Sept., 1963
KERRCOK'GTAXTS OF a-Aiv~moACIDSOLUTIOXS
1911
THE KERR CONSTANTS OF SOME U-AMINO ACID SOLUTIONS1'2 BY
w.H.
O R T T U N G 3 AND
J. A.
MEYERS3e4
Department of Chemistry, Stanford University, Stanford, California Received May 11, 1963 Thc Kerr constants, refractive indices, and densities of aqueous solutions of glycine, dl-alanine, and a-aminoisobutyric acid were measured a t 25' over a range of concentrations. Values of B and n were determined at 436, 546, and 578 mb. The dielectric constants were analyzed by Scholte'n ellipsoidal model theory and the results discussed. The same theory was extended to obtain optical anisotropies from the Kerr constants and auxiliary data. Part of the anisotropy in solution was caused by a shape effect. Optical data for crystalline glycine were analyeed to obtain polarizability tensor components and the results were compared with the anisotropy in solution.
Introduction Although considerable effort has been applied to the study of the Kerr effect of organic molecules in nonconducting soIvents,~6only a few attempts have been made to study solutions of small molecules in aqueous solution. Pau thenier' studied alcohol-water mixtures, and Krauses investigated the urea-water system. The amino acid-water system was of interest to us for several reasons. First, the Born-Langevin theory of the Kerr effect5 is only satisfactory for gases or nonpolar liquids, so that it was of interest to test the extension of more refined dielectric theories such as Scholte's ellipsoidal model theory' to the Kerr effect of polar liquids. Steps in this direction have been taken by Klageslo and Steppuhnl' for polar but nonaqueous solutions, and Le Feme and Williams12have discussed difficulties occurring with measurements of nonpolar solutes in polar solvents. Second, the direction of the dipole moment in the amino acids is known approximately irom the position of the zwitteiionic charges, allowing more definite conclusions to be drawn about the polarizability tensor components. The latter quantities also may be estimated from crystal refractive index data and compared with the solution results as a further check on the methods of analyzing the solution data. Since the amino acids have no planes or axes of symmetry, no simplifications may be made in the polarizability tensor, and the techniques that are used should also be applicable to more complicated systems such a8 protein solutions. Third, the possibility of estimating the anisotropies of ionic groups such as -CO2- was presented, and fourth, hydration effects could be indicated in the measured anisotropies. Experimental Apparatus and Methods.-The density, refractive index, and Kerr effect apparatus and methods have been described previously .13J4 (1) This work was supported b y Grant No. G 15555 from the National Science Foundation and by a n institutional research grant of the .American Cancer Society. (2) Presented in part a t the 144th National Meeting of the Binerican Chemical Society, Los Angeles, Calif., March 31-April 5, 1963. ( 3 ) Department of Chemistry, University of California, Riverside, California. (4) This investigation was supported in part b y a Public Health Service fellowship, GPM-18,279, from the Division of General Medical Sciences, Public Health Service. ( 5 ) C. G. Le Fevre and R. J . W. Le Fevre, Rev. Pure A p p l . Chem. (Australia), 6, 261 (1955). (6) R. J. W. Le Fevre, J . Proc. Roy. Soc. A'. S. Wales, 96, L (1961). (7) hl. Pauthenier, J. Phys. Radium, 2, 384 (1921). (8) S. Krause, Ph.D. Thesis, 'University of California, Berkeley. Calif., September, 1957. (9) T. G. Scholte, Physiea, 16, 436 (1949). (10) G. Klapes, Z . Naturforsch., 7a, 669 (1952). (11) A. Steppuhn, ibid., l l a , 912 (1966). (12) R. J. W. Le Fevre and A. J. Williams, J . Chem. SOL, 1671 (1961).
Materials.-Calbiochem A grade glycine, dl-alanine, and 01aminoisobutyric acid were recrystallized three times from warm water by adding alcohol and chilling. The crystals were dried a t 60-80' and kept in a desiccator. Both solids and water were weighed out in making up solutions and weighings were corrected to vacuum. Solutions were always used within a few hours of preparation and then discarded. The concentration range for the Kerr effect measurements was set by the conductivity of the solutions. The lowest usable resistance was about 1000 ohms. The specific conductivity, K , was calculated from the measured resistances of the solutions a t 1 kc. by the relation, ti = d / A R = (25 X lO-S)/R, where A and d a r e the area and separation of the electrodes. The results were lo6 ( K - tiO)/m== 16, 54, and 13 for glycine, alanine, and a-aminoisobutyric acid, respectively. The value for water was ti0 = 1.5 X 10-6. The glycine value is in agreement with the value, 17-18 X 10-8, obtained by Mehl and Schmidt,*6and the water value is close to that of ordinary conductivity water, 0.8 x 10-6.18
Results Densities.-The density data for dl-alanine and aaminoisobutyric acid are shown in Table I. The molar concentrations, e , and the apparent molal volumes of the solutes, &, mere calculated in the usual way,17 using M2 = 89.094 and 103.121 for alanine and LZaminoisobutyric acid, respectively. The data TTere fitted to the follo~vingexpressions, valid for m < 0.4 d(a1a.) = 0.99707 4-0.02864m - 0 . 0 0 2 2 8 ~4~A ~ (1) d(iso.) = 0.99707 f 0.0256Om - 0 . 0 0 1 8 2 ~f~A~ (2) The deviations are shown in Table I. The alanine data are in good agreement with the values reported by Gucker and Allen.l8 TABLEI DENSITY DATAAT 25" c
d (g./ml)
0 09970 .20195 .29369 ,39710
0 09882 .19893 28772 38664
Alanine 0 99990 1 00277 1 00528 1 00809
0 09762 18568
a-Aminoisobutyric acid 0 09660 +0 0 99956 18261 1 001i6 28725 1 00445 38483 1 00697
m
.29467 .39785
(ml./mole)
lO5A
0 0 6 -1 0 +O 6
+
+
5 0 6 2
60 60 60 60
60 63 75 77
77 77 77 77
60 65 66 63
(13) W.H. Orttung, J . P h p . Chem., 67, 1102 (1963). (14) W. H. Orttung and J. A. Meyers, zbzd., 67, 1905 (1963). (15) J . W. Mehl and C. L. A. Schmidt, J. Gen. P ~ ~ E W JIS, Z .467 . (1935). (16) S. Glasstone, "An Introduction to Electrochemistry," D. Van Nostrand Co., New York, N. P., 1942, p. 44. (17) H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd Ed., Reinhold Publishing Gorp., New York, N. Y., 1958. (18) F. T. Gucker, Jr., and T . W. Allen, J . Am. Chem. Soc., 64, l Q l (1942).
Refractive Indices.-The
TABLE IV KERECONSTANTS AT 25’
data are shown in Table
I1 and were fitted to fuiictions of the form 10sAn = am
+ bm2
TABLE I1 REFRACTIVE INDEX DATAAT 25’ X 108-----546 mp 578 mp
7--y--An 436 mp
c
172
Alanine 1548 3106 4494 5945
0.09932 .ZOO78 .29274 ,39026
0.09844 ,19779 .28680 .38013
1512 3039 4399 5814
1506 3023 4372 5784
0.09985 .18558 .28508 .38312
a-Aminoisobutyric acid 1708 1745 0.09879 3142 3214 ,18242 4793 4898 ,27811 6391 6531 .37100
1699 3131 4770 6366
Kerr Constants.-The data for B = An;XE2 are shown in Table IV. I n order to calculate molarities for the glycine data, the available density datal3 in the range 0 < m < 3.4 mere fitted by the equation =
0.99707
.-.
x 107 546 m r
B
(3) Values of a and b are shown in Table 111. The precision of the data was consistent with the limiting precision of the refractometer (f3 X in An).
d
Vol. 67
W, H. ORTTUNGAND J. A. ~ I E Y E R S
1912
+ 0.03196m - 0.00228m2+ 0.000115m3 (4)
to & I - 2 x 10-5 below m = 1 and to i l X abovem = 1. TABLE I11
m
C
0.1119 ,2002 ,2655 ,3990 ,4805 .4962 .6084 ,7600 ,8562 ,9566 ,9900 1.202 1.210 1.408 1 ,483 1,720 1.809 2.007
0 0.1111 ,1979 ,2617 ,3910 ,4693 ,4843 .5909 ,7334 ,8229 ,9154 ,9460 1.138 1.146 1.322 1.387 1.593 1.669 1.836
0
!4h
a
436 546 578
Alanine 15715 15385 15300
1240 1250 1230
436 546 578
a-Aminoisobutyric acid 17600 17210 17130
1450 1400 1350
Glycine 3.72 5.86 7.53 8.64 11.8 12.6 13.2 15.6 18.1 20.3
578 mp
2.89 4.75 5.60 6.90 8.8 9.5
2.72
...
5.61 6.54 8.6
...
...
10.0 11.7 13.8
11.5
... 15.9
...
...
...
21.2
16.7 20.8
17.1 16.0 20.5
...
...
25.2
24.2 30.6 36.4 36.2
27.8
...
32.9
23.6 26.4 28.3 31.2
Alanine 4.83 5.36
0.0731 .lo04 ,1035 .1436 ,1837 .2159
0.0726 3.86 ,0995 ... ,1026 ... 4.20 ,1419 6.14 4.84 ,1812 ... 5.32 ,2125 7.46 6.23 a-Aminoisobutyric acid 4.22 5.07 0.1019 5.55 7.12 ,2168 6.96 8.60 ,3169
0.1030 ,2211 ,3258
PARAYETERS FOR REFRACTIVE INDEX DATA
436 m#
3.72 4.23
... 4.85
... 6.14 3.84 5.06 6.45
TABLEV GLYCINE HERRCONSTAKT PARAMETERS
-b h, m#
436 546 578
a’
a“
h’
h”
+
1 9 . 3 1 0 . 4 - 0 . 7 f 0 . 7 19.4 f 0 . 3 + 0 . 2 0.7 1 4 . 7 f .4 . O f . 9 14.8 f . 5 .7 f .9 1 4 . 7 f . 2 - .41 . 4 1 4 . 8 f . 3 .2 f . 5
+ +
TABLEVI
Least squares fits19 were made to ( B - B l ) / m vs. m and to IB - B,)/c vs. c for the glycine data at the three wave lengths (using the B1 values14shown for m = 0 in Table IV). The results, with standard deviations, are given in Table V. a is the intercept and b is the slope, so that
KERRCONSTANT PARAMETERS 107 x ( B - BUC A, m p
G1yein e
Alanine
436 546 578
19.4 f 0 . 3 14.8 f . 5 14.8 f . 3
1 6 . 5 f 0.7 13.7 f .7 14.9 i .6
0-Aminoisobutyric acid
14.6 1.0 12.6 f 0 . 3 11.1 f 0 . 4
(5b) The average uncertainty of a‘ or a“ is about 2%. Negative values of b’ and positive values of b” seem consistent with the relationship betn een m and C. Since the alanine and a-aminoisobutyric acid data were not complete enough for a least squares treatment, the average of ( B - Bl),/mand (B - Bl/c for each point was averaged for each wave length. The results Lvith average deviations are shown iii Table TrI. It is thought that this treatment gives ai1 appropriate value for infinite dilution.
Calculations Mean Polarizability.-The refractive index data for dl-alanine and a-aminoisobutyric acid were analyzed by a procedure already discussed for glycine data.13 The new dat,a were not extensive enough to just,ify application of Bottcher’s method, so the LorenzLorentz equation was used to calculate a polarizability, a2,for each point. The results are shown in Table VII. Analysis of the dispersion data as beforel3 led to the results shown in Table 1‘111. The glycine results are included for comparison. I t . is interestilzg that t,he addition of a methyl group to glycine or alanine causes the same iilcrement in the three parameters: 012, XO, and f. For alanine, a molar refraction of 21.04 ml. was calculated for the Ka D line, in good agreement with the value 20.88 f 0.15 of McMeekin, et aLZ0
(19) W. J. Youden, “Statlstical Methods f u r Chemists,” John Wiley and Sons, Ino., New York, N. Y., 1951.
(20) T. L. MeMeckin, M. Wilensky, and 31. L. Groves, Biochent. B i o p k u ~ . Res. Cummun., 7 , 151 11962).
- B1) = a’m + b’m2
lO’(B - B1) = a’lc
+ b”c2
KERRCONSTANTS OF a-A~rrsoACID SOLCTIONS
Sept., 1963 TABLE 1-11
---
MCAh POLARIZABILITIES
x 546 xril
6 669 518
6 558 8 350
Glycine Alanine a-Aminoisobut) rii: acid
Ej
10 356
(aan/am) x 1021--476 mil 546 mil 578 m p
1024-578 m u
----(az)m-o 486 mil
6 534 0 022 0 020 0 020 8 350 035 040 040
10 188 10 157
050
050
049
TABLE VI11 PARAMETERS
DIsmRjIoN
A0
GI) h e
(A)
940 925 910
Alanmtl a-Aminoisobutyric acid
f 10 0 15 4 16 8
Dipole Moments.-Values for the dielectric increment of amino acid solutioiis in water have been available for some time,21$22 but only a few interpretations have been atternpted.23+6 For reasons similar to those given in the preceding discussion,l4 the ellipsoidal model theory of Scholteg seems most appropriate. For a pure liquid of anisotropic molecules26
1913
which is the equation used in the preceding paper.I4 The approximations in the first term of eq. 10 are not important for the polar molecules under consideration. The extension to a binary solution is accomplished by replacing the right-hand side of eq. 10 with a sum over solute and solvent molecules. The factor, pol2, may be replaced by po12g, vhere g is Kirkwood’s correlation parameter, to account for hydrogen bonding effects. The theoretical value of Pople is used as before.I4 A correlation parameter is not used for the solute, SO that any effects such as hydration show up in p02. ‘In order to apply Scholte’s theory, it is necessary to estimate values of the form factors, Ai, from the ratios, a : b:c. It may be assumed that the cavity has the same shape as the molecule. The principal molecular diameters, a’, b’, and c’, have been estimated with the help of crystal data (for glycine27 and alanine28), Stuart radii,29and molecular models. The results are shown in Table IX. The estimate for glycine agrees with that of Buckiiigl~am.~~ I n all three cases, the u’ axis is approximately along the assumed dipole direction (from the N atom to the midpoint of the two 0 atoms), the b’ axis is almost in the iY-C-C plane, and the c’ axis is perpendicular to it. The Ai factors were estimated from the curves of O~born.~O TABLE IX ~ ~ O L E C U L ADIWXSIONS R AND
where gi is the ratio of the cavity field to the average field and fi is the ratio of the reaction field to the total moment, both quantities referring to the direction of the ith principal molecular axis. ao, and poi are the polarizability and dipole moment of the molecule along the ith axis in vacuo. N is the number of molecules per milliliter. For Scholte’s ellipsoidal model9
(7)
a,bc ne2- 1 (9) 3 1 (ne2 - l)Ai where a, b, and c are the semiprincipal cavity axes and A,, Ab, and A , are the ellipsoidal form factors for these axes. Equation 9 implies that the molecule fills the cavity and defines ne2as an effective isotropic dielectric constant of the molecule. If it is zssumed in the first term of eq. 6 that the molecules are spherical (Ai = I/J,isotropic (aoi = a0),and fill all space (47rabc/3 = l/N), and that the dipole is along the a-axis, one obtains CVoi
e
- 1 = 4n.N €[E
[
abc
+ (1 .-
[e
+
=
1 + + ne2 3kT x
€(ne*- 1) 26
€)A,][l
+
(n,2
--
-
+ (ne2 - €)A,]*
(21) J. Wyman and T. L. McMeekin, J . A m . Chem. Soc., 55, 908 (1933). (22) W. P. Connor, R. P. Clarke, and C. P. Smyth, ibid., 64, 1379 (1942). (23) J. G. Kirkwood, in “Proteins, Amino Acids, and Peptides,” by E. J. Cohn and J. T. Edsall, Reinhold Publishing Gorp., New- York, N. Y . , 1943, pp. 294-296. (24) A. D. Buckingham, Australian J . Chem., 6 , 323 (1953). (26) J. T. Edsall and J. Wyman, “Biophysical Chemistry,” Vol. I, Academic Press, Inc., New York, N. T.,1958, Chapter 6. (26) W. F. Brown, Jr., “Handbuch der Physik,” Vol. 17, SpringsrVerlag, Berlin, 1956, pp. 60, 61.
2a’
2h’
6.1 4 7 Glycine Alanine 6 1 6 0 a-Aminoisobutyric acid 6 . 1 6 0
2c’
FORM FACTORS
(A
3 8 4 5 6 0
As,
Ab
Ac
0 24 0.34 0.42 .29 42 .20 33 34 .33
The use of eq. 10 (extended to binary solutions) also requires ai1 estimate of the effective dielectric constant, ne22,of the solute molecule. One estimate is given by the square of the average crystal refractive index, which is 2.52 and 2.38 for glycine and alanine, re~pectively.~~ If aoi and abc are knonn, eq. 9 provides another estimate. uoamay be taken as 1.05 times the mean optical polarizability a t infinite wave length, as is often done (neglecting intrinsic anisotropy of the molecule). For glycine, a’b’c’ = (2.4)3, and it is reasonable to assume that abc = (2.7)3, as suggested by several lines of evieq. 9 then predicts ne2 = 2.64 dence. With A, =: for glycine. For alanine, a’b’c’ = (2.7)3. Using the same increment as for glycine (0.3 A*), abc = (3.0)3, and eq. 9 gives nez2= 2.40. For a-aminoisobutyric and neZ2 acid, the same procedure gives abc = = 2.23. The agreement between the crystal and calculated values is good, and the latter are used in the following calculations. The more recent dielectric increments for glycine and alanine22have been used to estimate the vacuum dipole moments from the extension of eq. 10 to binary solutions. As before,l4 al = 1.928 A. and ne12 = 1.86 were used. The results are shown in Table X. A dielectric increment was assumed for a-aminoisobutyric acid to provide a rough estimate of the dipole moment. The results are fairly sensitive to the assumed shape, as evidenced by a prediction of p02 = 11.2 D. if glycine is (27) R. E. Marsh, Acta Cryst , 11, 664 (1958). J. Donohue, J . Am. Chem. Soc., 72, 949 (1950). H. A. Stuart, 2. physzlc. Chem. (Leipslg), B27, 350 (1934). J. A. Osborn, Phys. Rev., 67, 361 (1946). A. N. Winchell, “The Optical Propertmb of Organic Compounds,” Academic Press, Inc., New York, N. Y., 1954, pp. 66, 67. (28) (29) (30) (31)
W. H. ORTTUNG AND J. A. MEYERS
1914
assumed to be spherical. The values were essentially constant over the concentration range of the calculations (m < 1). The vacuum dipoles may be compared with the zwitterion contribution of 4.8 X e.s.u. separated by 3.0 A.27(14.4 D.) if a correction is made for the fact that the zwitterion is embedded in the molecule. For Scholte’s ellipsoidal model, the correction involves the factor f’
=
1
+
(ne22
TABLE X (abc)’/a
ne#
poz,
D.
14.4f‘, D.
2.7 3.0
2.54 2.40
14.2 12.9
10.5 10.2
3.3
2.23
(12.9)
10.2
Optical Anisotropy from Kerr Ed ect.-The extension to binary solutions of eq. 17 of the preceding paper14 was used to estimate the optical anisotropy of the amino acids. The induced moment orientation of the solute and the p and y terms were neglected. The orientational average for the solute was done in a coordinate system whose a-axis was taken along the permanent dipole. The resulting equation was
Nz (an (12) where all, a22, and 0133 are the diagonal elements of the polarizability tensor of the solute in the dipole oriented axis system, and not the prjncipal polarizabilities. The values obtained for the optical factors of the solutes are shown in Table XI along with an estimate of their uncertainty. Although the dispersion appears more complicated than might be expected, the precision is too low to justify detailed interpretation. Any Concentration dependence is also within the uncertainty of measurement. TABLE XI OPTICAL FACTORS - 1/2(0(22 aaa)i x 1024
+
A, m r
436 546 578
Glycine
0.75 f 0.02 .72 f .05 .76 f .03
Alanine
0 . 6 4 f 0.06 .67 f .07 .77 f .06
1
n2
+ (nz2- n2)Aa-
1
DIPOLEMOMENTS Ae/m
nyn2 - 1)
(11)
- 1)Aaz
as shown in Table X. The differences between pa2 and 14.4f’ for glycine and alanine (3.7 and 2.7 D.) are reasonable in terms of the bond dipoles involved.32 An orientation of the hydrated water might also be involved. The difference of 1.0 D. in po2 - 14.4f’ between glycine and alanine could be due to slightly different hydration effects.
Glycine 22.65 Alanine 23.20 a-Aminoisobut yric acid (23.75)
Since the effective solute refractive index is different from that of the solvent, deviations from spherical shape are responsible for part of the observed anisotropy of the solute. On the basis of Scholte’s model of a uniform isotropic ellipsoidal molecule, this effect is given by the followjng expression for comparison with the values in Table X I abc
1
Vol. 67
a-Aminoisobutyric acid
(0.51) f 0.07 ( . 5 5 ) f .03 ( .52) f .04
(32) C. P. Srnyth, “Dielectric Behavior and Structure,” McGraw-Hill Book Co., Inc., New York, N. Y..1955, pp. 242-245.
1/2{?%z
+ (nZ2 -
?%‘)Ab + n2
+ (nz2 - n 2 ) )’t]~ ~ (13)
For the solutes of interest, the largest principal axis of the shape ellipsoid is approximately parallel to the dipole direction, so that the shape effect contribution to the tensor components as predicted by eq. 13 is additive in the factors shown in Table XI. From eq. 13, using n2 = 1.80 and taking the n2 equal to nez2 as given in Table X, values of the shape effect were for glycine, found to be 0.37, 0.18, and 0.01 X alanine, and a-aminoisobutyric acid, respectively. The wave length dependence was negligible. Optical Anisotropy from Crystal Data.-Sufficiently complete information is available only for a-glycine, and the following calculations are limited to this system. The crystal is monoclinic with space group P21/n and four molecules per unit cell.27.33*34 The principal refractive indices are known to the fourth decimal at 486 and 686.7 mp.31 The smallest index is along the b-axis of the crystal, and the largest is approximately along the a-axis. The four molecules in the unit cell are related in pairs by inversion and by reflection in the a-c plane. The C-C02 plane is roughly parallel to the a-c crystal plane, and the C-C direction is roughly parallel to the c-axis. The C-N bond is rotated 18.6’ from the C-C02 plane. The coordinates of molecule A (notation of Albrecht and C ~ r e y ~were ~ ) obtained from Marsh’s results.27 A molecular reference system was taken with the x-axis along the C-C direction, the y-axis in the C-CO2 plane, and the x-axis perpendicular to the C-CO2 plane. The direction cosines of the molecular axes with respect to the principal axes of the crystal, a, b, c* (perpendicular to a, b), denoted by yc, xc, xc, respectively, are shown in the matrix
(
0,9284 0.2905 -0.2317
TA =
-0.3359 0.9227 -0.1893
0.1588 0.2536 (14) 0.9542
)
that operates on the components of a vector in the crystal system to give the components in the molecular system. Similarly, a dipole oriented coordinate system, xd, y d , ~ d may , be taken in which the xd-axis is along the line joining the nitrogen nucleus with the mid-point of the oxygen nuclei, y d is parallel to the C-CO2 plane, and xd is perpendicular to x d and yd. The matrix
SA =
0.8873 -0.4361 -0.1501
(
0.4411 0.8975 0.0000
1
0.1348 -0.0662 0.9887
(15)
(33) G.Albrecht and R. E. Carey, J. Am. Chem. Soc., 61, 1087 (1939). (34) “International Tables for X-Ray Crystallography, Vol. I, Symmetry Groups,” ed. by N. F. M. Henry and K. Lonsdale, The Kynoch Press, Birmingham, England, 1952.
KERRCONSTANTS OF ~ - A ~ I I N ACID O SOLUTIONS
Sept., 1963
gives the molecular coordinates, x, y, z from rd,y d , zd for an arbitrary vector. The particular rotation of y d , z d about x d is simply a convenient one and has no particular significance. If ( L I ~denotes the polarizability tensor of molecule A in the molecular reference system, the polarizability tensor in the crystal and dipole systems is given by
TB-'CYATA
(UA' =
(LIA'
= SA-'
(16) (17)
aASA
Molecules A and B are related by inversion (as are C and D). Thus the polarizability tensors are the same. Molecules A and C are related by reflection in the a-c crystal plane, so that (UC'
=
R-~QA'R
(18)
:)
R = ( 0l O1 0 0
ff
(
Axx
Ol'Axy
-1
ff'Axy ff'Ayy
0
:)
(20)
OLA ' ea
It is desired to determine the six components of the polarizabili ty tensor in the molecular reference system (or alternatively, the three principal values and their orientations). From eq. 16 and 20 it can be seen that the crystal data provide four relations, if the polarizabilities along the crystal axes can be calculated from the principal refractive indices. The Lorenz-Lorentz equation may be used to calculate polarizabilities corresponding to the principal axes of the crystal if the internal field in the crystal js represented by the Lorentz field. This approximation has been used because a fairly rough estimate of the molecular anisotropy will be sufficient. Mlethods for a more accurate treatment are available. 35 Values of the polarizabilities a t wave lengths other than those given are readily obtained by linear interpolation of 1/a us. 1 / X 2 , 1 3 and the following calculations were carried out for 546 mp. Equations 14, 16, and 20 were used to obtain a,,, ayy, aZz,and atxy as functions of a,, and ayc. The first column of Table XI1 shows the values obtained for the crystal axes from the Lorenz-Lorentz equation. The second column shows the results for the molecular coordinates on the assumption that a,, = ays = 0, which would hie exactly true if the C-N bond were in (35) B. R. A. Nijboer and F.
W.de Wette,
TABLE XI1 POLARIZABILITY COMPOSENTS AT 546 mfi x 1024 ij
zz
(19)
so that the polarizability of the four molecules in the unit cell may be represented by (LI'(U.c.) = 4
the C-C02 plane. The results in the third column were obtained on the same assumption. The value adxx - 1'/2(adyy4- adZz) = 0.57 X corresponds closely to the anisotropy predicted for a-aminoisobutyrjc acid, but not to that for glycine after correction for the shape effect. The discrepancy probably is not caused by the assumption that axz = ayr = 0. Thus it seems likely that Scholte's model does not provide an accurate value of the shape effect for the optical anisotropy of glycine. The second column of Table XI1 also indicates that the molecular reference system is a good approximation to the principal axis system of the molecule.
xx YY
where
Physzca, 24, 422 (1958).
1915
XY
-
acij
aij
6.50 6.91 15.48 0
6.57 6.97 5.35 -0.07
,d..
6.67 6.84 5.38 -0.20
Discussion The agreement between the calculated crystal anisotropy of glycine anid the solution anisotropy of a-aminoisobutyric acid is of interest because the shape effect should be small in both cases. If the effect of the two methyl groups js small, it may be concluded that the solution and crystal calculations are in agreement. It is also implied that Scholte's model does not predict the correct shape effect contribution for glycine in solution, as might be expected from the fact that glycine is only approximatlely ellipsoidal and has a quite different effective refractive index a t opposite ends of the molecule. Some qualitative comments in terms of bond polarizabilities are of interest. Le Fevre6 assembled a table giving the three principal values for most of the conimonly found bonds. He advised caution in their use because of the importance of bond environment and interactions between neighboring bonds.36 Within this framework, the intrinsic anisotropy of glycine may be attributed to four bonds: the two CO, the CC, and the CN bonds. Assuming that the CO bonds are equivalent to double bonds, as suggested by bond dist a n c e ~ , ~direct ' use of Le Fevre's table leads t o values in qualitative agreement with Table XI1 ( a y y> a,, :> aZz). From the bond polarizability point of view, the contribution of the CC bond to the anisotropy is as large as the net contribution of the two CO bonds, and the CN bond makes a smaller contribution. (36) L. Silberstein, Phil. Mag., 33, 92, 215, 521 (1917). (37) C. R. Noller, "Chemistry of Organic Compounds," 2nd Ed., W. B. Saunders Co., Philadelphia, Pa., 1958, p. 155.
