7830
J. Phys. Chem. 1990, 94, 7830-7834
that began in 1952, and J.K.H. thanks Frofessor Pitzer for his present advice and guidance as well as providing the facilities that have been used for preparation of part of this paper. We are grateful to Dr. Donald Palmer for providing us with a copy of ref 1 prior to publication. We also thank Dr. Peter Tremaine for helpful discussions and for allowing us to use the calorimeter and densimeter at the Alberta Research Council. Finally we thank
the National Sciences and Engineering Research Council of Canada for a postgraduate scholarship for J.K.H. and for continuing support of our research on the thermodynamic properties of solutions. Registry No. GO:-, 13907-45-4; Cr207*-, 13907-47-6; HCrOL, I 5596-54-0.
Entropy and Crystal Structure of Hydrates of Disodium Hydrogen Phosphate David H. Templeton,* Helena W. Ruben, and AIlan Zalkin Department of Chemistry, University of California, Berkeley, California 94720 (Received: January I I, 1990; In Final Form: March 19. 19901
Structures determined by X-ray diffraction ar_e reported for crystals of Na2HP0 - x H 2 0( x = I , 2, 7, and 12). The new c = 5.735 ( I ) b;, a = 103.83 (I)', @ monohydrate phase is triclinic, space group PI, a = 5.565 ( I ) A, b = 7.949 ( I ) = 102.15 (I)', y = 107.47 (I)', 2 = 2, R = 0.020 for 3329 independent reflections. The other structures are in agreement with independent determinations already published. For x = I , 2, or 7 no disorder is observed, and there are unique configurations of protons in hydrogen bonds. The dodecahydrate ( x = 12) has two kinds of disorder, randomness of phosphate ion orientations and of proton positions in hydrogen bonds. A model for this disorder gives 2.65 cal K-'mol-' as the residual entropy at low temperature, compared with 3.1 f 0.5 and 3.5 f 0.4 cal K-l mol-' derived from two thermodynamic cycles.
1,
Introduction One of Kenneth Pitzer's earliest papers described the crystal structure of tetraamminecadmium perrhenate.' A little later Pitzer and Coulter2 discovered in sodium sulfate decahydrate the first example of a salt hydrate that retains residual entropy when cooled to very low temperature. They attributed this entropy, like that in ice, to the disorder of proton positions in hydrogen bonds. The molecular details of this disorder were revealed when the crystal structure was determir~ed.~This history prompted us to honor Pitzer by reporting some further studies of crystal structure, disorder, and entropy of salt hydrates. Waterfield and Staveley4 found another example of low-temperature disorder in disodium hydrogen phosphate dodecahydrate, Na2HP04.12H20 (12). Their entropy data indicate lack of disorder a t 0 K in two other hydrates, Na2HP04.2H20 (2)and N a 2 H P 0 4 . 7 H 2 0(7).The crystal structures of these salts were unknown a t the time, and we started to study t h e m s Here we report the structures which we determined for 2,7,12,and a new phase N a 2 H P 0 4 . H 2 0 (I), and discuss them in relation to the entropy data. After finishing the structures, but before completing the entropy calculations described below, we learned of the prior determination of 7 by Baur and Khan,6 and subsequent ones of 2 by Catti, Ferraris, and Franchini-Angela' and 12 by Catti, Ferraris, and Ivaldi.s Experimental Section Reagent crystals of 7 from Allied Chemical were used for X-ray diffraction without recrystallization. Crystals of 2 were grown from a saturated solution of 7 in water by slow evaporation at 59 'C. Crystals of 1 were found in one such experiment. Cooling of the saturated solution to room temperature produced crystals of 12. ( I ) Pitzer, K. S. 2.Krisfallogr. 1935,92, 131. (2) Pitzer, K. S.;Coulter, L. V . J. A m . Chem. Soc. 1938,60, 1310. (3) Ruben, H. W.; Templeton, D. H.; Rosenstein, R. D.; Olovsson, I. J . A m . Chem. SOC.1961,83, 820. (4) Waterfield, C.G.; Staveley, L. A. K. Trans. Faraduy SOC.1967,63,
2349.
( 5 ) Zalkin, A.; Ruben, H.; Templeton, D. H. Am. Crystallop. Assoc. Prog.. Absrr. 1972,52. (6) Baur, W. H.; Khan, A. A. Acta Crystallogr., Serf. B 1970,26, 1584. (7) Catti. M.; Ferraris. G.;Franchini-Angela, - M. Acfa Crvsfallogr., Sect.
B i9i7?33, 3449. (8) Catti, M.; Ferraris. G.;Ivaldi, G .Acta Crystallogr.. Sect. B 1978.34,
369.
0022-3654/90/2094-7830$02.50/0
TABLE I: Atomic Parameters for Disodium Hydrogen Phosphate Monohvdrate (1) with Esd's in Parentheses X z B, or B Y 0.13251 (3) 0.29404 (2) 0.08857 (3) 0.76 0.33178 ( 5 ) 0.47910 (6) 0.78254 (4) I .27 0.72450 (6) 1.60 0,26803 (6) 0.56420 (4) 0.1983 ( I ) 0.28856 (9) 1.32 0.47805 (6) -0.16261 (9) 0.17455 (6) 0.00837 (8) 1.12 0.2331 ( I ) 0.31 152 (7) -0.13327 (9) 1.44 0.2932 ( I ) 0.2327 (1) 0.19061 (7) 1.37 0.2294 ( I ) 0.5588 (1) 0.90416 (8) 1.48 0.154 (3) 0.261 (3) 0.097 (2) 2.5 (3) 0.694 (3) 0.186 (2) 0.884 (2) 2.2 (3) 0.854 (2) 0.450 (3) 0.090 (3) 2.6 (3)
A Picker FACS-1 diffractometer was used with Mo K a radiation and a graphite monochromator, X(LY,) = 0.70926 A. Crystal dimensions were 0.24 X 0.16 X 0.24 mm for 1, 0.25 X 0.18 X 0.13 mm for 2,0.26 X 0.22 X 0.091 mm for 7,and 0.28 X 0.26 X 0.23 mm for 12. Integrated intensities were measured with 0 - 26 scans for 1, full sphere to 0 = 39.8'; 2,one-eighth sphere to 0 = 30'; 7, half sphere to 0 = 25'; 12, full sphere to 0 = 27.5'. N o correction was made for absorption; p = 7.0 (l), 5.7 (2),3.7 (7), 2.9 cm-l (12). The structures were solved by Patterson and Fourier methods and refined by least squares. Atomic scattering factors9 were corrected for dispersion.I0 Coordinates and isotropic thermal parameters were refined for hydrogen atoms in 1,2,and 7. Hydrogen atoms were omitted from the calculations for 12. All other atoms were given anisotropic temperature factors T = exp(-0).25(B,,h2a**+ 2B12hka*b*
+ ...))
(I)
Atomic parameters are listed in Tables I-IV, where the equivalent isotropic thermal parameter is defined as Be, = CCB,a*,a*,a,.a,
(2)
r J
Crystal Structures Monohydrate ( 1 ) . Pi,a = 5.565 ( I ) A, b = 7.949 (1) A, c = 5.735 ( I ) A, CY = 103.83 (I)', /3 = 102.15 (I)', y = 107.47 (9) Doyle, P. A.; Turner, P. S. Acta Crystallogr., Sect. A 1968,24, 390. (10) Cromer, D. T.; Liberman. D. J . Chem. Phys. 1970,53, 1891.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7831
Structures of N a 2 H P 0 4 . x H 2 0
TABLE 11: Atomic Parameters for the Dihydrate (2) with Esd's in Parentheses'
P Na( 1 ) Na(2)
O(l) O(2) O(3) o(4) o(5) O(6) H(1) H(2) H(3) H(4) H(5)
X
Y
Z
0.01708 (5) [ I ] 0.26795 (8) [-7) 0.35019 (9) [-81 -0.0213 ( I ) [-31 0.1535 ( I ) [-21 -0.0104 ( I ) [-I] -0.0788 ( I ) [4] 0.3163 (2) [O] 0.1795 (2) [I] 0.387 (4) [2] 0.269 (4) [-I] 0.1 17 (4) [-21 0.130 (4) [3] -0.037 (4) [-I]
0.12172 (8) [9] 0.3296 ( I ) [O] 0.3579 ( I ) [2] 0.3441 (2) [3] 0.0999 (2) [I] -0.0059 (2) [O] 0.0492 (2) [3] 0.0505 (3) [ I ] 0.1817 (3) [-31 -0.018 (6) [-91 0.064 (6) [ 111 0.283 (6) [5] 0.063 (6) [-31 -0.001 (7) [201
0.16262 (3) [6] 0.27109 (5) [4] 0.00013 (5) [-81 0.14902 (9) [-81 0.19388 (8) [I81 0.09017 (8) [-I31 0.23177 (9) [-231 0.0674 ( I ) [-21 0.3957 ( I ) [-I] 0.086 (2) [O] 0.111 (2) [6] 0.402 (2) [I] 0.387 (2) [-21 0.260 (2) [-I21
B, or B 0.89 1.47 1.83 1.44 1.24 I .49 1.32 1.86 1.95 3.8 (8) 4.1 (9) 3.6 (8) 4.9 (IO) 5.3 (12)
z
B, or B
0.10245 (5) [5] 0.14094 (9) [54] 0.13373 (9) [-71 0.2539 ( I ) [-251 0.0996 ( I ) [-I] -0.0084 (2) [I71 0.0631 (2) [I81 -0.0767 (2) [21] -0.0227 (2) [I41 0.1932 (2) [-I21 0.3060 (2) [-31 0.3028 (2) [-I31 -0.0916 (2) [2] 0.3195 (2) [-241 -0.026 (3) -0.146 (3) -0.075 (3) -0.107 (3) -0.027 (3) 0.157 (3) 0.292 (4) 0.334 (3) 0.271 (3) 0.293 (3) 0.401 (3) -0.157 (3) -0.093 (3) 0.353 (3) 0.397 (3)
1.40 2.3 1 2.16 2.10 2.03 2.26 2.49 2.83 2.31 2.24 2.3 1 2.62 2.41 2.50 3.3 (7) 2.8 (7) 4.2 (8) 3.1 (6) 4.2 (7) 3.5 (7) 6.4 (1 0) 3.6 (7) 2.3 (6) 2.9 (6) 3.6 (6) 5.1 (8) 2.7 (7) 3.2 (7) 3.8 (7)
'Square brackets enclose shifts from structure determined by Catti et al.' TABLE 111: Atomic Parameters for the Heptahydrate (7) with Esd's in Parentheses' X Y P 0.39545 (4) [-51 0.23730 (4) [60] 0.39618 (7) [48] Na( 1) 0.97556 (8) [-41 0.10539 (7) [9] Na(2) 0.97721 (8) [ I t ] O(1) 0.3537 ( I ) [-31 0.2486 ( I ) [8] O(2) 0.5402 ( I ) [-I] 0.2511 ( I ) [3] O(3) 0.3199 ( I ) [9] 0.3170 (1) [6] o(4) 0.3620 (2) [8] 0.0984 ( I ) [I61 O(5) 0.8512 (2) [-61 0.4771 (2) [3] O(6) 0.0560 (2) [-I21 0.2536 (2) [-61 o(7) 0.8116 ( I ) [I81 0.2493 (2) [-41 0.8710 (2) [30] 0.5109 (2) [-I91 O(8) o(9) 0.0938 (2) [-I31 0.2500 (2) [-71 O( 10) 0.8623 (2) [5] 0.0359 (2) [ 121 O(11) 0.8900 (2) [-151 -0.0046 (2) [O] H(1) 0.368 (3) 0.077 (2) H(2) 0.833 (2) 0.425 (2) 0.525 (3) H(3) 0.783 (3) H(4) 0.007 (3) 0.256 (2) H(5) 0.139 (3) 0.270 (3) H(6) 0.733 (3) 0.258 (2) H(7) 0.806 (3) 0.235 (3) H(8) 0.895 (3) 0.583 (3) H(9) 0.794 (3) 0.514 (2) H( 10) 0.173 (3) 0.248 (2) H(1I) 0.086 (3) 0.251 (2) H(12) 0.851 (3) 0.104 (3) ~(13) 0.794 (3) -0.002 (2) ~(14) 0.918 (3) -0.071 (3) 0.050 (3) H( 15) 0.881 (3)
'Square brackets enclose shifts from structure determined by Baur and Khan.6 TABLE I V Atomic Parameters for the Dodecahydrate (12) with Esd's in Parentheses' V
Z
0.04764 (9) [-21 -0.0590 ( I ) [-I] -0.0483 (4) [-31 0.0557 (4) [2] 0.1996 (4) [-31 -0.0374 (4) [-41 0.1607 (2) [5] 0.0580 (2) [4] 0.1570 (2) [I] 0.1661 (2) [ l ] 0.0461 (2) [2] 0.7216 (2) [-SI
114 0.08669 (9) [5] 0.2199 (3) [ I ] 0.1563 (3) [O] 0.2900 (3) [7] 0.3511 (3) [-21 0.0217 (2) [2] 0.2676 (2) [3] 0.3979 (2) [-51 0.6125 (2) [3] 0.06 19 (2) [-21 0.1582 (2) [-I]
X
P
Na
0
0.32886 (8) [8] 0.0754 (3) [-41 O(2) [31 -0.0981 (2) [O] ow [41 0.0405 (3) [2] o(4) [21 -0.0187 (3) [-51 o(5) ~ 5 1 0.3897 ( I ) [-I] O(6) WI 0.3874 (2) [6] 0.2756 (2) [-61 o(7) ~ 4 1 O(8) [W61 0.4905 (2) [7] O(9) [WII 0.1603 ( I ) [-I] o(10)~ 3 1 0.2799 (1) [O] 'Square brackets enclose shifts from structure determined by Catti et aI.* and atom designations in that report. O(1)
(I)O, V = 223.8 A', Z = 2, D, = 2.373 g cm-', T= 23 'C, R = 0.020 for 3329 independent nonzero reflections.
In the crystal structure (Figure 1 ) each Na( 1 ) ion has six neighbors: two water molecules and four oxygen atoms from different phosphate groups. Each Na(2) ion has five oxygen atoms from four phosphate ions as neighbors. The distances to these neighbors are listed in Table V. Also listed there are 0-0
B, 1.4 3.2 2.8 2.7 3.0 2.6 3.3 3.5 3.8 3.8 3.0 3.2
distances for three hydrogen bonds and some bond angles. Lengths of P-0 bonds for all four structures are given in Table VI. The water molecule, 0(5),has two hydrogen bonds with noopportunity for disorder. The other hydrogen bond joins O(4) to O(2). The position found for H(1) [0.72 A from 0(4)] and the lengths of the P-0 bonds indicate that H(1) is more strongly bonded to O(4) than to O(2) and is not alternating between two sites. We expect
Templeton et al.
1832 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990
-
U
Figure 1 . View of the structure of Na2HP04.H20 TABLE V Some Interatomic Distances (A) and Angles (deg) in the MonohvdrateO Na(Z)-O(lv) 2.497 ( I ) Na(l)-O(l) 2.375 ( I ) Na(l)-0(2i) 2.501 ( I ) Na(2)-0(2v) 2.568 ( I ) Na(l)-0(3ii) 2.345 (1) Na(2)-0(3vi) 2.314 ( I ) Na(2)-0(4iii) 2.546 ( I ) Na(l)-0(4iii) 2.478 ( I ) Na(l)-0(5) 2.351 ( I ) 0(4)-0(2vii) 2.694 ( I ) Na(l)-O(Siv) 2.372 ( I ) O(5)-O(2v) 2.772 (1) O(5)-O(3i) 2.868 ( I ) Na(2)-0(1) 2.342 ( I ) O( 1)-P-0(2)
I 11.24 (4)
0(1)-P-0(3) O(I)-P-O(4) 0(2)-P-0(3) 0(2)-P-0(4) 0(3)-P-0(4)
114.31 (4) 103.53 (4) 112.77 (4) 107.60 (4) 106.63 (4)
P-0(4)-H( 1) H(2)-0(5)-H(3) 0(4)-H(1)-0(2vii) 0(5)-H(2)-0(2~) 0(5)-H(3)-0(3i)
110 ( I )
103 (2) 169 ( I ) 167 ( I ) 168 ( I )
OSymmetry code: (i) -x, 1 - y , -2; (ii) 1 - x, 1 - y , -2; (iii) I - x , 1 -2; (v) - x , 1 - y , 1 - z ; (vi) x, y , 1 + z ;
- y , 1 - 2 ; (iv) 1 - x , 2 - y , 1 (vii) -x. -.v, -2.
TABLE VI: P - 0 Distances (A) in the Hydrogen Phosphate Ions 1 2 7 12 128 P-O(l) 1.513 ( I ) 1.537 (2) 1.513 (2) 1.518 (4) 1.522 (3) P-0(2) 1.532 ( I ) 1.513 (2) 1.521 (2) 1.515 (3) 1.517 (3) P-0(3) 1.513 ( I ) 1.509 ( I ) 1.511 (2) 1.510 (3) 1.510 (2) P-O(4) 1.617 ( I ) 1.602 (2) 1.601 (2) 1.596 (3) 1.594 (3)
no disorder at low temperatures for this phase, but entropy data are lacking. Dihydrate ( 2 ) . Pbca, a = 10.343 (2) A, b = 6.596 (2) A, c = 16.833 (2) A, V = 1148.4 A3, Z = 8, D, = 2.059 g cmW3,T = 23 "C, R = 0.029 for 1173 independent nonzero reflections. After transformation from a different setting, the atomic coordinates found by Catti, Ferraris, and Franchini-Angela' agree with those we report in Table I1 within amounts consistent with the estimated standard deviations. The largest difference of atomic position is 0.2 8, for H ( 5 ) or, if hydrogen is excluded, 0.006 8, for O(4). Individual differences of parameters are listed in Table 11. There is no need to repeat here the details of the structure.' The configuration of the hydrogen bonds which may be deduced from the positions of the oxygen atoms is confirmed by the two independent refinements of hydrogen atom positions. The topology of these bonds offers no alternatives like those in ice or in sodium sulfate decahydrate for the locations of hydrogen atoms, and there is no evidence of any disorder. Heptahydrate ( 7 ) . P2,/n, a = 10.432 (6) A, b = 11.000 (6) A, c = 9.252 (6) A, @ = 95.62 (5)", V = 1056.5 A3, Z = 4, D, = 1.685 g ~ m - T~ = , 24 "C, R = 0.022 for 1247 independent nonzero reflections. Baur and Kahn6 determined this structure, but with lower precision ( R = 0.081, and esd's are larger by factors about 2 or 3). Hydrogen atoms were excluded from their refinement. After transformation to the same setting, differences of coordinates found in the two determinations are somewhat larger than predicted by the esd's. The largest shift of an atomic position is 0.04 A for
'e Figure 2. Disorder of the phosphate group in Na2HPO4.l2H20. Numbers identify types of oxygen atoms. The six O(8) and O(9) water molecules are hydrogen bonded to O(4) at the top and O(1) and O(2) at the bottom, or the reverse, according to the orientation of the 0( I ,2,3,4) tetrahedron. This phosphate is bonded to eight additional water neighbors which are omitted for simplicity.
O(8); most are about 0.02 8, or less. These differences have no significant effect on the description of the structure,6and it is not repeated here. Again, there is no ambiguity of the configuration of hydrogen bonds deduced from the heavy atom positions6 and confirmed by the hydrogen atom positions. Dodecahydrate (12). 12/c, a = 14.178 (6) A, b = 9.021 (4) A, c = 12.771 (5) A, @ = 108.88 (6)", V = 1545.5 A3, Z = 4, D, = 1.539 g T = 23 "C, R = 0.046 for 1429 independent reflections. Unlike the well-ordered structures of the three lower hydrates, this phase exhibits two kinds of disorder: positions of protons in hydrogen bonds and orientations of phosphate ions. Very similar models for the geometry of the disorder, including all the same hydrogen bonds, were deduced independently in our work and by Catti, Ferraris, and IvaldL8 The latter workers identified hydrogen atoms (including those with half occupancy) in a AF Fourier synthesis. Inclusion of hydrogen in the calculation of structure factors allowed them to reduce R to 0.030 and to achieve esd's somewhat smaller than ours. They also tested and rejected an ordered structure in space group Cc (or IC if transformed to our choice of unit cell). The two sets of final parameters are in excellent agreement relative to the esd's (Table IV). The structure is described elsewhere,s and only some details relevant to the entropy are given here. The P atom is located in a special position on a 2-fold axis. The tetrahedron defined by its oxygen neighbors has two orientations which are related by this 2-fold rotation (Figure 2). Each oxygen atom is in a general position with half-occupancy. For either orientation there are six hydrogen bonds with six water molecules, but which oxygen atom is involved in which bond depends on the orientation. Which oxygen atom of HPOd2-has the proton, and which hydrogen bond it is in, are important questions if the residual entropy is to be explained. The difference of long and short P-0 bond lengths found in both determinations of the structure of 12 is about 90% of that found in the other hydrates (Table VI). This fact suggests that the acid phosphate proton is bonded to atom O(4) about 90% of the time. All of the 25 hydrogen atoms per formula unit are in hydrogen bonds. A cluster of 12 water molecules is linked in a chain by 1 1 of these bonds. The other 14 bonds connect water to phosphate oxygen atoms whose designations and positions depend on the orientations of eight phosphate ions. Figure 3 shows the shape of this chain and one example of the configurations of its phosphate neighbors. One and only one of these phosphate ions must provide a proton to this group of water molecules to fill all the bonds. The topology of these bonds and alternatives for oxygen neighbors are shown in Figure 4.
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7833
Structures of Na2HP04.xH20 Q
0
0
0
0
0
0
0
0
0
0
Figure 3. Cluster of 12 water molecules surrounded by eight HP0:ions. Hydrogen bonds between water molecules are indicated by heavy lines to emphasize the chain. Lighter lines indicate hydrogen bonds to phosphate groups. Each phosphate ion is shown in one of its two possible orientations. 2.3
1.4
1,2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Figure 5. The two possible proton configurations in a group of water molecules if the acid hydrogen is confined to a half-occupancy site.
I
3 4
0
2,3
3
6
2,4
0
0
0
0
0
0
0
0
0
0
0
0
~
0
0
0
0
0
0
0
0
0
Figure 6. Examples of proton configurations with the acid hydrogen in other locations. Figure 4. Topology of hydrogen bonding in Na,HP04.12H20. Double circles indicate water molecules. Types of oxygen atoms are identified by numbers; a pair of numbers indicates the alternatives for two orientations of the phosphate ion.
probable. To calculate this entropy we imitate the method that Pauling12 used for ice. This method is not exact, but a more elaborate analysis by DiMarzio and Stillinger13 gave nearly the same result for ice. These authors assumed that only one proton at a time is in each hydrogen bond and that each water molecule has exactly two protons. That is, H 3 0 + and OH- ions are absent. We make the same assumption. We also assume that exactly one proton is located near an oxygen atom of each HP042- ion so that no PO:- or H2PO4- ions are present. The rigid tetrahedral shape of each phosphate group requires its four oxygen atoms to occupy their alternate sites in a coordinated manner. These restrictions greatly reduce the number of possible configurations. The locations of the 24 water protons in each group are determined as soon as one specifies which bond contains a proton from an acid phosphate. The identities and locations of the oxygen atoms which accept the peripheral hydrogen bonds are determined when the same choice is made for each neighboring phosphate ion. Thus the choices of alternate sites for protons and oxygen atoms are reduced to only one choice of configuration per formula unit: the location of the acid proton. Even this choice is restricted by the requirement that two phosphate ions cannot both provide the extra proton to the same cluster of water molecules. Two half-occupancy sites for the acid proton were identified by Catti et a1,* one for each orientation of the phosphate ion. To restrict this proton only to these sites is too severe. It limits the configurations of water clusters to only two choices (Figure 5 ) , and the way in which these clusters are linked through phosphate ions causes the configuration of one to control that of another, endlessly through the structure. Disorder remains in the relative configurations of different molecular chains, but the molar entropy is only ( R / n ) In 2, where n is the average number of formula units per chain. For chains of significant length this entropy is negligible relative to the experimental value. More alternatives must be provided for the acid proton. Each phosphate ion has two orientations, and for each one of them the O(4) atom is involved in three hydrogen bonds. If the acid proton is assigned part of the time to each of these hydrogen bonds, or if it is sometimes on another oxygen atom, configurations
Disorder and Entropy From vapor pressures, integral heats of solution, and the known enthalpy of vaporization of water, Waterfield and Stavelef derived the entropy changes at 298 K for the hydration reactions:
--
+ 2H20(g) Na2HP04.2H20(c) + 5 H 2 0 ( g ) Na2HP04.7H20(c) + 5H,O(g) Na2HP04(c)
Na2HP04.2H20(c)
(3)
Na2HP04.7H20(c)
(4)
Na2HP04.12H20(c)
(5)
These data, combined with the calorimetric absolute entropy of the anhydrous salt" and the well-known entropy of water vapor, gave them values of the absolute entropies of the hydrates. They calculated the same entropies by integration of C,/ T using heat capacities measured down to about 10 K. Comparison of the results gives the molar entropy that each hydrate retains when cooled nearly to 0 K under laboratory conditions: 2, 0.02 f 0.2 cal K-l mol-I; 7, 0.37 f 0.3 cal K-' mol-I; 12, 3.51 f 0.4 cal K-' mol-' The first two results are not significantly different from zero, in accordance with the well-ordered atomic positions and unique hydrogen-bond configurations found in the crystal structures. The finite entropy retained by the dodecahydrate was the first evidence of the disorder which is also found in the crystal structure determinations. We note that its magnitude is 3.1 f 0.5 cal K-l if calculated with the assumption of zero entropy for 7 and following the more direct thermodynamic path from that phase. This path eliminates possible error in the entropy changes measured for reactions 3 and 4. A molecular explanation of the entropy requires that disorder of atomic positions be frozen in as the crystals are cooled. The entropy is k In W , where the statistical weight W is the number of configurations accessible to the crystal if they are equally (1 1) Andon, R. J. L.; Counsell, J . F.; Martin, J. F.; Mash, C. J . J . Appl. Chem. 1967, 1 7 , 65.
,
(12) Pauling, L. J . Am. Chem. SOC.1935, 57, 2680. (13) DiMarzio, E. A,: Stillinger, F. H . J . Chem. Phys. 1964, 40, 1577.
J . Phys. Chem. 1990, 94, 7834-7839
7834
similar to those shown in Figure 6 can occur. Then the entropy is appreciable. Following Pauling, but with some new notation, we set W = W ,F, where Wl is the weight calculated neglecting some of the restrictions and the factor F is an estimate of the probability that a configuration is acceptable. Then for one mole, if SI = k In W , and S2 = k In F, S = k In ( W , F ) = SI + S 2 (6)
If the acid hydrogen is on O ( 4 ) with equal probability for two choices of phosphate orientation and three choices of hydrogen bond for each one, W, = 6Nand SI= R In 6 = 3.560 cal K-I mol-'. For each group of 12 water molecules there are 6 phosphate ions, each of which provides the acid proton with probability 1 / 6 . The fraction of these configurations that provide just one proton is ( 5 / 6 ) 5= 0.402. With the approximation that the probability for one group is independent of that for another, F = 0.402N,S2 = -1.81 I cal K-' mol-'. and S = 1.75 cal K-' mol-'. This value is about half thc experimental result. As mentioned above, there is evidence that each acid proton is on another oxygen atom a fraction of the time. More configurations like those in Figure 6 are possible. For molecular configurations of unequal probability it is convenient to express the molar entropy as S = - R X X i In Xi (7) I
where Xi is mole fraction of configuration i. With each acid proton distributed equally among 2 2 such structures 10% of the time, and 90% of the time on O ( 4 ) as above, SI = 4.464 cal K-I mol-I. A tabulation of the possible configurations and their probabilities for the eight neighboring acid phosphate groups of each water cluster (Figure 3 ) shows that 40.1 % of them provide exactly one acid proton to the water group. With the same approximation that this probability for one water group is independent of those for others, F = 0.401N,S2 = -1.816, and S = 2.65 cal K-' mol-'.
The agreement with the thermodynamic data is less than perfect, but not beyond possible experimental error. It is improved if the protons are more evenly distributed, but S cannot be increased above about 3 cal K-' mo1-I without serious conflict with the evidence of the bond distances. This model is not unique. The result is not very sensitive to modest variation of the various probabilities. Nor can we exclude the existence of other kinds of imperfection in small concentrations. However, we are confident that randomness among orientations of phosphate ions and among configurations like those in Figures 5 and 6 is the origin of most of the residual entropy. An item which invites further study is a report that the crystal structure of anhydrous N a 2 H P 0 , has proton disorder at room t e m p e r a t ~ r e , yet ' ~ the heat capacity for the range 10-320 K indicates no phase change or anomaly," and the evidence is convincing that there is no residual entropy in this phase at low temperature. Perhaps the crystals have a larger unit cell with additional reflections too weak to have been noticed in the diffraction experiment. For example, a doubled cell with a glide plane in place of the mirror plane (space group P 2 , / a , P2,/c, or P 2 , / n ) offers a structure that is free of disorder, but with nearly identical diffraction intensities for the reflections that were observed.
Acknowledgment. We thank Dr. Lieselotte Templeton for assistance in preparing the figures. The initial part of this work was supported by the U S . Atomic Energy Commission. Registry No. Na2HP04.H20,118830-14-1; Na2HP04.2H20,1002824-7; Na2HP04.7H20,7782-85-6; Na2HP04.12H20,10039-32-4. Supplementary Material Auailable: Tables of anisotropic thermal parameters ( 3 pages) and tables of structure factors ( 3 5 pages). Ordering information is given on any current masthead page. (14)
Wiench, D. M.; Jansen, M . 2.Anorg. Chem. 1983, 501, 9 5 .
Polarized and Depolarized Raman Spectra of Liquid Carbon Disulfide at 0-10 k b a d 3. Interaction-Induced v2 and v3 Scattering and the Fluctuation of the Local Field S. Ikawa* Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060, Japan
and Edward Whalley* Division of Chemistry, National Research Council of Canada, Ottawa, Ontario K1 A OR9, Canada (Received: January 11, 1990; In Final Form: April 12, 1990)
The effect of pressure on the interaction-induced Raman scattering of the forbidden u2 and u3 vibrations of liquid carbon disulfide has been measured up to 10 kbar at 295 K, and a simple model for the fluctuation of the local field has been used to describe the origin of the spectra. The scattering intensities increase with increasing pressure approximately in proportion to the density of the liquid. This is ascribed to the dominant contribution of the orientational fluctuations of the molecules to the fluctuation of the local field. The width of the u2 band decreases with increasing pressure, and its high-pressure limit is attributed to nondiffusional broadening. The diffusional line width, which was obtained by subtracting the nondiffusional part from the total line width, provided the diffusional time constant, which is linear in the viscosity of the liquid. A large part of the diffusion broadening is ascribed to the relaxation of the local field caused by rotational diffusion.
I. Introduction Pressure is an important variable for studies of the liquid phase because its effects help to elucidate the nature ofthe intermolecular interactions and to test theoretical models ofthe liquid. One direct manifestation of these interactions is the interaction-induced Raman scattering by molecular vibrations that are forbidden in
the isolated molecules and in the crystal. For example, the doubly degenerate bending u2 and the antisymmetric stretching u3 vibrations of carbon disulfide, which are Raman inactive in the isolated molecules, become weakly allowed in the liquid owing to molecular interactions,' but they are forbidden in the crystals because O f the inversion enter.^-^ Therefore, the interaction( 1 ) Evans, J . C.; Bernstein, H. J. Can. J . Chem. 1956, 34, 1127.
' N R C No. 31853.
0022-36S4/90/2094-7834$02.50/0
0 1990 American Chemical Society
