ENERGIES OF DISOR~ENTATIOK IN SOLID CO AND NzO
2301
Energies of Disorientation in Solid Carbon Monoxide and Nitrous Oxide'
by Marian Whitney Melhuish and Robert L. Scott Contribution No. 1660 f r o m the Department of Chemistry, University of California, Los Angeles EL$, California (Received April 3, 1964)
The f.c.c. structure of a-Nz, a-CO, COz, and KzO is a consequence of the quadrupolequadrupole interactions between the molecules. The classical electrostatic energies resulting from fixed dipoles and quadrupoles in these crystals have been calculated by summation over the first 449 shells on an electronic computer. Depending upon the somewhat uncertain values of the quadrupole moments, the quadrupole-quadrupole contribution ranges from 10 to 40% of the total lattice energy. The additional energy arising froni the dipoledipole interactions is very small, -4.5 and -10.7 cal. mole-l for a-CO and NzO, respectively. 'The Curie temperatures for dipole orientation in these substances are thus around 5 andl llOK., a t which temperatures the rate of the ordering process will be almost infinitesimally slow. Consequently, the disorientation entropy of R In 2 for these substances is always present above these temperatures and will in practice persist to the lowest temperatures.
Introduction It is well known that the low temperature crystalline phases of carbon monoxide and nitrous oxide retain a residual entropy of approximately R In 2 per mole to the lowest temperatures a t which heat capacities have been m e a s ~ r e d . ~The ! ~ universally accepted explanation of these residual entropies is that the linear hetero-. nuclear molecules CO and KiYO (which are isoelec-. tronic with NS and OCO, respectively) are distributed on their crystal lattices with no discrimination between their actually different opposite ends. However, carbon monoxide and nitrous oxide do have small dipole moments; and it is obvious that a t sufficiently low temperatures (e.g., below 1OK.) the equilibrium state will be one in which the dipoles are oriented. While it is obvious that a t these temperatures this equilibrium state would be approached a t an almost infinitesimally rate, it is nonetheless of interest to attempt the calculation of the oriented and disoriented lattice energies and to estimate the Curie temperatures for the order-disorder transitions. Crystal Structures and Parameters COS, SZO, and the low temperature phases a-Sz and a-CO all have essentially the same crystal structure, an f.c.c. lattice in which the linear molecules are located a t lattice sites and oriented along the body diagonals
(space group T h e or Pa3). This relatively complex structure arises from minimizing the electrostatic energy of the strong quadrupole-quadrupole interaction, which has the result of making the net contribution of the interaction between the weak dipoles even smaller. The opposing influences of the quadrupolar and dipolar interactions are clearly illustrated in Table I, which shows the relative magnitudes of the electrostatic interactions between two isolated molecules in different orientations. Table I : Electrostatic Energies for Different Pair Orientations Perpen-
Parallel
II
Dipoles Quadrupoles
u12/w2r-8
fl
ulz/@r
+'/4
-6
Skew
/.
0 +'/4
dicular I-
0 -3
Linear
__ f 2 $6
The state of minimum energy (greatest attraction) for an isolated pair of point quadrupoles is the perpen(1) Paper presented before the Division of Physical Chemistry at the 146th National Meeting of the American Chemical Society, Denver, Colo. January, 1964. (2) W. F. Giauque and J. 0. Clayton, J . Am. Chem. SOC.,54, 2610 (1932); 5 5 , 5071 (1933). (3) R. W. Blue and W. F. Giauque, ibid.. 57, 991 (1935).
Volume 68, Number 8 August, I964
2302
b f A R I A S W H I T N E Y hfELHU1SI-I AXD
dicular orientation; but if the molecules have dipoles oriented along the principal axes of the quadrupoles, the dipolar energy is exactly zero since the vectors are orthogonal. Conversely, if the molecules assume the linear orientation most favorable for two dipoles (-+ +), the quadrupolar energy is repulsive. For a molecule like CO, with the parameters given in Table 11, the dipolar energy in the most favorable (linear) Table I1 : Lattice Parameters and Molecular Constants" adl0-8 om.
2
fJ
/L/lO-'s e.8.u.
e/lo-ze 0.8.U.
(0.039)b 0 . 0530e (0.042)' 0.117'
0
(+l,
coz
( 5 . 6 7 ) * (0.070)' 5. 644d 0.0530" 5.63' (0.067)' 5.540' 0.117'
0.112' 0
N2O
5 . 6 ~ 5 ~ 0.117k
0.117k
0.166'
(fl)h -4. l i (f4.8)' (-2.5)O
a-Nz
a-CO
a Where possible extrapolated to O0K. * 1,. Vegard, 2. Physik., 58, 497 (1929). A. D. Buckingham, Quart. Rev., 13, 183 (1959). L. H. Bolz, M. E. Boyd, F. A. Mauer, and H. S. Peiser, Acta Cryst., 12, 247 (1959). ' J. Donohue, ibid., 14, 1000 (1961). L. Vegard, 2. Phgslsik., 61, 185 (1930). ' C. A. Burrus, J . Chem. Phys., 28,427 (1958); 31,1270 (1959). A. D. W. H. Buckingham, Ann. Rept. Chem. Soc., 57, 53 (1961). Keesom and J. W. L. Kohler, Physica, 1, 167 (1935); J. de S nedt and W. H. Keesom, Proc. Roy. Acad. Sei. Amsterdam, 27, 839 (1924). 7 Ref. 7 . L. Vegard, 2. Physik, 71, 465 (1931). ' R. G. Shulman, B. P. Dailey, and C. H. Townes, Phys. Rev., 78, 145 (1950).
'
The Journal of Physicnl Chemistry
SCOTT
amined the effect of displacing the dipoles and quadrupoles from the lattice sites.
Electrostatic Energy of the Lattices Instead of calculating the electrostatic interaction energies of point dipoles and point quadrupoles in appropriate positions and orientations (as was done in an earlier paper by Jansen, Michels, and Lupton*), we iiistead assumed discrete dipolar and quadrupolar charge distributions for each molecule and summed over all the electrostatic interactions. These were ultimately expanded into inverse powers of the 1att)ice parameter ao, so the final results are equivalent to one which starts with a multipole expansion. The central dipole is assumed to consist of a charge 6 f at position x, 5 , x corresponding to one end iiucleus and a charge 6- at position -yI -y, -y corresponding to the other end nucleus; other dipoles are equivalently located at the other lattice sites. Similarly the central quadrupole is assumed to arise from two charges q - a t x, x, x and - y, -y, -y, and a charge 2qf a t (x - y)/2, (x - y)/2, (x - y)/2, with other quadrupoles equivalently orbited. Because of the symmetry of the lattice there is no net interaction between the dipoles and the quadrupoles. The energy of interaction between the central dipole and the rest of the (ordered) lattice is 62 Udipole
=
-
a0
[A(x
+ y)2 + B ( x + Y)'((x - y)2 + C(X
orientation is only 10% of the quadrupolar energy in its most favorable (perpendicular) orientation (both a t the same distance of separation). Thus, it is not surprising that the quadrupolar interaction dominates and determines the structure. In the crystal, with its three-dimensional large lattice, the exact cancellation of the dipolar energy no longer obtains, but the qualitative feature remains: if the orientation is determined by minimizing the quadrupole-quadrupole energy, the dipole-dipole energy becomes very unimportant. Table I1 summarizes the lattice parameters for the four crystals as determined from X-ray diffraction; a0 is the length of the cubic unit cell, while the terminal nuclei of the molecule are a t positions x, x , x and -y, -y, -y (in units of ao) from the lattice point. It is interesting to note that the space group T h 6 requires that the molecules Kzand COz be centered on the lattice sites. Vegard reported that 1C2 and CO were not so centered (space group T4 in which x f y) but more recent work has largely disproved this for Kz and we suspect that the similar data for CO were probably also in error. Nonetheless in our calculations we have ex-
ROBERT L.
+ YI4 + . .
'
'
1 (1)
while the corresponding energy of interaction between the central quadrupole and the rest of the lattice is
R(s
+ YP + ... .I
(2)
The terms A , B , C, P, &, R of the expansion of the energies (see Appendix) were evaluated for each of the first 449 shells6 with the aid of high speed electronic computers (a CDC 1640 a t the University of California, San Diego, and an IBNI 7090 a t the University of California, Los hngeles) ; the coefficients for the first few shells (and the number n, in each shell) and the sums are shown in Table 111. (4) L. Jansen. A. Michels, and J. M. Lupton, Physica, 20, 1235 (1954). (We were unaware of this paper until this research was nearly romplete.) (5) By shell number, we mean 2ro*/ao2 where ro is the distance between a member of the shell and the central molecule. T h e 450th shell is a t distanre T O = 15ua (indices 15, 0, 0). Some shells are empty (14, 30, 46, etc.). while others contaln two or more subshells which are nonequivalent (9, 13, 17, etc.).
ENERGIES O F DISORIENTATION
IN SOLID
co AND NzO
2303
Table 111: Energy Coefficients of Vnriouu Shells Shell
12 6 24 12 24 8 48 6
1 2 3 4 5 6 7 8 9 10
12
B
A
ng
+ 24
24
Sum 1-449 “Best value” Estimated uncertainty
-33.94 0 13.06 0 -6.07 0 -1.50 0 2.93 0 26,046 26,Ol 10.04
-1272.8 0 10.9 0 -4.7 0 14.6 0 -9.5 0 1260,43 1260.43 f O .01
I n Table 111, the “best value” of each coefficient was selected after studying the variation of the sum over several hundred shells. There are fluctuations in the sums with long periods; thus, the sum over the 449 shells is not necessarily the best value. I n particular the dipolar coefficient A converges slowly; the first shell contributes 130’% of the total, and the sum over 10 shells is 96% of the total. If we convert to the point multipoles, we may write y), for the quadfor the dipole moment p = d&ao(z rupole moment 8 = 3qu0’(z ~ ) ~ / : 2for , the octopole y)3/4. == 3paoz(z y)2/4, moment Q = 3d361203(2 and for the hexadecapole moment = 9qao4(z 2/)4 = 30a02(z ~)~/4. If we substitute these and the numerical values of the coefficients into eq. 1 and 2, we obtain for the energy of a lattice of N molecules
+
+
+
+ +
+
13.3@
ao5
+
..
.] (3)
+
..
.] (4)
0%
257.1-a07
With the lattice parameters and inoments of Table I1 we calculate the energies shown in Table IV. The lattice energy (El,tt,,,.)is essentially the negative of the energy of sublimation a t 0°K.6 If Vegard’s unequal values of IL: and y were correct for CO, the factor { l 4 8 . 5 ( ~- y)21 == 1.03 and E d l p o l e s would be -4.64 cal. mole-I. Similarly for the quad-
+
C
P
-- 101.82
-241.83 -33..50 -2,oo 2.78 1.55 1.80 1.52 -0.98 -1.23 -0.45 -269.245 -269.246 *o. 002
42.00 6.29 -3.71 -3.64 -2.40 2.84 1.31 -1.53 0.60 60.214 60,214 *o. 001
Q
R
14,223 0 -325 0 - 18 0 13 0 5 0 13,894,20 13,894.2
827.3 67.5 -11.2 -19.4 -0.2 3.4 0.2 0.5 0.3 -0.7 867,841 867.84 *o. 01
f O .1
Table IV : Lattice Energies and Electrostatic Contributions, Gal. Mole-‘, Assuming z = y
u-co
a-Nn ElatLioe Edipois-dipole Edipole-ootopde
Total Edipolea Equediupoie-quadiupoie
Esuadrupole-heaadsospole
Total Esuadrupoiea
-1650
-1900
0 0 0
coz -6450
-4.39 -0.12 -4.51
6 - 160) (f6)
( -180)
+7 0-150) (-170)
N20
0 0 0 -2800 +500 -2300
-5800 -9.54 -1.20 -10.72 ( -940)
(Jr170) (-770)
rupolar energy ( 1 - 51.4(x - Y)~)would be 0.97 for
co.
Jansen, Michels, and Lupton4 reported the dipolar energy for a-CO as -3.4 cal. mo1ep1. Our value of -4.51 cal. mole-1 differs from theirs (a) because they summed over the first four shells only and the coefficients (especially that for p 2 ) converge slowly as the number of shells increases and (b) because they neglected the dipole-octopole term which contributes 3% of the total (llyoin N20). With the exception of COZ, whose quadrupole moment has recently been determined by a direct measurement,’ all quadrupole moments are indirectly deduced and are consequently not very accurate. Nonetheless, it is apparent that the classical electrostatic interaction between point quadrupoles accounts for only a small part of the lattice energies of these crystals. The principal contribution to the energy must be the Lon(6) K. K. Kelley, U. S. Department of the Interior Bureau, of Mines Bulletin No. 383, 1935. (7) A. D. Buokingham and R. L. Disch, Proc. R o y . Soc., ( L o n d o n ) , A 2 7 3 , 2 7 5 (1963).
Volume 68, Number 8
August, 1964
2304
M A R I A N WHITKEY MELHUJSH AND
don dispersion effect ("van der Waals attraction") although there must be a quadrupole-induced quadrupole effect as well.
Order-Disorder Effects When the orientations of the dipoles are distributed randomly between positive and negative directions, E d i p o l e s = 0, but E q u a d r u p o i e s is unchanged. I n other words the energy of disorientation is simply AE
=
Edisoriented
-
Eoriented =
O -
Edipoles = -EdiBole8
(5)
I n the zeroth approximation to the order-disorder problem (e.g., the Bragg-Williams theory8), the Curie temperature is related to the disorientation energy by the equation
NLT,
=
2AE
(6)
which gives T , for CO and N 2 0 as approximately 5 and 11"K., respectively. Since the energy of activation for rotating a dipole in the crystal lattice must be large (e.g., several kcal.), the rate at which the crystals might orient at temperatures below T , must be utterly negligible. One concludes therefore that a residual entropy of R In 2 = 1.38 cal. deg.-l mole-' is to be expected in crystalline CO and K20 at low temperatures. The residual entropies determined by calorimetric s t ~ d i e s(1.10 ~ , ~ cal. deg.-' mole-' for CO and 1.14 cal. deg.-' mole-1 for N 2 0 ) are close to R In 2 but somewhat lower. Only a careful examination of the data (and probably additional experimental studies) can determine whether these discrepancies of about 0.1R are due to experimental error or to a slight amount of ordering in the crystals at low temperatures. If additional examples of residual entropy of dipolar substances like CO and S20are to be found, they will be found only in crystals where a structure is essentially fixed by factors other than the dipole-dipole interactions and where, because of this, the dipoledipole energy is very nearly negligible. Acknowledgments. This research was supported in the greater part by the Petroleum Research Fund of the American Chemical Society and in a lesser part by the Xational Science Foundation. We thank Miss Judith Edwards for assistance with the computer work a t the University of California, Los hngeles. We also thank the University of California, San Diego (where one of us (M.W.11.) was temporarily resident) for the use of the CDC 1640 computer.
The Journal of Phgsical Chemistry
ROBERT L.
SCOTT
Appendix Equations $01. the Shell Coeficients. I n the unit cell of the f.c.c. lattice there are four equivalent lattice points: ( H , K , L), ( H K L), (H I/Z, L where l/2, K , L '/d, and ( H , K H , K , and L are integers (not to be confused with the indices h, IC, 1 of the reciprocal lattice). The evennumbered shells around a central molecule a t (0, 0, 0) arise from the first of these while odd-numbered shells arise from the permutations of the other three. I n developing the power series which lead to the coefficients A , B , C, P , Q, R, one must distinguish between these two sets. The following equations do not yield the energy of interaction between a central molecule and a molecule a t lattice point ( H , K , L ) , etc., because the expressions have already been summed by permuting the indices through all the n, members (6, 8, 1 2 , 2 4 , 4 8 ) of a particular subshell ; a single representative set of indices suffices. Even-Numbered Shells.
+
+
+
+
+
+
A=B=O C
P
+ H 2 L 2+ K2L2)- p 4 ) / p Q = 2 1 n , { 5 ( H 2 K 2+ H 2 L 2+ K 2 L 2 )- p 4 ) / 4 p 9 =
-7n,(8(H2K2
Q = O
R
=
45n,{231H2K2L2-
+ H 2 L 2+ K 2 L 2 ) p 2+ 2 ~ ~ ] / 8 p ' ~ where p2 = ro2/ao2= H 2 + K 2 + L2. 21(H2K2
Odd-Numbered Shells.
A
=
n,(3H2 -
p2)/p5
B = -15n,f7(H2 C
=
+ ~K'X')
- 4p2H2- p 4 ] / 2 p 9
+ 3p4)/2pg
-n,(35H4 - 30H2p2
+
P = - 3 n , { 3 5 ~ ~ X ~ 5H2p2- 4p4)/4p9 Q = 45n,{231H6 - 3 0 0 3 H 2 ~ 2 X-2 (588H4 -
+
273ti2X2)p2 378H2p4- 3 1 ~ ~ \ / 1 6 p ' ~
R
=
-15n,{231H2ti2X2
+ 21(H4 -
ti2X2)p2
-
+
2 1 ~ 2 ~ 44p6)/8p13 where K H 2 K~
=
K
+ + X2.
+ 1/2, X
=
L
+ 1/2,
p2 =
ro2/ao2=
(8) W. L. Bragg and E. J. Williams, Proc. Roy. Soc. (London), A145, 699 (1934); A151, 540 (1935).
