STUDIES ON CRYSTALLINE HYDRATES
2033
Studies on Crystalline Hydrates by M. F. C. Ladd and W. H. Lee Departments of Chemical Physics and of Chemistry, University of Surrey, Guildford, Surrey, England (Received September 83, 1 9 6 8 )
The process of hydrate formation is considered in terms of an electrostatic model. It is suggested that hydrate formation can be considered in two stages: (1) the expansion of the anhydrous crystal and the accommodation of gaseous water molecules and (2) the water-ion interaction. Results which are presented for a number of dihydrates extend our previous thermodynamic studies.
Introduction In an earlier paper,' we discussed some aspects of the thermodynamics of crystalline ionic hydrates. Crystal energies were calculated for a variety of compounds, using a modified Born-Haber cycle. The free energy changes, AGw, for reactions of the type MX(c)
+ nHzO(1) -+
MX.nHzO(c)
(1)
were deduced, and the component enthalpies and entropies were discussed. In this paper, we consider further the factors which govern hydrate formation. A series of dihydrates has been chosen for this investigation.
Calculations The crystal energy, U ( L ) ,has been evaluated with equations based upon the simple Born-Mayer electrostatic model2
where A' is the Avogadro constant, J is the Joule equivalent, A ( L ) is the Madelung constant, e is the electronic charge, p is a repulsion constant, p is the isothermal compressibility, and uo is the volume occupied per formula weight. The Madelung term, A ( L )/ L , was calculated by the method of Bertaut3 as a single absolutely convergent infinite series in which the terms are functions of reciprocal lattice vectors. A consideration of the extensions of this calculation by Templeton* and by Jones and Templetons led to the adoption of a linear charge distribution. Adequate convergence was obtained with a summation limit of 1/R reciprocal units, where R was taken as 0.495 times the shortest interatomic distance, L, in the crystal structure. Increasing the summation limit to 3/2R altered the Madelung constant by less than 0.01%; accordingly, the lower summation limit was used.
Among the compounds studied, compressibility data are available for only sodium bromide16barium chloride dihydrate,' calcium sulfate,8 and calcium sulfate dihydrate.8 The results for the crystal energies, obtained through these compressibilities and eq 2 and 3, were underestimated to the extent of 3-8%, as compared with the corresponding Born-Haber cycle values. Such discrepancies are not unreasonable, since dipole-dipole and dipole-quadrupole interactions have been neglected. In order to produce consistent models for discussing hydrate formation, we have evaluated p/L from eq 2 and the thermodynamic values for crystal energies reported in our earlier paper^.^!^ For both the hydrates and the expanded structures, p/L has been deduced from the corresponding values for the anhydrous compounds. The approximate proportionality between compressibility and molar volumelo has been used. This is particularly appropriate for structures of similar degree of ionicity and containing similar atomic species, Thus, uo/p in eq 3 is a constant for different structures. Given ( p / L )1 for the anhydrous compound, 1, ( P / L for )~ the related hydrate, 2, is obtained from
(4) A ( L ) e 2 / Lfor structures 1 and 2, respectively. It is noteworthy that a change of &15% in p produces a consequent change of only about 1% in the crystal energy. We define the hypothetical crystal MX" as having a12is the ratio of the terms
(1) 1L.I.F. C. Ladd and W. H . Lee, J . Phys. Chem.. 69, 1840 (1965). (2) M. Born and J. E. Mayer, Z . Physik., 7 5 , 1 (1932). (3) F. Bertaut. J. Phys. Radium. 13, 499 (1952). (4) D.H.Templeton, J . Chem. Phys., 2 3 , 1629 (1955). (5) R. E.Jones and D. H . Templeton. ibid., 2 5 , 1062 (1956). (6) M. F. 0. Ladd and W. H. Lee, Progr. Solid State Chem., 2 , 378 (1965). (7) E. Saerens, Bull. SOC.Chim. Belges, 3 3 , 17 (1924). (8) E . Madelung and R. Fuchs. Ann. Phys., 6 5 , 289 (1921). (9) M . F. C. Ladd and W. H. Lee, Progr. Solid Stale Chem.. 1, 37 (1964). (10) J. R. Partington, "An Advanced Treatise on Physical Chemistry," Vol. 111, Longmans, Green and Co., London, 1052.
Volume YS, Number 6 June 1969
M. F. C. LADDAND W. H. LEE
2034 Table I: Energetice of Hydrates A ( L ) / L ,- 4 - 1
L,
11
- U , kcal/mol (Born-Haber)
NaBr XaBr.2HzO NaBr*
0.6028 0.7060 0.4817
2.90 0.99 2.89
NaCK NaCK*2H10 NaCN*
0.5931 0.6638 0.4804
2.95 0.99 2.30
I78 202
Cas08
3.20 0.99 3.70
613“ 63ia
CaS04*
2.007 2.107 1.855
BaCl2 BaCln. 2Hr0 BaCl?*
1.585 1.671 1.409
3.18 0.99 3.13
494 521
CuFz CuFS. 2H20 CuFr*
2.391 2.336 2.024
1.93 0.99 1.90
722 753h
CaS04.2HgO
a The value of AHr[SOa2-(g)] = value.
Table 11: Compressibility Data l 0 6 ~ ( e x p t l ) , lOep (calcd), bar-! bar-’
CaS04.2H2O BaC12.2H20
CuF2.2H20 The Journal of Physical Chemistry
...
...
...
1024vo,cm-a
(eq 3)
0.13 (I) 0.14 (6) 0.11 (0) io.11 (3) 0.096 0.10 (5)
53.3 106 106
200 142
51.2 103 103
197 147
76.4 126 126
642 ,568
85.9 131 131
519 444
34.4 78.3 78.3
701 619
0,080
0.083 10.076 10.080 0.061 0.064
...
...
0.091 0.089 io. 079 10,079
...
...
... ...
169 kcal/g-ion was used: ?VI.F. C. Ladd and W. H. Lee, J. Iiiorg. Nucl. Chern., 30, 330 (1968).
the structure of the corresponding hydrate but lacking the water molecule-ion interactions. Since MX* and MX*nHsO then have the same molar volume, we shall assume that they have similar compressibilities. This postulate is supported by the apparent incompressible nature of coordinated water molecules, as determined in several studies on ion solvation.11J2 Hydration numbers of ions, derived from measurements of ultrasonic velocities in electrolyte solutions and the assumption that water molecules in the primary solvation sheaths are incompressible, are in agreement with those obtained by other methods. Crystal structure data have been taken from Wyckoff ,I3-l5 except for sodium bromide dihydrate; for this compound, the hydrogen atom positions were located by a method due to Ladd.16 A simple dipolar model for the water molecule, taking 0-H = 0.99A and a dipole moment of 1.84 D, led to a fractional charge of -0.31 on the oxygen atom and consequent charges of S O . l . 5 (5) on each of the hydrogen atoms. The results of the calculations are listed in Table I and
NaBr * 2H20 N a C S . 2HgO Cas04
174 198
- L7(L),kcal/mol P/L
...
...
(P/L)‘
...
...
0.15 (6) 0.084
1.7 2.5 2.8
1.5 2.8 2.4
0,090
...
...
...
0.061 0.031
* Corrected
are correct to within 2-37,. It may be noted that once p / L has been obtained for both the hydrate and the anhydrous compound, p/L for the expanded structure (MX*) may be obtained by two routes indicated by using ff13 and a3in eq 4;the two results for p / L so obtained are bracketed in column 5 of Table I. Table I1 lists experimental and calculated (eq 3) compressibilities, and ( p / L )’ for the hydrate evaluated from eq 2 and the thermodynamic crystal energies.’ Comparisons of @(exptl)and p(ca1cd) and of ( p / L ) ’ and p / L (Table I) are reasonable, except for copper(I1) fluoride dihydrate. This substance is exceptional also in that its Madelung term is lower than that for the corresponding anhydrous compound, although its Born-Haber cycle crystal energy is numerically larger.
Model for Hydrate Formation We suggest that the process of hydrate formation may be regarded in two stages. The anhydrous crystal structure (MX) is first expanded and gaseous water molecules accommodated, so that the resulting structure (MX*) is equivalent to that of the hydrate, but without water molecule-ion interaction. This process requires an expenditure of enthalpy ( A H r ) on the (11) A. Pasynskii, Acta Physicochim. U R S S , 8, 385 (1938). (12) D.S. Allam and W. H . Lee, J. Chem. Soc., 426 (1966). (13) R. W. G. Wyckoff, “Crystal Structures,” Vol. I, Interscience Publishers, Inc., Kew York, N. Y., 1963. (14) R. W. G. Wyckoff, “Crystal Structures,” Vol. 11, Interscience Publishers, Inc., New York, N. Y..1964. (15) R. W.G.Wyckoff, “Crystal Structures,” Vol. 111, Interscience Publishers, Inc., New York, N. Y.. 1965. (16) M .F. C. Ladd, Z.Krist., 126, 147 (1968).
2035
STUDIES ON CRYSTALLINE HYDRATES system. Then, the water molecules and ions interact, and enthalpy ( A H + ) is released by the system. This is shown schematically as expansion + nHzO(g) MX (c) --A MX* AH t water-ion interaction
' MX*nHzO(c)
Table 111: Enthalpy Changes in Hydrate Formation AH(,
NaBr-water NaCN-water CaSOa-water BaClz-water CuFz-water
-AH+*
-AHwT
kcal /mol
kcal /mol
kcal /mol
53 52 66 71 124
57 56 70 78 135
4 4 4 7 11
AH4
The relationships among the hydrate, the anhydrous compound, and the expanded structure are shown diagrammatically in Figure 1, in terms of enthalpy
L
MX.nHzO(c)
1
Figure 1. Enthalpy levels in the formation of hydrates.
changes : for any process, the corresponding internal energy change is given by A U = AH
+ nRT
(5)
Equation 1 represents the practical case of the formation of a hydrate from the anhydrous crystal and liquid water; in a crystalline hydrate, the water is present as discrete molecules. We call AH+ and AHr the water interaction enthalpy and the expansion enthalpy, respectively; they are related by the equation AHw = A H +
+ AHc
(6) It should be noted that the molar heat of vaporization of liquid water, AH,, is a part of the term AHe, The results of the calculations of AH+, AHc, and AHw are listed in Table 111.
zation of 2 mol of water requires an expenditure of enthalpy of 53 kcal/mol of NaBr. The water interaction enthalpy is -57 kcal/mol of NaBr, so that AHw is -4 kcal/mol of NaBr. It may appear from Figure 1 that there are two routes to the evaluation of AH,. This is not so, because the crystal energies are interrelated through the calculation of p / L for MX*, From the point of view of enthalpy changes, a hydrate is formed when the enthalpy of expansion is more than compensated by the water interaction enthalpy. This is the case if A H , is negative.' The water interaction term is about -29 kcal/mol of HzO for the 1:l electrovalent salts studied, and about -37 kcal/mol of HzO in the case of 2:2 electrovalent compounds. Copper (11) fluoride has the correspondingly large value of -68 kcal/mol of HzO. This is not related to a possible discrepancy in the Madelung constant for CuFz.2H20, to which reference has been made. It is more probable that the large value of AH+ reflects strong F-H-0 hydrogen bonds in the s t r u c t ~ r e . ' ~ -The ~ ~ term AHc is correspondingly large. Copper(I1) fluoride has a compact structure: the efficiency of packing of its ions is 72%, compared with 65% for NaBr (similar mass) and with 67% for BaClz (similar charges). Relatively more work is required to expand the CuFz structure, since the hydrates do not show the same variations in packing efficiency: CuF202H20, 60%; NaBr. 2H20, 55%; BaClZ-2Hz0, 62%The formation of the hydrate is governed mainly by AH,. Since the magnitude of this term depends upon the difference between two relatively large terms, A H f and A H + , the differences in the water interaction enthalpy in different types of hydrates are not revealed clearly by thermodynamic calculations alone.'
Discussion In the particular case of sodium bromide, for example, the expansion of the anhydrous crystal and the vapori-
(17) 9. Geller and W. L. Bond, J. Chem. P h y s . . 29, 925 (1958). (18) S. C . Abrahams, ibid., 36, 56 (1962). (19) 9. C . Abrahams and E. Prince, i b i d . , 36, 50 (1962).
Volume 7S,Number 6 June 1969
