D. E. O'Reilly
1674
the estimate 1 5 x cm3 for 8,and setting W(M0 ry) equal t o its average value Mo2/6, then yields A[P] = 2.4 X cm3/g for comparison with the experimental jump in [PI ai the transition of 5.2 X cm3/g. This order-of-magnitude agreement with the theoretical estimate lends some confidence to the explanation advanced. If it is correct, the sulfur system would appear to be the first system in which the effect of composition fluctuation on dielectric properties has been directly observed.
equation, the validity of which in assessing effects as delicate as those considered in this work may be questioned. We have not attempted to take into account species other than the SS ring and the ( S S ) chain, ~ though other ring species, at least, are known to be present.2a It is evident that many points remain to be clarified before full understanding of the physical properties of liquid sulfur is attained.
Sumniary The careful examination of the dielectric properties of liquid sulfur reveals new phenomena calling for refinement of the interpretation of the physical properties of this system. Tn particular the low-temperature phase, which we have assumed to consist entirely of Ss rings, exhibits a temperature-dependent molar polarization; the high-temperature phase, consisting of a complex mixture of rings and chains, exhibits a molar polarization increasing proportional to PV; and extrapolation of the molar polarization to the transition temperature indicates a discontinuity a t that point. The temperature dependence of the dielectric properties of the low-temperature phase can be interpreted in terms of a postulated polar Ss ring conformer, whose predicted properties are reasonable. The strict proportionalit) of molar polarization to W in the high-temperature phase raises some interesting questions on relative symmetry and conformational flexibility of SS units in ring and chain. Further study of this matter is certainly desirrible. Fbsults of a simple fluctuation analysis are consistent with the magnitude of the discontinuity in molar pola~~izatioinat the transition. Finally, no evidence was found for any dependence of static dielectric constant on Yrequency up to 10 kHz. The interpretation of our results involves many uncertainties. We have made heavy use of the Clausius-Mosotti
References and Notes (a) Supported in part by the Public Health Service, National Institutes of Health under Grant No. GM l 1125-03. (b) Contribution No. 3216. (a) J. A. Pouiis and C. H. Massen in "Elemental Sulfur," B. Meyer, Ed.,?nterscience, New York, N. Y., 1965, p 109; (b) E. D. West, J. Amer. Chem. Soc., 81, 29 (1959). W. J. MacKnight and A. V. Tobolsky in ref 1, p 95. G. Gee, Trans. FaradaySoc., 48, 515 (1952). A. V. Tobolsky, J. Polym. Sci., 25, 220 (1957); A. V . Toboisky and A. Eisenberg, J. Amer. Chem. Soc., 81, 780 (1959); 82, 289 (1960). G. E. Sauer and L. B. Borst, Science, 158, 1567 (1967). J. Curtis, J, Chem. Phys., 1, 160 (1933). C. P. Smyth, "Dielectric Behavior and Structure," McGraw-Hill, New York, N. Y., 1955, Chapter Vi. A. M . Keilas, J. Chem. SOC.,113, 903 (1918). J. F. Bottcher, "Theory of Electric Polarisation," Elsevier, New York, N. Y., 1952. H. L. Strauss and J. A. Greenhouse in ref 2a, p 241. A. S. Cooper, W. L. Bond, and S. C. Abrahams, Acta. Crystallogr., 14,1008 (1961). J. A. Semlyen, Trans. Faraday Soc., 63, 743 (1967). We are indebted to the referee for calling our attention to this paper. Reference 8, Chapter XI. H. Gerding and R. Westrik, Recl. Trav. Chim. Pays-Bas, 62, 68 (1943). M. Kubo, Sci. Papers Inst. Phys.-Chem. Res. (Tokyoj, 32, 26 (1937); A. L. McClellan, "Tables of Experimental Dipole Moments," W. A. Freeman, San Francisco, Calif.. 1963, p 212. M. Schmidt, Osterr. Chem. Z.,64, 236 (1963). M. Davies, "Electrical and Optical Aspects 011 Molecular Behavior," Pergamon, New York, N. Y., 1965, p 20. M. E. Baur, manuscript in preparation. H. Frohiich, "Theory of Dielectrics," Oxford University Press, London, 1949, Chapter li.
eridence of Viscosity and Nuclear Relaxation Time in Water and
D. E. O'Reilly AryonrseNational Laboratory, Argonne, lllfnois 60439 (Received February 7, 1974) Piibliciition costs assisted by Argonne National Laboratory
The pressure dependence of viscosity and nuclear relaxation times in water and deuterium oxide may be readily explained in terms of a new model for water that has been formulated recently. Effective values of &hedeuteron quadrupolar coupling constant in D20 are given.
Water is unusual among liquids as evidenced by the fact that, ai, low temperatures and pressures, the coefficient of viscosity (7)decreases with increase in pressure. The Journal of Phy:sicai Chi?mistry, Vol. 78, No. 16, 1974
Likewise, the nuclear relaxation time ( T I )of deuterium in D20 and the proton in HzO increase with increase in pressure. A quantitative explanation of these effects has been
Viscosity arid PJuclear Relaxation Time in Water and Deuterium Oxide
1675
ence between ( 3 In TI/aP), for HzO and DzO (Figure 1)is considerably larger and the variation of Q with pressure and the intermolecular contribution to the proton in HzO must be taken into account. The variation of Q with pressure i s readily estimated from the following semiempirical equation4
.-0.20
L
I
20
0
40
60
\
80
J
100
t,"C
(-a-, ref 3a) and for D 2 0 ( - -
Figure 1. (it In q/aP)* for H20 El- -, ref 3c); la In T , / c ? P ) ~for H20 ( - -#--, ref 3c).
(-a-,ref 3b) and for D20
proposed in recent communications2 with the aid of a model which is based on the ice VI1 and ice ICstructures. It may easily be shown (from eq 6, ref 2a) that the pressure coefficients of 7'1 and q are related as follows
where (ez) is Ihe mean square angle of rotation of a water molecule during a hard collision and w is the mean work required to form a vacancy in water. Equation 1 is written with the approximation that the water molecule is considered to be a spherical top molecule. In addition, it was shownzbthat the pressure coefficient of q is given by
1
-t T
Q = 310 - ( 5 7 2 / R 3 ) (31 where Q is in kHz and R is the H - - - 0 hydrogen bond length in Angstrom units. From eq 3 and the compressibilityZa of HzO we obtain (a In = -0.031 kbar-' at 0". The fraction of intact hydrogen bonds increases slightly with increase in p r e ~ s u r e ,but ~ this is a small effect and will be neglected. Using the approximation2b for protons in HzO that ( T ~ ) , ~ t ~=, -lh(TdIntra-l, l the difference between (a In T I / d P ) T for HzO and D z 0 is accounted for accurately by eq 1 with (a In ( 8 2 ) 1 / 2 / 8 P ) T = -0.021 kbar-1 (10") where we have used the result2bJ that (ez) = 1.1 which was assumed to be the same for H20 and DzO. That is, the root mean square angle of rotation upon a hard collision decreases slightly with increase in pressure. The magnitude of the coefficient (a In ( 6 2 ) 1 / 2 / a P ) given ~ above appears to be a reasonable value. In the interpretation of the temperature and pressure dependence of the relaxation times of deuterium and oxygen-17 in water the variation of Q with the fraction of hydrogen bonds that are intact ( f ) as well as the change of Q with the mean hydrogen bond length must be taken into account. Denoting the quadrupolar coupling constants for the deuteron in a ruptured and intact hydrogen bond by QM and QHB, respectively, the effective value of Q is given by Q = f & H B + (1 - ~ ) Q M . In the present model the effective value of Q for deuterium in D2O at 0" is 237 kHz and at 100" Q = 245 kHz; in the calculation we have placed QM = 318 kHze and QHB = 213 kHz7 and used values off recently reportedS (at 0" f = 0.774; at 100"f = 0.694). The behavior of water with temperature and pressure can be explained readily with the aid of the quasilattice model with reasonable values of the parameters that occur in the model and that are consistent with molecular dynamics calculations8 on water.
where A73 is the mean rotational energy barrier for molecReferences and Notes ular reorientation.zt'.c For deuterium oxide, the term (1) Based on work performed under the auspices of the U S Atomic -2(a In Q/dP) r must be added to the right-hand side of eq 1 Energy Commission where Q is the effective quadrupolar coupling constant of (2) (a) D E O'Reilly, Phys Rev A , 7, 1659 (1973), (b) D E O'Reilly, E M Peterson and C E Scheie Chem Phys Lett in press (c) deuterium iin liquid deuterium oxide. D E O'Reilly, J Chem Phys., 60, 1607 (1974) Experimental3 values of the pressure coefficients (a In (3) (a) K E Bett and J B Cappi, Nature (London), 207, 620 (1965), (b) H G Hertz and C Radle Z Phys Chem (Frankfurt am Mam) q/aP)r and ( 8 h ?;/dP)T for liquid HzO and DzO are 68, 324 (1969), (c) Y Lee and J Jonas J Chem Phys 57, 4233 shown in Figure 1 over the temperature range 0-100". (1972) Water and deuterium oxide exhibit differences, particu(4) G Soda and T Chiba, J Chem Phys , 50,439 (1969) ( 5 ) D E 0 Reilly J Chem Phys , in press larly with regard to (a In T,/aP),. Small differences in (6) P Thaddeus, L C Kirsher, and T H Loubser. J Chem Phys, 40, the energy .parametersZbyct o and €1, the equilibrium con257 (1964) (7) P Waldstein, S W Rabideau, and J A Jackson, J Chem Phys, siantXb*C K, and ( a w l a P ) between ~ HzO and DzO can rea41,3407 (1964) sonably account for the difference between (a In q / a P ) ~ (8) (a) F H Stillinger and A Rahman, J Chem Phys, 57, 1281 for HzO and I)& shown in Figure 1. However the differ(1972), (b) A Rahman and F H Stillinger ib/d 55, 336 (1971)
The Journal of Physical Chemistry, Vol. 78, No. 16, 1974
