This method also enahles u s to compare various analytical cuhic equations of state (by simply changing the coefflcients of cuhic equations in eqs 3 and 4, and expressions for areas i n eas 5 and 6). The students are encouraged to compare the results for various gad laws with c~xperitncntnlresults. As an examvle. tht: Rcdlich-Kwonpeauat~onof state is much hetter than the VDW equation. A run takes around 1-2 min on a 486 PC, depending on how closely the pressure i s scanned. The program is compatible with both DOS and UNIX, and a copy is available from the author. Acknowledgment We are very thankful to the Centre for Science Education and Communication (University of Delhi), for making available the computational and plotting facilities.
Addition (superposition) of left and right circularly polarized light waves (components) generates plane-polarized light (Fig. 12) that enters the chiral sample. When the plane-polarized wave passes through the isotropic medium containing chiral molecules, one of the circularly polarized components (waves) will interact more strongly with the "one-handed electron distribution of a particular enantiomer. As a result, i t is delayed (phase-shifted) with respect to the other component. The result is a n elliptically polarized light emanating from the sample (Fig. 13). The ratio of lengths of axes of the polarization ellipsis will depend on the magnitude of the phase shift a s demonstrated helow. The sign of [a1also depends on the magnitude of the shift (see Fig. 13).
ORD through the Eyes of Mathernatica
h.lathematica input: ParametricPlot3D[(u*Sin[t],u*Sin[t+Pi/2],~), It,0,15l,(u,-l.ll,PlotPoints~l30,30l,ViewPoint~(J.-4.-4]]
lgor Novak Department of Chemistry and Computational Science Programme National University of Singapore Singapore 0511 Interactions of polarized light with chiral (optically active) molecules form the basis of several physical methods used i n t h e study of structure and properties of chiral chemical compounds (13). The two most important methods are ORD and CD. In the ORD method, we measure the rotation of a plane of linearly polarized light after i t passes through a samvle (a solution containing.. a n achiral solvent and ,I chiral sdutt:.. The specific rotation angle lo1 1s def i t i d for unit concentration and opt~calthtckness as
where n, and n, are refractive indices of left and right circularly polarized components of the plane-polarized wave of wavelength h. Figure 11. Len circularly polarized light wave component Hill and Whatley (14) have discussed difficulties of explaining the phenomenon of optical rotation a t the undergraduate level and suggested the use of a mechanical model. Circularly ~ o l a r - M a t h m i c a input: P~me~cPlot3D[(2*u*Sin[tl,u*Sin[t+Pi/2l+u*Sin[t-Pi~2].t}.(t.o,1s],(u,-i,i}, ized light can he generated by considering PlotPoints+(30,30).ViewPoint~{OO0,-411 two sine waves t h a t lie i n perpendicular planes; one wave describes the propagation -o, -2 -1 o 1 2 of an electric field vector E, and the other describes the propagation of magnetic field vec-o.zs o tor B. If the waves are phase-shifted by W 2 , one obtains two (left and right) circularly po14 2 larized light waves. Ordinarily, unpolarized light has no phase shift; E and B are inphase. Vector A can be defined as Figure 12. Plane-polarized light wave entering the chiral solution viewed along the direction of the light beam.
o~Zz
Mathematica input:
where t is the time.
ParamemcPlot3D[(2*u*Sin[t],u*Sin[t+P'd2-~]+u*Sin[t-Pi/2],t/31,( t,0,15], (u,-1,1),PlotPoints~[30,30],ViewPoint+[00,-4] q?= - 0.5
E = E,sin (t) B = Basin
I:+
left circular wave
-2 -0.5
-1
0
-0.25 0
E = E,sin(t)
0.25
B = B,sin [t -?) 2
right circular wave
0.5
Aleft circularly polarized wave is shown i n Figure 13. Elliptically polarized wave leaving the chiral solution viewed along the direction Figure 11; the surface swept by A is plotted. of the light beam. 1084
Journal of Chemical Education
Mathematica Implementation Mathematica software package (15) with its powerful graphics and plotting capabilities can illustrate the ORD phenomenon. A useful sketch of circularly polarized light can be found in a textbook by Cotton and Wilkinson (16). The important point of the Mathematica exercise is to convey a central point: The electronic structure (distribution) in an individual chiral molecule determines the strength of interactions with E and B, their phase shifts, and thus the polarization ratio and angle lul. 'I'he studcni; may. experiment with produeingelliptically . polarized waves by varying phase shifts from q = 0(plane polarized)
q = - P a (elliptically polarized) rp =-Pi ( c i r c u l a r l y p o l a r i z e d ) rp = -3Pi12 ( e l l i p t i c a l l y p o l a r i z e d )
to ~p = -2Pi ( p l a n e p o l a r i z e d )
ohvsical orwnic chemistrv. general oreanic chemistrv. and co;rses teaching the applf&ons of s&troscopic methods in oreanic chemistrv. This contribution was stimulated bv questions from several second-year undergraduates whb remained unconvinced by conventional explanations of ORD given in standard organic chemistry textbooks. Literature Cited 1. Haile, J. M. Mokculor Dynamics Simslofion: Wiley: New York. 1992. 2. Lo, C. W: Shuh,D. K:Charkalian,V: Durbin,T. D.:Varekamp,PR.:Yarmo& J.A. Phya Rev 6' l993,47,E648-I5659. 3. Garrison, B. J.;Coddad, W. A , 111. Phw. Reu. B 1987,36.9805-9808. 4. SchoolmaR, T.A,: Garrison, B. J.J.Am Chem. Sac 1991,113,82214228. Saunders Col5. Skoog, D. A ; Learn J. J. PrLncLpl~soflnstrumntoLAnolysis,4th d.; lege, 1992. 6. Ramachandran, B.:Kong, P C. J. Cham. Edve accepted for publiestion. 7. Coleman, W. F J. Cham. Educ IBW), 67.A203. 8. Zdravkovski, Z. J Chem. Educ 1991. 68,.495; 1992,69,A242. 9. Rioux. F.J Chem. Edue. 1992 69, A240. 10. Blizuela, G.P.;Juan, A. J. C h m . Educ 1993, 70, A2S6. 11. TheScientlficP~opersofJamesClerk Marwell; Camblidge University 189k Vol. 2, p 425 ~-~
This variation of phase shift corresponds to the passage of plane-polarized light through solutions containing different chiral molecules. The resulting waves can be seen from different angles (perspectives) by changing the ViewPoint coordinates. The whole demonstration may be carried out eauallv . " successfullv wine-M a.~ l Ve Release 3 software (17). It may be usefz for undergraduate courses in
12. No&. J. H.; W0od.R. H. J. C h m . Educ 1982. 69.810411. 13. Drago. R. S. Physimi Methods /or Chemists, 2nd ed.: Saunders College: Orlando. 1992; pp 137-141. 14. Hill, R. R.; WhatleyB.0. J Chem E d v e 1980.57, 306. IS. Mathematics. 22.1 for Windows, Walfrsm Research, Champsign, IL 61820-7237. Sthed.: John Wiley: New 16. Cotian,FA.: Wilkinson,G.AdvoneedlnorpnnieChamirfry. York, 1988: p 639 17. Maple V Release 3 (for MS Wmdowal, Waterloa Maple SoRware and University of Waterloo. 1994.
Volume 72 Number 12 December 1995
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