Combining Residual Entropy and Jack M. Williams' Argonne Notionol Loborotory Argonne, lllinois 60439
I
Diffraction Results to Understand Crystal Structure
Residual entropy measurements2 offer great potential as an aid in the interpretation of molecular structure of crystalline substances. However, in order to understand the relationship between molecular structure derived directly from crystal structure analysis, and residual entropy derived from statistical thermodvnamics and heat caoacitv measurements, we must first consider the Third i a w df Thermodvnamics and its "structural" imdications. Probably the'most vivid expression of the ~ h i r dLaw is the statement (2) that "every substance has a finite positive entropy, hut at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances." Similarly the Nernst heat theorem states that in the case of a crystalline material if lim S G O T+O
is obeyed, i t may he concluded that the system is "perfectly ordered" a t low temperature. However, having determined that lim S = 0 T-0
for any particular system, caution must be exercjsed in using such data to predict molecular geometries (uide infra). Ice and the Bifluoride Ion, (F-H-F)-, in KHF2 The most classic use of residual entropy measurements in predicting molecular structure was the correct interpretation by Pauling (3) of the disordered structure of ice. The discovery of a residual entropy in ordinary ice (-0.82 cal/deg/mole) a t low temperature by Giaque et al. (4) led Pauling to suggest that the discrepancy between the Third Law entropy3 and the experimental entropy was caused by the presence of water molecules "frozen in" one of many possible configurations, differing only in orientation, as illustrated in Figure 1. The successful interpretation of the structure in terms of the beat capacity data led numerous investigators to turn their attention to the pmto-
type of the strong hydrogen bond:. the bifluoride ion (FH-F)-. This anion contains the shortest. stronaest - hvdrogen hond known and if Lim S + 0 T+ 0 it would suggest the presence of a disorder indicative of a non-centered H-atom in (F-H-F)-. Such a disorder could in turn indicate the presence of a double minimum potential energy surface for the bridging proton. Arguing from rather incomplete infrared data, two independent investigators (8, 9) concluded that a double minimum exists in (F-H-F)- in KHFz. It is important to note that a t that time (1942) this conclusion was partially due to the incorrect but prevailing belief that H-X distances were fixed from one system to another; i.e., since the gas phase H-F separation is -0.95 A (A = 10-8 cm) and the F...F separation in ( F - H Z - is -2.26 A, then one might expect two H-F separations of -0.95 and -1.31 A, respectively, assuming the anion is linear. From more extensive infrared measurements of KHFz, Ketelaar (10) also concluded that the anion was asvmmetric. However. Westrum and Pitzer (11) reassignedUKetelaar's spectra: scooic data and concluded there was no evidence of double minimum character in the anion. Added evidence for single minimum character came from the low temperature heat capacity measurements of KHF2, combined with a statistical calculation of the change in entropy of the reaction KFisj HFI,I.= KHFz is), by Westrum and Pitzer (12) in 1949. They concluded that KHFz reaches zero entropy when cooled to 0°K; hence there is no residual randomness, which was interpreted in terms of a "centered" single minimum hydrogen hond for (F-H-F)-. However, this particular finding indicated only that the crystal lattice was ordered; i.e., any conclusion about the precise molecular geometry of the ( H F z ) ion could only he inferred. For example, two very different ( H F z ) ion configurations (centered and noncentered) which if ordered in a perfect crystal would both yield
+
Possible Linear (HX& Ion Configurations and Associated lim S Values T-0
1 Work performed under the auspices of the US. Atomic Energy Commission. 2 By using the Third Law one can compute the absolute entropy (SST)of a crystalline material, at temperatures other than 0°K. from its heat capacity at constant pressure ((2,). The total A S change from liquid helium temperature up to the point of vaporization is a quantity that can be evaluated independently (if malecular interactions in the vapor are small) using the methods of statistical thermadvnamies. The residual entrow at O"K, So", is derived by comparing SOT determined experimentally with that computed using statistical thermodynamics (See Ref. ( 2 ) ) . 3 Pauling's calculation (3) of the residual entropy of ice (So' = 0.805 e.u.) is, in fact, not exact. Onsager and Dupuis (5) have shown that Pauling's result is actually a lower hound for the number. The work of DiMarzio and Stillinger (6), who applied methods of lattice statistics to the problem, has been extended hy Nagle (7) who proved that the correct value of So' is 0.8145 e.u. for ordinarv * ice. Thus the value derived bv Pauline " is within 1.2% of the correct value. ~
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/ Journal of Chemical Education
CO"fi9"ration ~attice Con%-
Lattice lim S T-O
(X-H-XI - (X-H-X) X-H-x X-H-X X-H-X X-H-X X-H-X X-H-x ordered -0
X-H-x X-H-X X-H-x X-H-X X-H-X X-H-x ordered -0
- (X-H-X) X-H-x X-H-X X-H-X X-H-X X-H-X X-H-x ordered -0
-
(b)
+ (e)
(X-Ha-X)
X-H-x X-H-X
X-H~-X X-H"-X
X-H-X X-H-X X-H-X X-H-x
X-Ha-X X-HLX
disordered
>o
-
X-Ha-X X-H"-x disordered
>o
In thin ease the H-atom ia distributed bstween two sites, described by a double well potential, and it is equally likely that it will be in either site a t an given instant in time. TThis type of departure from perfect periodicity in the crystal lattice ia sometimes referred to as a stotic disorder OL.a ''diorder in space" and arises in this example simply because two ion orientations, differing only by a 180rotation, exist in the lattice. ( A dynamic disorder, referred to as a "disorder in time," arises via atom "motion," e.g. the tunnel effect in ploton transfer.
Figure 1 . A s t e r e o s c o p i c illustration of t h e structure of ordinary i c e v i e w e d a l o n g t h e c a x i s (16). lndicaled proton s i t e positions (small c i r c l e s ) are half-fiiied.
lim
S =0
T+O are illustrated in the tahle as configurations a, b, or c. Therefore, when in the course of a heat capacity study i t is observed that lim S = 0
T+O then a centered hydrogen atom configuration such as (Ia) is always allowed hut (Ih) or (Ic) only if all equivalent (HF2)- anion dipoles are identically oriented in the lattice unit cells. In the case of the diffraction studies of KHFz the lattice imposed (crystallographic) center of symmetry of the (FH-F)- ion required the hridging H-atom he located at, or svmmetricallv dis~laced (disordered) about. the hond center. ~ m amneutron diffraction (13) investigation, in which H-atoms are easily "visible," it was concluded that the H-atom was pmhahly located a t the centered position and could not in anv case he more than 0.05 A hevond it even if H-atom diso;der existed. However, if config&ation (Ih) were present its centrosymmetric equivalent (Ic) must also he; i.e., an "orientational" disorder of (F-H-F ) and (F--H-F)- ions as depicted in (Id), would exist and result in lim
S>O
T+ 0 Since the anion is constrained by a center of symmetry, and experimentally
T-0 not because the ion was centered, hut rather because the
ordered-asymmetric ion is always oriented in an identical fashion from one unit cell to another in the crystal as depicted in (Ib) or (Ic). A random arrangement of asymmetric (CI-H--C1) ions in two possible statically ordered configurations, as depicted in (Id), would yield lim S =Rln2
T+O Conclusion I t is honed that the examdes discussed here convev to required to understandgnd the reade; the basic relate residual entrouy .- and diffraction findings. Clearly heat capacity data are useful in predicting~molecul& structure, hut must he used cautiously. While diffraction data alone are usually sufficient to settle questions of molecular geometry, there are often occasions when an interpretation is not unique. Thus it is obvious that the marriage of heat capacity and diffraction data is extremely powerful in elucidating molecular structure. Literature Cited (1) W&rum,E.F.,JI.. J.CHEM.EDUC..39.C13(19621. (2) Lewis, G. N., and Randsll, M.. "Thermodynamics and the Free E n e m of Chemical Substances," 1st Ed.. Mecraw-HillBmk Co., lnc., New York, 1923, p. M8. 13) Pading, L . , d Amer Chem. Soc, 57.2680119&5). (41 Giaque, W. F., and Ashley, M., Phyr. Re"., 43, 81 11933): Giaque, W. F.. and Stout, J. W..J Amrr Chem. Soc.. 58. Ilk4 (1936). (5) Onsagor, L., and Dupuis, M., Rc. Sru. Int. Pis. ''ERnco Rrmi" 10, 294 (19Ml; "Eleetmlytd' (Editor Pesee, B.1, P e r g a m o n P ~ ~ Iandon a. (1962). (61 OiManio. E.A.,andStiilinger,F.H.. Jr.. J. Cham.Phys.. 40.1517 119641. (7) Naeio, J. F., J. Moth. Phys.. 7,1484 (19661. ( 8 ) Buswell, A. M., Maycock. R. L..and Radebush. W. H.. J. Chem. Phyr.. 8, 362
