5818
J . Phys. Chem. 1987, 91, 5818-5825
along the polymer backbone. In the fully deprotonated polymer of an intermediate oxidation state, no translation of quinoidal structures is possible. However, as soon as an imine nitrogen becomes protonated, translation over a distance of two rings becomes possible by valence resonance. Further translation then requires a deprotonation/protonation cycle as illustrated in Figure 9. Of course, if both imine nitrogens are protonated, translation over a larger number of rings becomes possible. However, any deprotonated imine nitrogen or protonated amine nitrogen will act as a barrier for translation along the polymer backbone. It follows that considerable mobility along the polymer chain is possible on protonation of only one of the imine nitrogens of the quinoidal structures. For interchain transport a double protonation (at least initially to allow the formation of radical cations) is required (see Figure 10). The model implies a coupling of electronic and ionic transport which increases as the degree of protonation of the imine nitrogens decreases. Since proton exchange depends on the presence of a source of protons, these concepts may explain the considerable decrease in resistivity that is observed when initially dry PAn is exposed to moisture. Also, H 2 0 molecules may aid in proton transport (via the hydronium ion) between chains. It is anticipated that the resistivity will be more sensitive to the degree of protonation in the “dry” state vs the “wet” state, and in fact this is what is found (Figure 8). The fact that the effect of moisture on the resistivity increases with increasing pH is also in agreement with the predictions of the proton-exchange model: At low pH a majority of imine nitrogens are protonated and only a few barriers (unprotonated imine nitrogens) exist. Therefore proton-exchange
reactions will have a relatively small impact on resistivity. At high pH many barriers to conduction exist and proton exchange has a greater influence on mobility along, as well as between, chains
Conclusions The conductive state of polyaniline is associated with an intermediate oxidation state. The electrochemical data obtained by cyclic voltammetry show that, at low pH, this state is consistent with radical cations. The radical cations are stable over a limited potential range only and, as a result, only low resistivity exists over the same limited potential range. As the stability range of the intermediate state decreases with increasing pH, the potential range of low resistivity also decreases with increasing pH. The protonation of quinone diimine units is incomplete at higher pH. However, even partial protonation is apparently sufficient to decrease resistivity by more than 3 orders of magnitude in the presence of electrolyte. In addition the sensitivity of the resistivity to moisture content increases with increasing pH. These observations imply that proton-exchange reactions play a central role in the conduction mechanism in polyaniline. Acknowledgment. This work was supported in part by a grant from the Office of Naval Research and an A R C 0 Career Development Award (to G.E.W.). W.W.F. is grateful for a graduate fellowship from the NIMR/CSIR. Registry No. PAn, 25233-30-1; HCI, 7647-01-0; KCI, 7447-40-7; CH~C~HSOI-, 16722-51-3;BFd-, 14874-70-5;CFBCOO-, 14477-72-6; tartaric acid, 87-69-4;phosphoric acid, 7664-38-2;phthalic acid, 88-99-3.
Is Liquid Water Really Anomalous? M.-P. Bassez,? J. Lee, and G. W. Robinson* Picosecond and Quantum Radiation Laboratory, Texas Tech University, Lubbock, Texas 79409 (Received: May 12, 1987)
A temperature dependence of the effective potential barrier hindering rotational motions of water molecules in liquid water has recently been invoked as the key to a detailed understanding of water’s “anomalous” characteristics. In this paper we demonstrate in greater detail how a large number of thermodynamic and transport properties of this remarkable liquid, supercooled to superheated, can be interrelated through empirically determined temperature-dependent activation barriers. It is suggested that the height of these barriers is a quantitative measure of the departure from local ice-Hike structure of the hydrogen-bonded network. Influences of temperature and pressure in liquid water are explained, and deuterium effects are reproduced. Specific volumes at supercooled temperatures and elevated pressures are calculated, and the origin of the wayward heat capacities becomes clear. In fact, from the analysis, a “water-like substance“ emerges that in all respects seems capable of approaching the properties of real water within a few percent.
1. Introduction
For some
it has been thought that rotational motions
rotational barriers decrease.1° It immediately follows that contributions to the partition function from the temperature-dependent
of water molecules play an essential role in chemical and dy-
namical effects in liquid water. Dynamical properties of pure liquid water, such as viscosity, dielectric relaxation, and N M R spin-lattice relaxation, are known to be related to the rates of molecular r e ~ r i e n t a t i o n . ~ ~ A theory described considers liquid water to be a network of hydrogen-bonded hindered rotor molecules that experience librations below a temperature-dependent activation barrier, AH( T ) , and rotational diffusion above it. As the temperature rises and the hydrogen bonds weaken, the forces between the molecules become more isotropic, and the heights of the ‘Permanent address: Institut Universitaire de Technologie, Laboratoire de Chimie-Physique, Universitt d’Angers, Belle-Beille, 49045 Angers Cedex,
France.
0022-365418712091-5818%01SO10
(1) Danneel, H. Z . Elektrochem. Angew. Phys. Chem. 1905, 1 1 , 249. (2) Bernal, J. D.; Fowler, R. H. J. Chem. Phys. 1933, 1 , 515. (3) Conway, B. E.; Bockris, J. OM.; Linton, H. J . Chem. Phys. 1956, 24,
834.
(4) Eisenberg, D.; Kauzmann, W. The Structure and Properties of Water, Oxford University Press: New York, 1969. ( 5 ) Hasted, J. B. Aqueous Dielectrics; Chapman and Hall: London, 1973. Collie, C. H.; Hasted, J. B.; Ritson, D. M. Proc. Phys. SOC.,London 1948, 60, 145. (6) Abragam, A. The Principles of Nuclear Magnetism; Oxford University Press: New York, 1961; pp 298-300. (7) Robinson, G. W.; Lee, J.; Casey, K. G.;Statman, D. Chem. Phys. Lett. 1986, 123,483. (8) Robinson, G. W.; Lee, J.; Bassez, M.-P. Trans. Faraday SOC.1986,
82, 2351. (9) Robinson, G. W.; Lee, J.; Bassez, M.-P. Chem. Phys. Lett., 1987, 137, 316.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vo1. 91, No. 22, 1987 5819
Is Liquid Water Really Anomalous? librational frequencies4Jl * l Zassociated with this kind of potential function can give rise to abnormal thermodynamics. Recently, a similar temperature dependence of the intermolecular "translational modes" has been carefully documented with respect to both temperature-dependent frequency changes and temperature-dependent intensity changes.I3 We believe it is these thermal characteristics of the partition function that constitute the essential ingredient for the existence of a "water-like substance". Interestingly, this is just the ingredient that has been missing in all previous theoretical attempts to understand the properties of liquid water. This ingredient has also not been accurately mimicked by any current molecular dynamics simulation. Coulomb interactions arising from a small number of distributed point charges simply do not possess the "fragile" angular dependence of the real interactions, and thus M D water is much stiffer and more "structured" than real liquid water. Other workers have been concerned about this same p ~ i n t . ' ~ . ~ ~ Experimental data on N M R relaxation times of H, D, and 1 7 0 have been measuredlbB for HZl60,DZI60,H 2 7 0 , and DZl7O.The inverse of the N M R spin-lattice relaxation time T I is proportional6 to the rotational correlation time, l/TI = A"T,. This process must be distinguished from the "hydrogen-bond lifetime", which is much shorter.z1 Similarly, experimental value^^^^^ of the dielectric relaxation times 7 4 of HzO and DzO can be related to this rotational time, Td = &,. Datazsz6 on the shear viscosity 7 measured from the supercooled region to the boiling point for HzO and D20 can be included in this same formalism, since T , = AflvV,, where V ,is a molecular volume and fl = (kT)-'. The activation energy for all these processes depends on the effective potential barrier hindering the molecular rotation. A central feature is that this barrier is temperature-dependent, giving rise to highly non-Arrhenius b e h a ~ i o r . ~ , ' ~
2. Mode Softening about the low-frequency intermolecular modes is essential for reaching an understanding of the properties of liquid water. For example, assumption of a quadratic formz7
where i = H 2 0 or DzO, AEr( T ) c [AHi(T ) - APv
+ l/Zh~L'(T ) ]
(10) Fowler, R. H. Statisrical Mechanics, 2nd ed.; University Press: Cambridge, 1966; p 811. (11) Larsson, K. E.; Dahlborg, U. React. Sci. Technol. 1962, 16, 81. (12) Chen, S. H.; Teixeira, J., not fully published results for temperatures down to -15 OC. See: Chen, S.-H.; Toukan, K.; Loong, C. K.; Price, D. L.; Teixeira, J. Phys. Rev. Lett. 1984, 53, 1360. (13) Walrafen, G. E.; Fisher, M. R.;Hokmabadi, M. S.; Yang, W.-H. J . Chem. Phys. 1986, 12, 6970. Also, a private discussion with Dr. Walrafen. (14) Chen, S.-H.; Teixeira, J. Adu. Chem. Phys. 1986, 64, 1. (15) Newton, M. D. J . Phys. Chem. 1983, 87, 4288. (16) Hindman, J. C.; Zielen, A. J.; Svirmickas, A.; Wood, M. J. Chem. Phys. 1971, 54, 621. (17) Jonas, J.; DeFries, T.; Wilbur, D. J. J . Chem. Phys. 1976, 65, 582. Lang, E.; Liidemann, H. D. J . Chem. Phys. 1977, 67, 718. (18) Lang, E.; Liidemann, H.-D. Eer. Bunsen-Ges.Phys. Chem. 1980,84, 462. (19) Lang, E. W.; Liidemann, H.-D. Eer. Bunsen-Ges.Phys. Chem. 1981, 85, 603. (20) Liidemann, H.-D.; Lang, E. W. J . Phys. (Les Ulis, Fr.) 1984, (Suppl. No. 9), 41. (21) Bertolini, D.; Cassettari, M.; Ferrario, M.; Grigolini, P.; Salvetti, G. Adu. Chem. Phys. 1985, 62, 277. (22) Bertolini, D.; Cassettari, M.; Salvetti, G. J. Chem. Phys. 1982, 76, 3285. (23) Hallett, J. Proc. Phys. SOC.,London 1954, 82, 1046. (24) Stokes, R. H.; Mills, R. Viscosity of Electrolyres and Related Properties; Pergamon: Oxford, 1965; p 74. Kell, G. S. In Water, A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1972; Vol. 1, p 407. (25) Osipov, Yu.A.; Zheleznyi, B. V.; Bondarenko, N. F. Russ.J. Phys. Chem. (Engl. Transl.) 1977, 51, 1264. (26) Heiks, J. R.; Barnett, M. K.; Jones, L. V.; Orban, E. J . Phys. Chem. 1954, 58, 488. (27) Townes, C. H.; Schawlow, A. L. Microwaue Spectroscopy; McGraw-Hill: New York, 1955; pp 322-323.
TABLE I:" Temperature-Dependent Barriers for H20Calculated from "0 Spin-Lattice Relaxation Times T, K T I ,ms AH, J mol-l 238 243 248 253 258 263 268 273 283 299 309 323 353 383 403
0.40 (0.26) 0.57 (0.48) 0.83 (0.75) 1.23 (1.09) 1.55 (1.48) 1.93 (1.92) 2.4 (2.41) 3.4 (2.95) 4.6 (4.21) 7.4 (6.72) 9.2 (8.59) 12.5 (11.60) 21 (19.28) 30.5 (27.96) 36 (33.89)
10440 (11350) 9938 (10330) 9350 (9567) 8690 (8969) 8369 (8482) 8049 (8072) 7722 (7714) 7028 (7394) 6590 (6833) 5783 (6067) 5438 (5643) 4856 (5100) 3768 (4090) 2857 (3241) 2447 (2745)
"The values of A H in parentheses are polynomial fits to the experimental data in Figure 1 averaged over all available data in Tables IIV. See eq 3. The T, data in parentheses, following the experimental T, data,lg are calculated from the averaged AH values via eq 2. As can be seen in Figure 1, the actual I7O spin-lattice A H values mostly lie lower than the averaged values, while the actual viscosity data in Table 111 mainly lie higher.
TABLE 11:" Temperature-DependentBarriers for H20Calculated from Proton Spin-Lattice Relaxation Times AH,J mol-' Ti,bs T, K 237 243 248 254.4 263 272 283 293 303 363
0.20 (0.10) 0.32 (0.22) 0.47 (0.35) 0.75 (0.54) 1.14 (0.89) 1.60 (1.32) 2.10 (1.96) 3.10 (2.65) 4.15 (3.45) 11.1 (10.27)
10220 (11598) 9513 (10330) 8904 (9567) 8093 (8863) 7458 (8072) 6955 (7455) 6640 (6833) 5899 (6340) 5348 (5893) 3479 (3791)
"See footnote a, Table I. bThe proton spin-lattice relaxation times are measured between 237 and 293 K at 50 bar and for 303 and 363 K at 1 bar.I7
with AE: denoting the potential energy of the hindered rotational barrier, maximum minus minimum, and Bi a constant, allows the librational frequency and the activation barrier for rotational diffusion to be approximately related. The experimental relaxation data then provide a self-consistent set7 of empirical activation enthalpies AHi( T ) and associated librational frequencies vLi( 7') for both HzO and DzO from -240 to 400 K. The pressurevolume contribution APv to this librational enthalpy term is not large. It is important to keep in mind that there are three unresolved or marginally resolved reorientation librations in liquid water.4J3,z8 Unfortunately, because of overlapping structure and poor resolution, these modes are difficult to analyze. Our contention is that it is the lowest frequency vibration of this type that corresponds to the lowest activation barrier and thus would be the mode primarily contributing to the relaxation processes. This mode is to become "softer" as the temperature rises. We have estimated its frequency from AH( T) data using eq 1, and we find it to range from -280 cm-' at 360 K to -445 cm-' at 240 K for Hz0.7 These frequencies match reasonably well those given by the old inelastic neutron scattering data" b u t seem a bit low
compared with modern Raman work.13 Because of the current uncertainty in these frequencies, for the determination of AH(T), we have used7 eq 1 with the ratio q ( 2 7 3 K)/vL(373 K ) = 1.4. This ratio was obtained from the temperature shift of the vL(T) maxima in the old neutron data." When more modern temperature shifts become available, the (28) Ramsay, J. D. F.; Lauter, H. J.; Tompkinson, J. J . Phys. (Les. (ilis, Fr.) 1984, 45 (Suppl. No. 9 ) , 73-80.
5820
Bassez et al.
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
TABLE III:" Temperature-Dependent Barriers for H20 Calculated from Shear Viscosity Datab T, K n, cp AH, J mo1-I 238 243 248 253 258 263 268 273 278 283 288 293 298 303 308 313 318 323 328 333 338 343 348 353 358 363 368 373
18.7 (16.87) 10.2 (9.53) 6.45 (6.19) 4.33 (4.38) 3.34 (3.31) 2.66 (2.60) 2.16 (2.11) 1.79 (1.76) 1.52 (1.49) 1.31 (1.28) 1.14 (1.11) 1.00 (0.974) 0.890 (0.862) 0.798 (0.770) 0.720 (0.693) 0.653 (0.627) 0.596 (0.571) 0.547 (0.522) 0.505 (0.481) 0.467 (0.444) 0.434 (0.412) 0.405 (0.385) 0.379 (0.360) 0.356 (0.338) 0.334 (0.319) 0.315 (0.337) 0.298 (0.286) 0.283 (0.272)
11570 (11350) 10490 (10330) 9309 (9567) 8945 (8969) 8512 (8482) 8124 (8072) 7771 (7714) 7444 (7394) 7153 (7103) 6897 (6833) 6649 (6579) 6419 (6340) 6207 (61 11) 5995 (5893) 5801 (5684) 5606 (5482) 5430 (5288) 5245 (5100) 5077 (4918) 4910 (4743) 4751 (4572) 4593 (4407) 4435 (4246) 4277 (4090) 4129 (3939) 3972 (3791) 3824 (3648) 3676 (3509)
TABLE I V " Temperature-DependentBarriers for H 2 0 Calculated from Experimental Dielectric Relaxation Timesb T, K Td, PS AH, J mol-! 253 273 283 293 303 313 323 333 348
3. The Activation Barriers Activation enthalpies resulting from the above analysis of relaxation data at atmospheric pressure are listed in Tables I-IV for H 2 0 and Tables V-VI1 for D20, and are depicted as a function
8669 (8969) 7386 (7394) 6834 (6833) 6343 (6340) 5922 (5893) 5570 (5482) 5260 (5100) 4845 (4743) 4530 (4246)
"See footnote a, Table I. b ~ d ( 2 5K), 3 ref 22; ~ ~ ( 2 7 3 - 3 4K), 8 ref 5. TABLE V " Temperature-DependentBarriers for D20 Calculated from Deuteron Spin-Lattice Relaxation Times T, K TI, ms AH, J mol-' 239 24 1 242 243 246 250 253 255 257 260 263 264 269 273 283 303 363
"See footnote a, Table I. bq(238-268 K), ref 25; q(253-268 K), ref 23; q(273-373 K), ref 24. accuracy of this ratio, and of the frequencies themselves, will no doubt improve. Incidently, the same frequency ratio of 1.4 appliesI3 to the -60-cm-' "restricted translational mode", which is related to the librational mode since it is classed as "hydrogen-bond bending" perpendicular to the hydrogen bond. As explained in ref 7, the ratio vL(273 K)/vL(373 K) provides the single constraint required to determine AH( T ) vs T for the entire range of available relaxation data. Use of this ratio instead of the absolute vL( T ) values reduces uncertainties inherent in the Bi value of eq 1, which depends, for example, on the number of minima and the detailed shape- of the highly anharmonic quasiperiodic potential function. As will be seen in the following sections, the librational/rotational motions do seem to give rise to trends that correlate with the inherent properties of liquid water. Low-frequency translational modes correspond to such large barriers compared with their frequency, the mass of the entire molecule having to move, that the character of these motions remains purely vibrational throughout the normal thermodynamic regime. Thus, the translational modes cannot contribute very much to the dynamical properties; but, because of their temperature dependence, they may contribute to the abnormal partition function and thermodynamics of liquid water. They also act as an independent guide to the state of hydrogen bonding in water.13 On the other hand, higher frequency intramolecular vibrations are not sufficiently populated to contribute much to the liquid water partition function, even though some of them too are known to be temperature s e n ~ i t i v e . ' ~ On . ~ ~closing this section, it is interesting to note that a separation, in effect, of "anomalous" and normally behaving modes was utilized e ~ p e r i m e n t a l l ybefore ~ ~ any theoretical justification was advanced for its plausibility.
42.7 (48.6) 17.9 (18.0) 12.6 (12.6) 9.3 (9.3) 7.2 (7.1) 5.8 (5.6) 4.8 (4.6) 3.9 (3.8) 3.2 (3.0)
10.3 (8.1) 14 (12.0) 17 (14.3) 18.6 (16.9) 32 (26.0) 42 (41.6) 54 (55.6) 63 (66.0) 76.3 (77.0) 85 (95.0) 110 (114.3) 117 (121.1) 143 (157.4) 175 (189.0) 275 (279.4) 508 (508.1) 1490 (1523.7)
13250 (1 3760) 12710 (13050) 12350 (12730) 12220 (12430) 11180 (11650) 10780 (10810) 10360 (10290) 10100 (9993) 9743 (9723) 9632 (9361) 9142 (9042) 9031 (8944) 8745 (8501) 8394 (8192) 7586 (7537) 6507 (6504) 4367 (4279)
"These data for D20are treated in an analogous way as the H20 data, the averaged AH values being obtained from a fit of all data, Tables V-VII, to eq 3. See footnote a, Table I. Experimental data are from ref 18. TABLE VI:" Temperature-DependentBarriers for D 2 0 Calculated from Experimental Shear Viscosity Datab AH,J mol-' T, K 7 , cp 243 248 253 258 263 268 273 303 318 333 348 363 373
23.2 (21.80) 12.10 (11.37) 7.40 (7.01) 5.04 (4.83) 3.75 (3.58) 2.95 (2.79) 2.39 (2.25) 0.969 (0.930) 0.713 (0.684) 0.551 (0.532) 0.445 (0.430) 0.366 (0.361) 0.327 (0.326)
12570 (1 2430) 11340 (11200) 10420 (10290) 9698 (9579) 9164 (9042) 8732 (8584) 8347 (8192) 6633 (6504) 6000 (5858) 5414 (5281) 4885 (4757) 4342 (4279) 3996 (3981)
"See footnote a, Table V. bq(243-273 K), ref 25; q(303-373 K), ref 26.
TABLE VII:" Temperature-DependentBarriers for D20 Calculated from Experimental Dielectric Relaxation Timesb T.K T A . DS AH. J mol-' 273 278 283 293 303 313 323 333
23.27 (26.58) 20.37 (21.64) 16.55 (18.00) 12.26 (13.06) 9.34 (9.90) 7.22 (7.78) 5.89 (6.29) 4.90 (5.21)
7840 7685 7306 6808 6328 5825 5434 5061
(8192) (7847) (7537) (6988) (6504) (6065) (5659) (5281)
"See footnote a, Table V. b~d(273-333K), ref 5. (29) D'Arrigo, G.; Maisano, G.; Mallamice, F.; Migliardo, P.; Wanderlingh, F.J . Chem. Phys. 1981, 75, 4264. (30) Oguni, M.; Angell, C. A. J . Chem. Phys. 1983, 78, 7334.
of temperature in Figures 1 and 2. These values are similar to, but supersede, those reported earlier.' The entries in the tables
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987 5821
Is Liquid Water Really Anomalous?
molecules are subjected become more isotropic.I0 In the absence9 of an ab initio theoretical description of AH( T), a purely empirical fit to the AH(T) data can be expressed as
i 4‘
0“
AH(T) = AH(To)[ A
1
I
I
270
I
I
I
I
310 350 Temperature
I
(to
I
390
I
* I
Figure 1. AH(T) vs T for H20: X, viscosity, A, dielectric relaxation; 0,”0spin-lattice relaxation. The solid line is an average calculated from eq 3. I
4 k
i
tL
L L U , 270
310
I 350
Temperature
390
(I.0
Figure 2. AH( T ) vs T for D20. Symbols have same meaning as in Figure 1.
show the experimental rate data and the AiY(T) obtained for each data point. These values of A H ( T ) , when used in the pseudoArrhenius equation 1;7 = kOe-BAWT) (2) exactly reproduce to within a constant scale factor the experimental values of the viscosity and relaxation data.7 The AH( T ) and rate data in parentheses are smoothed values that are discussed below. It can be seen from Tables I-VI1 and Figures 1 and 2 that the activation energies for rotational diffusion are close to being the same for all experimental relaxation processes considered. The deviations are partly caused by experimental uncertainty in the data points, but partly also through uncertainties introduced by the oversimplified nature of our picture. For example, a greater or lesser contribution from the intermolecular translational modes in the various relaxation processes would give rise to variations in the AH( T ) curves and in the effective parameters derived from the data. Nonetheless, the reasonable agreement confirms that a single mechanism is able to explain the main trends in these properties of liquid water. It has already been ~ u g g e s t e d ~that - ~ the thermal behavior of the enthalpy parameter reflects a cooperativity within the network of hydrogen bonds in liquid water. As the temperature is raised, and the hydrogen bonds weaken, fluctuations give rise to rotational diffusion of some of the molecules. The presence of a rotating molecule in a local region further weakens the bonding in that region because of the loss of directionality of the bonding with its neighbors. The intermolecular forces to which surrounding
+ B(
):
+ C(
I“):
(3)
for H 2 0 (D,O)at atmospheric pressure. A nonlinear least-squares fit to the relaxation and viscosity data gives n = 20 (20), A = -0.4552 (-0.4519), B = 1.1320 (1.1322), C = 0.3231 (0.3196), and AH(To) = 14816 (15053) J mol-’. The reference temperatures Toare somewhat arbitrary. We choose 228 (236) K, which closely match those of the thermodynamic singularities of liquid water proposed by Speedy and Angell.31 The power n = 20 may seem odd. It merely reflects one of a number of possible empirical representations of the extreme steepness of the temperature dependence of AH(T) as the limit T -,To is approached. The accuracy of A, B, and Cis better than 0.5%. The constant AH(To) is the maximum value of AH( T ) attained in the extreme lowTo. Averaged AH( T ) values calculated temperature limit, T from eq 3 and the calculated relaxation data derived from these averaged AH( T ) values are listed in parentheses in Tables I-VII. Since the experimental data all lie above To, they are not too sensitive to the form of eq 3 or to concurrent variations of the fitting parameters, including To and AH( To). Good fits to AH( T ) can be obtained, for example, with 12 5 AH( To)5 18 kJ mol-’, together with other variations of eq 3. The connection between AH( T ) and the dynamic and transport properties of liquid water is thus seen to be direct: AH(T) is the activation enthalpy for these processes.
-
4. AH( T ) Connections The connection between AH(T) and “static” properties such as molar volume, compressibility, etc., is not so direct. However, the following reasoning based on work in an earlier paper9 traces this connection. AH( T ) is the enthalpy difference between the zero-point level in the well and the barrier maximum for the intermolecular hindering potential at temperature T. Therefore, its negative should be exactly proportional to the small, nearly constant APVterm plus an internal energy U ( r ) . The proportionality factor (S) depends on the number of hydrogen bonds that must be ”excited” in order for a molecule to be able to rotate, probably S = 2-4. This U( T ) is derived from a single degree of freedom per molecule, which, analogous to a spin system in a f e r r ~ m a g n e t ,describes ~~ the collection of torsional oscillators (librators) representing the liquid water network. At sufficiently high temperatures, both AH( T ) and U( r ) approach zero since many molecules are “rotating”, and the angular dependence of the intermolecular potential tends to disappear. Since -AH( T ) is related to U( T ) for the librational degree of freedom, AH(T) becomes a direct link to important thermodynamic quantities, such as “anomalous” contributions to the heat capacity. It is also a measure of the concentrations of librating molecules (intact or hydrogen bonded) and “rotating” molecules (broken or “unbonded”). A simple way-of seeing this is to consider the Ising lattice gas problem,32where nearest-neighbor molecular pairs, their number denoted by N++,interact attractively with energy -to, and remaining pairs do not interact. The total energy U(T ) of the lattice gas is therefore given rigorously by the quantity -N++eo,irrespective of the details of the cooperativity or the distribution of molecules. The result depends only on the assumption of pairwise additivity and the nearest-neighbor restriction. Thus, the key quantity N,, is equal to -U(T)/to, which would lead to the result 2SN++/q(T)N = AH( T)/AH(To) in our scheme, where q( 7‘) = 4 is the nearest-neighbor number density given by the integral over the first maximum in the radial distribution function at temperature T of liquid water. (31) Speedy, R. J.; Angell, C. A. J . Chem. Phys. 1976,65, 851. Angell, C. A. In Wafer,A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1982; Vol. 7, p 66. Speedy, R. J. J . Phys. Chem. 1987, 91, 3354. (32) Lee, T. D.; Yang, C. H. Phys. Reu. 1952, 87, 410. Pathria, R. K. Statistical Mechanics; Pergamon: New York, 1972; Chapter 12.
5822 The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
In this picture, the quantity N,, represents the number of molecular pairs that are hydrogen-bonded. The calculation of N,, thus becomes equivalent to the calculation of AH(T)/AH( To) and may not be straightforward because the distribution of hydrogen bonds appears as if it is far from randomS9If this notion ~.’~ proves correct, random network theories of liquid ~ a t e r ~would not be very accurate and in fact would seem to omit an “essential ingredient” leading to water-like properties. Thus, the ratio AH( T)/AH(To)serves as a direct measure of the fraction of strongly hydrogen bonded molecular pairs or “intact” structure at each temperature T, and [ 1 - AH(T)/AH(To)]is a measure of the fraction of weakly hydrogen bonded molecules or “broken” structure. Depending on how quantitative these relationships are, a knowledge of the activation parameter AH(T,P) as a function of temperature and pressure might then allow the computation of thermodynamic functions as well as the relaxation data and transport properties discussed earlier. AH(T,P),which is conceivably amenable to a theoretical eval~ation,~ would then supply an important tieline between the structure, the thermodynamics, and the relaxation processes of liquid water.
Bassez et al.
TABLE VIII: Specific Volume Parameters for Liauid Water H20a
T I data vb(To),b’C cm3 g-’ 1.0858 K-’ 1.14 X VU(To),c.d cm3 g-’ 0.858 f 0.041 a,,d K-I 1.37 (f0.50) x 10-3
D,O’
av 1.0858 1.14 X lo4 0.853 f 0.025 1.40 (f0.32) x 10-3
av 0.9788 1.19 X IO4 0.771 f 0.038 1.43 (f0.62)
x 10-3
In the case of H20, the listed coefficients are calculated from AH( r ) values obtained from the 170spin-lattice relaxation times and also a
from the averaged values of AH(r) calculated with eq 3; while in the case of D20, only the averaged AH(r) values calculated with eq 3 are used. bvband a b are from ice Ih experimental data, ref 45. ‘The reference temperature Tois 228 K for H20and 236 K for D20.dThe coefficients V,(To) and au are determined with a least-squares fit of experimental specific volume data to eq 5.
SP?
V I S --
-- 0
x
,
c
5. Thermal Expansion We have proposed here that a librational molecule implies more or less intact hydrogen bonding with its neighbors and that AH(T)/AH(To)is a measure of the number of such hydrogen bonds. Since hydrogen bonds compose more volume than linkages where the hydrogen-bond structure has been broken down, the specific volume of liquid water, hydrogen bonded plus “unbonded”, as a function of temperature should then depend on AH(T)/AH(To). To test this proposition further, we consider the experimental densities of H 2 0 and D 2 0 at atmospheric pressure, which show maxima a t 3.98 and 11.185 OC,respectively.’’ The specific volume of liquid water in terms of bonded (b) and “unbonded” (u) components may be written as V(T) = f v b + (l -nVu (4) where f, Vb, and Vu are all temperature-dependent. Similar procedures have been proposed and criticized2v3- by many authors. Here, however, following the discussion in section 4, we employ a different approach than has theretofore been used. Instead of requiring a theoretical calculation of the fraction f of molecules comprising the bonded structure, or using f as a fitting parameter, we employ the experimental ratio AH(T)/AH(To)as a measure of this fraction. This parameter automatically takes account of important temperature effects, not only on the population but also on the structure and state density, of the hydrogen-bonded energy levels. Thus VT)= [M(T)/AH(To)]Vb(T) + 11 - M ( T ) / m ( T o ) I V u ( T ) ( 5 ) with vb(T) = Vb(TO)[l + a b ( T Vu(V = Vu(To)[l + 4 T - To)] The reference temperatures To are chosen as before to be 228 (236) K, and the corresponding AH(To)are taken from the eq (33) Stanley, H.E.;Teixeira, J. J . Chem. Phys. 1980, 73,3404. (34) Sceats, M.G.;Rice, S . A. In Water, A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1982;Vol. 7,p 83. (35) Weast, R.C.; Astle, M. J.; Beyer, W. H. CRC Handbook of Chemistry and Physics; CRC: Boca Raton, FL, 1986-87; pp F-4, F-10. (36) Rontgen, W. C. Ann. Phys. 1892,45, 91. (37)Vand, V.; Senior, W. A. J . Chem. Phys. 1965,43, 1869. (38) Kell, G.S.;Whalley, E. Philos. Tram. R. SOC.London, A 1965,258, 565. (39) Davis, Jr., C. M.; Jarzynski, J. Adv. Mol. Relax. Processes 1967,I , 155. (40) Davis, C. M.;Jarzynski, J. Water and Aqueous Solutiom. Structure, Thermodynamics and Transport Processes; Horne, R. A., Ed.; Wiley-Interscience: New York, 1972;p 377. (41)Kauzmann, W. L’Eau Syst. Biol., Colloq. Int. C.N.R.S.1975,246, 63. (42) Endo, H.J . Chem. Phys. 1980,72,4324. (43)Haghibi, H.; Dec, S . F.; Gill, S . J. J . Phys. Chem. 1986,90, 4621. (44) Gill, S.J.; Dec, S. F.; Olofsson, G.;Wadso, I. J. Phys. Chem. 1985, 89,3758.
I
1
I
275
I
I
I
315 355 Temperature
I
(I.0
I
1
395
Figure 3. Correlation with specific volume of AH(r)/AH(To)from spin-lattice relaxation time (0)and viscosity measurements ( X ) for H20. Calculated (symbols) vs experimental (solid line).
It-
020
Figure 4. Correlation with specific volume of A H ( T ) / A H ( T o )from averaged dynamical measurements for D20.Calculated (symbols) vs experimental (solid line).
3 fitting procedure to be and 14 8 16 (1 5 053) J mol-’ for H 2 0 (D20), respectively. The specific volume V, and To and the coefficient of thermal expansion (Yb are obtained from density data on normal ice Ih.45 Equation 5 is then fit to density data for HzO and DzO from the supercooled r e g i ~ n ~ above ~ ~ ~ the ~ t boiling o point48 to,obtain the temperature-dependent specific volume of (45)Lonsdale, D. K. Proc. R . SOC.London, A 1958,247,424. (46)Zheleznyi, B. V. Russ. J . Phys. Chem. (Engl. Trawl.) 1969,43,1311. (47)Hare, D. E.; Sorensen, C. M. J . Chem. Phys. 1986,84,5085.
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987 5823
Is Liquid Water Really Anomalous? TABLE IX' Activation Enthalpies (J mol-') of H20 as a Function of Temperature and Pressure P, MPa T. K 0.1 50 100 200 238 243 248 253 258 263 268 273 283 299 309 323 353 383 403 423
10440 (11200) 9350 (9412) 9938 (10170) 9020 (8929) 9350 (9400) 8665 (8510) 8690 (8792) 8164 (8137) 8369 (8293) 7640 (7797) 8049 (7870) 7419 (7482) 7722 (7499) 7256 (7186) 7028 (7166) 6639 (6906) 6590 (6578) 6452 (6386) 5783 (5774) 5494 (5637) 5438 (5437) 5286 (5212) 4856 (4754) 4681 (4662) 3768 (3688) 3729 (3632) 2857 (2792) 2720 (2764) 2447 (2268) 2447 (2257) 1546 (1795) 1626 (1798)
8937 (8900) 8435 (8472) 8156 (8097) 7771 (7759) 7428 (7448) 7110 (7159) 6890 (6886) 6549 (6628) 6129 (6145) 5342 (5450) 4999 (5054) 4554 (4542) 3659 (3584) 2842 (2776) 2463 (2304) 1684 (1877)
8451 (8318) 8016 (8008) 7697 (7713) 7419 (7432) 6988 (7164) 6598 (6906) 6541 (6659) 6339 (6421) 6000 (5971) 5334 (5314) 5118 (4939) 4531 (4452) 3659 (3539) 2972 (2769) 2463 (2320) 1597 (1913)
'The values in parentheses are smoothed values of AH obtained from fits to eq 3. the "unbonded" structure V,,(T)and, within the above linear approximation, the values of Vu(To)and the coefficient of thermal expansion a,,.These results are summarized in Table VI11 and Figures 3 and 4. As can be seen from the figures, the agreement is not bad. For both H 2 0 and D20, however, the calculated volumes are about 0.1-0.2% too small in the region of the minimum, and they are somewhat too large at the high- and lowtemperature extremes. This could be caused by the relative simplicity of the model, the inadequacy of the linear approximations to Vb(7') and V,,(T),or the uncertainty in the AH(To) value from the eq 3 fit. It is interesting to note that, in the case of H 2 0 , V,,(273 K) and a, are similar to the values found by NBmethy and Sheraga for the unbonded The density of the "unbonded" structure at 228 K is -1.17 g cm-3 for H 2 0 , which suggests a denser form of the substance, as already proposed by many authors. See, for example, ref 2 and 50. Vuis 79% that of ice Ih. In fact, the peak at 3.5 A in the radial distribution function from X-ray diffraction data14351 for liquid H 2 0 suggests an oxygen atom network resembling that found in the high-pressure polymorphs of i ~ e , ~such , ~as~ice * 11, ~ ~111, and V, where p ranges from 1.15 to 1.26 g ~ m - and ~ , the distance of closest 0-0 approach of non-hydrogen-bonded neighbors is 3.24-3.45 A. These structural properties are also suggestive of those found for the high-density ~ , second form of amorphous solid water,34p i= 1.1 g ~ m - with nearest neighbors at 3.3 A. Considering that density and X-ray experiments depend mainly on the oxygen atom positions, these correlations mean only that some of the oxygen atom network in liquid water may resemble that in ice 11, 111, and V or of highdensity amorphous solid water. The correlations say little about the state of the hydrogen atom structure in liquid water. 6. Pressure Effects Experimental values of the spin-lattice relaxation times in H2170are available up to 250 MPa from supercooled to superheated temperatures.'"20 By use of the same method described in sections 1 and 2, the enthalpies of activation M(T)for reorientation in compressed water can be calculated by assuming a pressure-independent frequency factor ko in eq 2. They can then be fit to an equation of the form (3). These values are listed in (48) Kell, G. S. J. Chem. Eng. Data 1975, 20, 97. (49) Nemethy, G.; Scheraga, H. A. J . Chem. Phys. 1962, 36, 3382; J. Chem. Phys. 1964,41,680. Scheraga, H. A. Ann. N.Y.Acad. Sei. 1965,125,
249. (50) Kamb, B. In Structural Chemistry and Molecular Biology;Rich, A., Davidson, N., Eds.; W.H. Freeman: San Francisco, 1968; pp 507-542. (51) Narten, A. B.;Danford, M. D.; Levy, H. A. Discuss. Faraday Soc. 1967, 43, 97. Narten, A. H.; Levy, H.A. J. Chem. Phys. 1971, 55, 2263. Triolo, R.; Narten, A. H. J . Chem. Phys. 1975, 63, 3524. (52) See ref 4, Table 3.3, p 83.
TABLE X ' Higb-Pressure I'O Spin-Lattice Relaxation Times (ms) for H 2 0
P. MPa T, K 238 243 248 253 258 263 268 273 283 299 309 323 353 383 403 423
0.1 50 0.40 (0.28) 0.66 (0.64) 0.57 (0.51) 0.86 (0.90) 0.83 (0.81) 1.12 (1.20) 1.23 (1.18) 1.54 (1.56) 1.55 (1.60) 2.10 (1.97) 1.93 (2.08) 2.50 (2.43) 2.4 (2.62) 2.89 (2.97) 3.95 (3.56) 3.4 (3.22) 4.6 (4.62) 4.84 (4.96) 7.4 (7.42) 8.16 (7.78) 9.2 (9.53) 9.65 (9.89) 12.5 (12.90) 13.2 (13.26) 21 (21.44) 21.2 (21.75) 30.5 (30.93) 31.4 (31.11) 36 (37.26) 36 (37.34) 44.5 (43.25) 45 (43.29)
100 0.80 (.81) 1.17 (1.10) 1.40 (1.43) 1.82 (1.83) 2.29 (2.27) 2.83 (2.77) 3.33 (3.34) 4.09 (3.97) 5.46 (5.42) 8.59 (8.29) 10.6 (10.41) 13.7 (13.75) 21.6 (22.03) 30.6 (31.04) 35.9 (37.01) 44.1 (42.70)
200 1.00 (1.06) 1.35 (1.35) 1.71 (1.70) 2.11 (2.10) 2.75 (2.55) 3.48 (3.07) 3.83 (3.65) 4.43 (4.30) 5.72 (5.78) 8.62 (8.68) 10.2 (10.81) 13.8 (14.13) 21.6 (22.29) 29.70 (31.08) 35.9 (36.89) 44.7 (42.45)
'The listed values are taken from ref 19, and the values in parentheses are calculated from the averaged AH data of Table IX.
c
I
I I
270
310 350 Temperature
(K) (4)
390
Figure 5. Effect of pressure on AH(T) for H20.Top to bottom: 0.1, 50, 100, 200 MPa. The solid lines are averages calculated from eq 3.
Table IX, and Figure 5 shows their dependence on pressure and temperature. Using the smoothed values of AH( T ) from eq 3, one can calculate spin-lattice relaxation times for 170.These are listed within parentheses in Table X in order to illustrate the level of accuracy of this procedure. The isothermal M(7') values decrease with increasing pressure. This behavior is more pronounced in the supercooled region than at higher temperatures: at 238 K, AH(200 MPa)/M(O.l MPa) = 0.78. This observation shows that the rotation of water molecules becomes less hindered as pressure is applied. Because of compression, hydrogen bonds with their "fragile"53 angular dependence are distorted from the ideal tetrahedral geometry, and rotation becomes easier. In the supercooled regime, where the ice-I-like oxygen atom structure is abundant, the effect of pressure on the hydrogen-bond network is more important. Above 273 K more water molecules are assembled in "unbonded" structures, where the hydrogen bonds are already strongly distorted even at atmospheric pressure. Under these circumstances, the effect of compression is less. As can be seen from Figure 5 , the AH(T) curves for the various pressures converge near 370 K, confirming water's approach to a more normally behaving liquid under these conditions." Using, in eq 5, the parameters AH( r ) from Table IX, one can reproduce the specific volume as a function of temperature at different pressures. Since no information about V,, ab,or Tois (53) Dore, J. C. J . Phys. (Les Ulis, Fr.) 1984, 45 (Suppl. No. 9), 49.
5824 The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
Bassez et al.
TABLE X I Thermal Expansion Coefficients of the Components of Liquid H20under Pressure To, K P , MPa AH(To),J mol-’ Vb(TO), cm3 g-I 104ab,K-’ V,(To),cm3 g-’ 0.1 50 100 200 a
228 225 222 210
14816 11 279 10929 10 646
Values experimentally determined, ref 45.
1.086“ 1.017 0.988 0.938
K-1
0.855 0.846 0.836 0.81 1
1.14a 2.64 3.59 4.64
VI,b cm3 g-’
TI,*K
1.113 1.074 1.075 1.071
445 439 466 516
1.40 1.27 1.17 1.05
* VI is the intersection of Vb and Vu, and TI is the corresponding temperature.
I
1
272
-
I
350
3!0
empe-ata-e
1
(4)
I
I
,
390
Figure 6. Calculated (symbols) vs experimental (lines) specific volumes for H20at various pressures. Top to bottom: 0.1, 50, 100, 200 MPa.
Figure 7. V,(T,P) (upper four lines) and V,(T,P) (lower four lines) for P = 0.1, 50, 100, and 200 MPa.
available at high pressures, they must also be treated as variables. In order to obtain accurate fitting parameters, eq 5 is simultaneously fit to the specific volumes and compression ratios (AV/AP) at each temperature, where AVare volume changes with pressure relative to the atmospheric pressure (0.1 MPa) data. These results are shown in Figure 6, and the pertinent parameters are summarized in Table XI. Note in Table X I that Todecreases substantially with increasing pressure in this pressure range, a result that is fully consistent with the spin-lattice relaxation measurements.20 As seen, the calculations yield not yet measured values of the specific volumes at low temperatures and elevated pressures (Figure 6). The minima of these curves seem displaced toward lower temperatures as the pressure rises, a behavior supporting suggestions made by Kanno and AngelLs4 Figure 7 shows that the isothermal specific volumes of both the ice-I-like structure and the “unbonded” structure decrease with increasing pressure. The sense and magnitude of the variations are similar to those determined by BridgmanS5 for ice I and high-pressure ice forms. On the other hand, as the pressure increases, the thermal expansion coefficent ab of the ice-I-like structure increases, while that of the “unbonded” structure (a,) decreases (Table XI). Since the ice-I-like structure is more sensitive to pressure variations than the “unbonded” structure, these two components will converge at a sufficiently high pressure: 500 MPa, a b = a, i= 7.6 X lo4 K-’. At this pressure, water is therefore expected to behave like a normal liquid, all the hydrogen bonds being “broken”. A Raman spectrums6 of liquid water taken at 400 MPa confirms this expectation. It is also of interest to examine the temperature TIas a function of pressure. TI is the temperature above which v b = Vu = VI, water in this range behaving like a “normal” liquid. The values of VI and TI are listed in the last two columns of Table XI. The indication is that TI increases as a function of pressure, while VI remains relatively constant. A linear extrapolation up to 5 0 0 MPa gives TI = 666 K. This value is very close to the critical temperature of water, 647.15 K.4 In spite of such simple concepts
and insufficiencies of the experimental data, the methods predict a reasonable upper limit of temperature and pressure, beyond which all the hydrogen bonds are “broken” and water becomes a “normal” liquid. In the above context, the Raman spectrum of water under transient (- 10” s) ultrahigh pressures ( 6 2 6 GPa) created by a shock wave becomes a bit of a puzzle. Perhaps new structural properties or phases of water emerge at these ultrahigh pressures.s8 In any case, the steady-state resultss6 at 400 MPa seem definitive: the hydrogen-bonded structure is absent at this pressure.
(54) Kanno, H.; Angell, C. A. J. Chew. Phys. 1979, 70, 4008. (55) Bridgman, P. W. Proc. Am. Aced. Arts Sci. 1912, 47, 441.
(56) Lindner, H. Ph.D. Thesis, University of Karlsruhe, 1970 (unpublished). See ref 57 for a description. (57) Holmes, N. C.;Nellis, W. J.; Graham, W. B. Phys. Rev. Lett. 1985, 55, 2433.
7. Isothermal Compressibilities Using the ratio AH(T)/AH(To), one can also reproduce the behavior of the compressibility factor, K(T) = (-l/V‘)(8V/8P)T, a t 0.1 MPa. According to eq 5, K(T) may be written as a sum of three terms
where
The term K ~ T( ) is the so-called “structural” or “relaxational” part of the compressibility factor.39 The absence of experimental relaxation data for liquid water in the neighborhood of 1 atm precludes an exact calculation of &( T). Considering Kell’s polynomial expressions,48we decomposed the compressibility factors of the ice-I-like structure Kb( T), the “unbonded” structure K,( T), and K ~ T( ) as follows: Kb(T) = ~b(273)[1+ & ( T - 273)] K,( T) = ~ ~ ( 2 7[ 1 3 ) @,( T - 273)] K ~T ( )
+ + Ps(T - 273)]
= ~ ~ ( 2 7[31 )
The compressibility factor K ( T of ) liquid water48at 0 O C and 1 (58) Polian, A,; Grimsditch, M . Phys. Reo. Lett. 1984, 52, 1312.
The Journal of Physical Chemistry, Vol. 91, No. 22, 1987
Is Liquid Water Really Anomalous?
5825
wz:x1 L% r
n’I CD n CD
I
Compressibility
\
m
\
::
x
I--.I
c
X Y
1
61
m
,--;
-
i
x
I
I
270
I
I
I
I
350 Temperature ( 310
I
K
I
I
390
I
1
240
)
I
1
I
1
250 260 TemperatLre
I
(vi)
I
I
270
Figure 8. Calculated ( X ) vs experimental compressibility factors, K(T), for H20at P = 0.1 MPa.
Figure 9. Temperature derivatives of M ( q data in Figure 1 (H20, lower curve) and in Figure 2 (D20, upper curve).
atm is 50.885 X 10” bar-’. The preferred value59 of the compressibility factor of ice I is 13 X 10-6 bar-’, the value determined by Richard and S p e y e r ~rather , ~ ~ ~than ~ the Bridgman value,5s 35 X 10” bar-’. The thermal coefficient Pb is calculated from Bridgmao’s measurements of the isothermal compressibilitySSto K-I. A least-squares fit of eq 6 to Kell’s data,48 be 4.5 X using 16 values of A H ( T ) / A H ( T o )from spin-lattice relaxation times between 243 and 423 K, reproduces the known minimum near 320 K (Figure 8). The fitted coefficients are ~ ~ ( 2 7= 3 )33 X bar-’, 6, = 5.6 lo-* K-’
Figure 9, deuterium substitution should enhance the contribution to C, from the librational/rotational motion, particularly in the supercooled regime. This conclusion seems in good accord with the experimental findings; cf. Figure 4 of ref 65. The scale factor S (section 4) connecting AH( T ) and U( T ) is not known, but one would expect that 2-4 hydrogen bonds would have to be “excited” before a molecule could rotate. Comparison between our curves in Figure 9 and the experimental data for CJ1) - Cp(s)in ref 65 does indicate that S is of this general size. However, a quantitative comparison depends on a number of additional factors. For example, there are “anomalous” contributions to C, besides the one directly related to AH( T). The most notable of these are expected to arise from the low-frequency translational modes, which are also temperature-dependent.13It is important also to realize that the C,(1) - C,(s) difference data plotted in ref 65 do not provide an exact measure of the C, cofltribution from the temperature-dependent frequencies: C,(1) contains contributions from m temperature-dependent intermolecular modes and (6 - m ) normal intermolecular modes, while C,(s) contains contributions from 6 normal intermolecular modes. Thus, the difference data underestimate the temperature-dependent contributions to C,( T). In spite of these uncertainties, the similarity between the calculated M(T)slopes in this paper and the C,(1) - CJs) results in ref 65 is rather remarkable. A full report of the heat capacity calculations for liquid water will be reported at a later time.66
~~(27= 3 )31 X 10” bar-’,
8, = -5.6
K-I
Admittedly, uncertainties in the compressibility data of ice-I weakens the analysis here. The trends are probably correct, but the absolute values of the fitting parameters may be in error. However, the contribution of the “structural” part of the compressibility factor at 0 O C does agree with the estimates of other workers.6144 The sign difference of P, and P, points to the different physical content of K, and K,. As the temperature rises, the compressibility K, increases in a normal manner, (dV/dP), becoming more negative with increasing temperature. However, both (negative) factors in K , trend closer to zero at high T, reducing the value of K , as the temperature rises.
8. A Note about the Heat Capacity Since we have proposed that AH(T) is proportional to the enthalpy change accompanying the transformation from a locally ordered to a rotationally disordered phase of liquid water, it is evident that -dAH( T ) / d T provides an important contribution to C, for this substance. Because of the steep dependence of AH(T) on temperature, this heat capacity contribution is large and anomalous appearing. A simple, but not altogether correct, way of seeing this is to examine the partition function 1 / 2 csch [hv(T)/2kT] of a harmonic oscillator with a temperature-dependent frequency v ( T ) . This would give rise to extra terms in C, containing the first and second temperature derivatives of v( T). Figure 9 depicts the numerically determined slopes for H 2 0 and D20of the AH( T ) curves in Figures 1 and 2. These slopes are plotted in cal g-’ so that they may be compared more easily with plotted experimental heat capacity data.65 According to (59) Dorsey, N. E. Properties of Ordinary Water-Substance; Reinhold: New York, 1940; pp 471-472. (60) R.ichard, T. W.; Speyers, C. L. J . Am. Chem. SOC.1914, 36, 491. (61) Litovitz, T. A,; Camevale, E. H. J. Appl. Phys. 1955, 26, 816. Davis, Jr., C. M.; Litovitz, T. A. J . Chem. Phys. 1965, 42, 2563. (62) Frank, H. S.;Quist, A. S.J . Chem. Phys. 1961, 34, 604. (63) Smith, A. H.; Lawson, A. W. J . Chem. Phys. 1954, 22, 351. (64) Slie, W. M.; Donfor, A. R.; Litovitz, T. A. J . Chem. Phys. 1966, 44, 3712. (65) Rasmussen, D. H.; MacKenzie, A. P. J . Chem. Phys. 1973,59, 5003.
9. Conclusions When the properties of liquid water are compared with those of other common substances, water seems in many ways unusual and mysterious. We believe its unusual properties are primarily caused by highly temperature sensitive intermolecular potentials and attendant temperature-dependent vibrational frequencies. When these features are correctly incorporated into rate equations and thermodynamic expressions, water seems conformable to all the well-established concepts to which other substances adhere. In this sense, liquid water is is not a “lawless”67substance. It is not anomalous.
Acknowledgment. Financial support at the PQRL has been shared jointly by the National Science Foundation (Grant CHE8611381) and the Robert A. Welch Foundation. Important discussions with Rufus Lumry and George Walrafen about t h e general problem of water and the consequences of the Raman data are acknowledged. The help of Francis Muguet with various concepts throughout the formulation of the work is also gratefully acknowledged. Special thanks go to Cheryl Starkey for her tireless efforts at the word processor. (66) Bassez, M.-P.; Robinson, G. W., to be published. (67) Webster’s Third New International Dictionary Unabridged; Gove, P. B., Ed.; G. and C. Merriam: Springfield, MA, 1966; pp 88-89.
