquired from Edmund Scientific Co. The emission spectrometer consisted of the analyzing monochromator, detector, and data acquisition components of a Spex Fluorolog fluorometer. A resolution of 0.1 nm was used in the experiment. To achieve an acceptable signal-to-noise ratio for the weaker emission lines (between 368 and 400 nm), we employed the multiple scan option, in which case 6 scans were averaged. One scan was used between 399 and 660 nm. In both cases the scan step size was 0.1 nm with an integration time of 1 s. The monochromator was calibrated by scanning five mercury lines between 362 and 548 nm. Under these conditions, we acquired the emission lines with a precision of 0.1 nm. Data were analyzed using Scientific Data Analysis Software (SDAS)1 (3), an add-on to Microsoft Excel 97, which was run on a Pentium-based PC operating with the Microsoft Windows 95 operating system. In principle, any software capable of performing nonlinear regression on user-
Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu
In the Laboratory
Figure 1. A schematic diagram showing the various emission series observed for hydrogen. The data for this diagram were obtained from the Handbook of Chemistry and Physics, 73rd ed; CRC Press, 1992–1993; p 10-41.
defined equations—for example RS/1 (BBN Domain, Cambridge, MA), Mathcad (Mathsoft, Cambridge, MA), or PSI-Plot (Polysoft International, Salt Lake City, UT)—can be used for the data analysis. Discussion We describe an approach in which students analyze their data using a three-parameter fit to “discover” that indeed n, = 2 (the Balmer series), and also that for the 656-nm line n, = 3 (i.e., there is no longer-wavelength line). To determine this information, students can tabulate their data by assigning a running index, n, beginning at 1 for the 656-nm line and increasing successively for the shorter-wavelength lines. They then fit these (n, λ) data to a modified form of eq 3, that is, 7
λ= RH
10 1 – 1 2 n, n +∆
(4) 2
where λ is in nanometers and ∆ is an offset integer that establishes the relationship between the running index, n, and nu; that is, nu = n + ∆. R H is the Rydberg constant (in wavenumbers). Using eq 4, students regress their (n, λ) data to obtain values of RH, n,, and ∆. Once it is established that in fact n, = 2 and ∆ = 2, students constrain these values in eq 4 and then perform a one-parameter regression on their data to obtain a refined value of R H. It is also possible for students to first invert their wavelength values to cm{1 units and to regress these data to the relationship
ν = RH
1 1 – 2 n, n +∆
2
(5)
We applied this treatment to 13 lines obtained from the Balmer spectrum of the D atom.2 The data are listed in Table 1. The success of the initial three-parameter fit depends on the proper choice of initial guesses for the parameters n, and ∆. Evidently some minimization algorithms will locate false minima in the error surface, or will even fail to complete the search, if the initial guess values are far removed from the true values. Using the user-defined function of the regression analysis feature of SDAS (3) we find that with initial guess
values of R D (the Rydberg constant for the D atom), nu, and ∆ of 1, 1, and 2, respectively, we obtain optimized values of 109,528 cm{1, 1.9981, and 1.9972. This result confirms that the spectrum is the Balmer series (n, = 2) and that the 656-nm line is the longest-wavelength feature (nu = 3). A subsequent one-parameter fit of the data using n, = 2 and ∆ = 2 with λ in namometers (eq 6) provides a value of RD of 109,740 cm{1 (σ = 3.8) with a standard deviation of regression of 0.05.
λ=
7
10
1 RD 1 – 4 n +2
(6) 2
Figure 2 shows the experimental data and the fit obtained. Students can try different initial guess values3 for the parameters to convince themselves that this result does represent the global minimum. It is interesting that the standard deviation in R D is smaller than the uncertainty in the input data (the wavelength values). Students can compare their value of RD with the values of R∞ and RH, where the latter are obtained from R∞ = RD(me/µD) and RH = RD(µH/µD), where me is the electron mass and µH, and µD are reduced masses of 1H and 2H, respectively. R∞ and RH values thus obtained are 109,710 and 109,770 cm{1, which compare well with literature values of 109,677.6 (4) and 109,737.31 (5). Notes W Supplementary materials for this article are available on JCE Online at http://jchemed.chem.wisc.edu/Journal/issues/1999/Sep/ abs1266.html. 1. For further information about SDAS contact the corresponding author at
[email protected]. 2. We used a deuterium-filled discharge tube because it had much lower background emission than the hydrogen tube that we acquired and we were able to obtain many more weak emission lines. See the Experimental Procedures section. 3. Because R D is greater than the wavenumber values of any of the observed emission lines as implied by eq 3, students can safely use a value of 104 or higher as the initial guess value for that parameter.
Literature Cited 1. Shoemaker, D. P; Garland, C. W; Nibler, J. W. Experiments in Physical Chemistry, 5th ed.; McGraw-Hill: New York 1989.
JChemEd.chem.wisc.edu • Vol. 76 No. 9 September 1999 • Journal of Chemical Education
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In the Laboratory Table 1. Positions of First 13 Balmer Emission Lines from a Deuterium Discharge Tube ∼ν λ /nm n /cm{1 1
656.1
15242
2
486.0
20576
3
433.9
23047
4
410.0
24390
5
396.9
25195
6
388.8
25720
7
383.4
26082
8
379.7
26337
9
377.0
26525
10
375.0
26667
11
373.4
26781
12
372.1
26874
13
371.0
26954
Figure 2. A plot of the line positions of the Balmer emission of deuterium as a function of the running index, n. (•) experimental data; (––) best fit obtained from a one-parameter regression analysis of the experimental data using eq 5. R D = 109740 (σ = 3.8) cm{1; standard deviation of regression = 4.9 × 10{2.
2. Khundkar, L. R. J. Chem. Educ. 1996, 73, 1055–1056. 3. Frye, S; Huber, M. SDA, v. 4.3; Scientific Data Experts. This Excel 97 add-on software pack is a data analysis supplement for Halpern, A. M.; Experimental Physical Chemistry, A Laboratory Textbook, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1997. 4. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995, p 563. 5. NIST Online Reference Databases; http://www.nist.gov/srd/ online.htm (accessed May 1999).
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Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu
