The Spectroscopy and Thermochemistry of Na and Ma2 H. D. McSwiney The Ohio State University, Lima Campus, Lima, OH 45804 Dam1 W. Peters1. Wllllam B. Grifflth.. Jr.=.. and C. Weldon Mathews The Ohio State University, Columbus, OH 43210
The visible emission spectrum of atomic hydrogen has been used for many years as a hasis for undergraduate spectroscopy experiments (I,2). Besides the ease of its photography in the visible region, this spectral series (Balmer) is not contaminated by overlap from others. Finally, the series shows the simple relationship between term values and principal numbers for hydrogen (3). Alkali metal spectra have also been used in emission. Although in each c&e means had t o be taken to correct for the series overlap that occurs in alkali metal spectra, Stafford and Wortman (4) designed a study of sodium and Miller (5) of lithium. These experiments showed as well as ionization notential a feature oeculiar t o ~olvelectronic atoms, the quantum defect. Ashbv and Gotthard (6)have eliminated the series overlap problem simply by usingsodium an absorption as well as emission spectrum in an experiment using sodium. They have designed a heated absorption cell that produces vapor of sufficient pressure for single pass absorption; the cell features water-cooled ends t o prevent window damage by sodium vapor. The experiment effectively demonstrates self-absorption, resonance lines, and temperature broadening and allows an accurate calculation of the ionization potential of sodium. Ashbv (7)has shown the usefulness of the experimental apparatus & for Li, K, Rb, and Cs. The nresent experiment is desiened for a student not onlv t o examine the spectrum of atom& sodium but also to ex&ine the soectrnm of diatomic sodium, the goal being the use of data from both studies to show the co ni. The nj and n, are integers, and RH is an experimentally determined constant that can also he obtained from fundamental physical considerations of quantum mechanics.' The success and sim~licitvof fittine the hvdroeen data to the above expressions ieads h a t u r a t~o ~a n~e i f o r t k describe the observed electronic transitions of other elements bv similar means. The most obvious extension is to othe; oneelectron systems (He+, Liz+,Be3+, etc.), where the modification of eq 3 t o
Ej= -PR,Inf
(4)
proves satisfactory. Z has the value of one plus the charge of the species under consideration, thus an internal value. Rw is the ~ y d b e r gconstant for a n atom of mass M (= mass of M + me)such that
+
R,
= R,
mass of Mt mass of Mt+ m,
R, has the value 103,737,315cm-' (111, which is the limiting value for successively larger nuclear masses. Note that RH is just one of many RM calculable by eq 5. There are two interesting modifications of eo 4 useful for accurately calculating the term values of systems other than one-electron systems. These modifications involve changes either in Z or in n (not both), the changes being in either instance of a simple and consistent nature applicable to observable spectra. The first modification is the replacement of Z in eq 4 withZ* ( = Z - a). The a are independent of
Table 1. Term Values for the S and P Levels ol Atomlc Sodlum Term vdue
n'
Z but not of either the principal or of the azimuthal quantum number. The term values thus calculated are useful in comparing isoelectronic spectra. The a is called the "screening constant". The second modification is the re~lacementof n in eq 4 with n* ( = n - 6). The term values caiculated in this case are useful for interpreting the Rydberg spectrum of any given atom, since 6 approaches a constant value for the successively larger values of n for a given atom and azimuthal quantum number (note Table 1for the S and P levels of atomic sodium (12)). The 6 is called the "quantum defect" (13,14). Since this is a study of many different energy levels in a single type of atom (specifically, sodium), the quantum defect modification of eq 4 is the appropriate form for our purpose. Hence the energy expression to be used in fitting observed transitions in sodium will be
where X is the wavelength (in cm) of the energy of transition between levels n; and nj, and where RN, is the Rydberg constant (in cm-') for sodium. For sodium atoms in the ground state, absorption of a photon will cause promotion of the outermost electron to any of many levels of increasing n'. If thesubscript idenotes theground state, one candefine
such that
+
-
-
As n* increases, (n>J = (n;) 1, since 6 approaches a m, one bas A E I E so constant value. Also note that as n; that A E is the ionization energy of sodium. Trealment ot Atomic Sodium: Experimental Procedure Data were ohtained on a Bausch and Lomb 1.5-m spectrograph. Either Tri-X or SA-1 film from Kodak is suitable; in this case SA-1 was used without a filter. The dispersion a t the center of the film was 7.4 A/mm in second order. The continuum source was a high-pressure xenon lamp (a hydrogen lamp can also he used), and a mercury lamp was used to superimpose standard emission limes for calibration. The absorption cell required only moderate heating to produce sufficient pressure of sodium vapor for single-pass ahsorption. A portion of the absorption spectrum near the ionization limit is shown in Figure 3. Notice the regular convergence of successive transitions (certainly expected
Recall that
"i
where 858
10
15
Flgure 3. Detail of absorption near the lonl.&lon limn. Journal of Chemical Education
20 25
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given the reciprocal square energy-level dependence) and the decreasine ease of observation as hieher excited states are attained. Gate also that the ionizationlimit is not directIv observed (althoueh its location is marked). . . hence the ckculation t o determine its value. T o obtain waveleneths the film was laced in a measurine device with a traveling microscope, and the positions of the mercurv lines and 15 or 20 sodium lines were recorded on the arbitrary linear scale of the microscope. The mercury wavelencrths ( 1 5 ) were fitted using- least-squares techniques (16) to & equation of the form X; = a
+ bx; + cx?
where Xi are the known tabulated wavelengths to 0.01 A and xi are the scale readings. The scale readings of the sodium lines were then used in the equation to determine the wavelength of each. Treatment of Atomlc Sodlum: The Calculations Table 2 shows the energies of successive Rydberg lines of atomic sodium as lvoicallv obtained in this exneriment. The first step in detendihing ihe ionization e n e r i f r o m the data is to make a rough estimate of the ionization energy from the observations in Tahle 2. A reasonable value for the data given in Table 2 is 41300 em-' (slightly above the observed transition limit). With IE = 41300 cm-', R N =~ 109,734.71 cm-I and the first observed energy (30,277.48 cm-I), eq 7 now yields a trial value of n: as 3.155. Make a new table (Tahle 3) that assumes each nj to increase by unity over the preceding one for successively hlgher energies, and use the above equation to calculate an ionization energy for each observation. From Tahle 3 chowe the hieher n; to select a new limiting value for the ionization energy. his improved value of the ionization enerev -.is now used in ea 7 with each of the observations and R ~ , t ocalculatea new set of n*. Table 4 shows the first set of these n*, the values of which are far more sensitive to the IE value than is the IE value to theirs. A plot of n* mod 1vs. n can be done a t this time. If the plot is horizontal, the best obtainable value of IE has been found. If the plot is not horizontal, the IE value is varied slightly and a new tahle such as Tahle 4 is prepared for t h e production of another n* mod 1 vs. n graph. A set of nj that fits the observed transition energies and the ionization energy over a
Table 2.
wide range of j and particularly a t high j will be obtained very quickly. Note that this dataset in the first cycle yielded an approximate ionization energy of 41,441 cm-' (8.1718 X 10-l9 J) for atomic sodium; the accepted value is 41,449.65 cm-1 (17). Although the primary goal of this part of the experiment is indeed the ionization energy, note that the value of n* yields also a good value for 6. Treatment of Dlatomlc Sodlum: The Energy Levels There are two kinds of energy levels that must be discussed for sodium molecules-electronic energy levels and vibrational energy levels. The electronic energy levels are successive Rydberg states of 'II, symmetry attainable by promoting the molecule from its '2; ground state. The vibrational energy levels are of successive u and of the ground state only. The treatment of the electronic energy levels is to obtain the ionization enerev IE(Nad of the sodium molecule. Successive 11-2 0-0-bands of the sodium molecule are Rydbere transitions that can be described by the modified eq 7
+
Invoking (fa;+,) = (n:) 1 along with eq 8 will allow an expression as useful for describing the molecular Rydberg transitions as similar expressions are useful for describing transitions in atomic sodium. The treatment of the vihrational energy levels is to obtain the dissociation energy of the sodium molecule. Vibrational
Table 3.
Flmi Cycle HI Tabulation To Calculate ImNa)
O b w n e d Energlea HI Rydberg TraMHlons In Atomlc Sodlum
Table 4.
n' Calculation Holding IEConsiant
Volume 66 Number 10 October 1989
859
energy differences, if Morse potential dependence is assumed, can be described by AG(o
+ %) = (we - 2o,x,) - 2w,x,u
where u is vibrational quantum number, AG is vibrational . . resnectivelv. enerm difference and the o. and w-a ~" rare. ".the coefficients of linear and qladratic terms in a power series describinevibrational enerw -.levels. The nrecedine"eouation . is the sGe as
.
AG,=mu+b where m = -2o& and b = o. - 2w&-. " .A .d o t of AG.. vs. u yields then astraigit line that can heextrapolated to AG = 0 (a Birae-Sponer plot). The dissociation enerev D is the sum of theindividual-AG, such that the area un&r the straight line is an approximation to D. This straight-line B i r g e Sponer extrapolation will yield a dissociation energy somewhat greater than the state actually uossesses as discussed by Hirzberg (18). One may carry outthe Rirge-Sponer extrapolation either graphically or by fitting the linear equation. finding u ,, (i.e., AG = 0 ) and integrating (191. Treatment of Diatomic Sodium: Exnerimental Procedure Although absorption bands of sodium molecule can be seen on the film containing the atomic sodium absorntion. . . the students are not asked identify band origins appropriate to determining the ionization energyof sodium molecule. Such identification is beyond the scope of juniorlsenior-level undergraduates; thus the students are supplied with the energies of two adjacent II-2 transitions. These two energies and the appropriate Rydberg constant will allow calculation of IE(Na2) by eq 8 as later discussed. The advent of lasers has made possible a new method of obtainine vibrational enereies in mound electronic states. The mo~ochromaticradiaGon avaylable from the laser can be used to populate a specific rotational-vibrational level in an upper electronic state of a molecule. The molecule in this excited electronic state has access to manv vibrational levels of the ground electronic state by the emission of radiation of the appropriate energy. This laser-induced fluorescence~ r o duces fewer transitions in toto than other types of electrbnic emission spectroscopy and at the same time provides a convenient means of obtaining more ground state vibrational energy differences than previously used methods (20). The apparatus used for observing electronic energy differences in atomic sodium can also be used for the observation of mound state vibrational differences in diatomic sodium. ~xierimentaldifferences are that the cell must be heated more strongly in order to produce a sufficient vapor phase concentration of Na2 and that an argon ion laser rather than a xenon continuum source is used for excitation. The areon ion laser beam is directed at a slight angle into the heated cell. Thesodium molecule fluorescence is monitored through the same window, but on-axis. Perfectly usable fluorescence data (again a Bausch and Lomb 1.5-m-the film should be Tri-X) can hereby be obtained, but in the interests of conserving laboratory time it may in fact he convenient to have the students measure an already prepared print (greatly enlarged) of the emission spectrum.
tb
Treatment of Diatomic Sodium: The Calculations The supplied energies of the two n-2 0-0 Rydberg bands of diatomic sodium along with the given Rydberg constant allow substitution of each energy into eq 8 to yield two simultaneous equations which can be solved for n;.and for IE(Na2). The solution requires reduction of a quartlc equa-
880
Journal of Chemical Education
tion by iteration methods; many of the variety of programmable calculators perform quite adequately here. In treating the vihrational data recall from the earlier discussion
A student given a fluorescence spectrum of sodium and one vibrational quantum number assignment from Demtroder et al. can assign the remaining vibrational quantum numbers (u) with vibrational energy differences (AG,). The student can then construct a Birge-Sponer plot to approximate D(Nad. . -. If the resolution of the svstem (inchdine the film measuring device) allown wavelength determinations to within 0.1 .h or better. the meat amount of r,ihrational enerw difference data available from the laser-induced fluor;cence urocedure allows one who so desires to substitute for the ~ l r g e - ~ p o n elinear r extrapolation a quadratic least squares fit in u to approximate the AG,. This naturallv leads td a cubic equatio~uponintegration, hut the "correction factor" invariably used with the linear extrapolation is no longer necessary--the greater amount of data needed for the nonlinear fit is obtainable from the fluorescence spectrum and will vield a result satisfvinelv close to the exnerimental value ( 2 5 . It should be noted &tin fitting a qiadratic for the purpose of extrapolating from it one might be accused of having unduly high regard for his data, but the data in this instance extended about halfwav UD the notential well of the vibrational mode. A student c&ld, of course, be asked to comment on the appropriateness of linear fitting vs. quadratic for this particular data set.
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Atomic and Diatomic Sodium: The Energy Diagram The dissociation energy of Naz (laser fluorescence experiment) is finally combined with the ionization energies of Na2 (0-0 band origins supplied to the student) and of Na (absorption experiment) in eq 1to determine the dissociation energy of the species Nal. Acknowledgment This work was supported in part by a grant from the Lima Campus Research Committee of The Ohio State University. Also, thanks are due to Pat Fleming for refining the computer-assisted processing procedure f i r student data. Literature Cited
4. Stafford,F E.: Wortman. J. H. J. Cham.Educ. 1962.39.630. 5. Miller, K. J. J. Chem. Edw. 1971.51.805. 6. Ashby, R. A.; Gotthaud, H.W. J.C k m . Ed=. 1974,51.408. 7. Ashby, R. A. J. Cham. Educ. 1978.55. WO. 8. Inneb,K. K. Am. J.Phys. 1)66,34,306. 9. Nucl.Sci.Abatr. 1969.23.11168. 10. The Royal Society of Chemistry. HomrdP in the Chomleol Lnbamlory: Bmlingfan House: London. 1981;p 484. 11. Cohen, E. R.;Taylor.B.N. J. Phya. Ref. Data lWS,2,711-721. 12. Kuhn,H. G. Atomic Speetro, 2nded.:Longman: London. 1969;p 169. 13. Kuhn, H. G. Atomic Specfro, 2nd 4.;Longman: London. 196%pp 150-163. 14. Mlen, B. Atomic Spectra, Handbuch der Phyaik XXVII; Springer: N w Yo&. 1864: pp 81-M.123-125. 15. Hsrriaan. G. R..Ed. W n v e k g l h Tab1os;M.I.T.: 1969. 16. Wesat,R C.,Ed.Handbaoh)rf Chhmiit'y'~'~dPhy~iii~49th9thd~;ChhicalR~bbaCo.: ",-,."A
1-*:
.
17. Str~mnav.A R Suenlirak#i.N S Tobls* olS,uifml ldnea of Neufmlond lonrmd Ar ms. I V IiVlmum Ucu York. 1968 18 H e r z b c r g . C 3 p r ~ r r o o f D ~ o , o m Mole ,r ~ l r l . 2 n d c d:VanNmrand Pnn-o. l9UI. ,.4,(9 I9 K8ng.C W Spsrrroarop) and Mol~ulorSfncrrurr.H o l ~ R m c h a nand W i n a m : Now York. It* pp M&lhl. 20 nm?trdr#. W . MrClmuxk. M :&re. H N J. Chsm I'h~a 1%9,51.5495 21. Verma. K K .Hahn
