J. Phys. Chem. 1996, 100, 3825-3827
3825
Simple Relationship between the Properties of Isotopic Water M. Vedamuthu,† S. Singh,† and G. W. Robinson*,†,‡ SubPicosecond and Quantum Radiation Laboratory, Departments of Chemistry and Physics, Texas Tech UniVersity, Lubbock, Texas 79409 ReceiVed: NoVember 2, 1995X
In an earlier paper on this subject, it was conjectured that if temperatures of H2O and D2O are scaled according to their temperatures of maximum density, then the structural properties of these two liquids should be very closely the same. In this paper it is found that the idea is valid for the densities within the four decimal point experimental precision of the D2O data. If this structural effect is not taken into account, a comparison of properties, such as viscosities or self-diffusion coefficients, of isotopic liquid water would have a reduced meaning. Comparing data at the same temperature, as has traditionally been done, simply gives rise to misleading isotope effects.
I. Introduction In a recent series of four papers in this journal concerning properties of liquid water, it was shown that a modernized mixture model for the liquid over a temperature range from the deep supercooled region to boiling temperatures is semiquantitatively consistent with the “anomalous” heat capacity and dynamic properties of water1 and is quantitatively consistent, in the temperature range t j 50 °C, within the 4-5 decimal point point precision of measurement with the densities2,3 of H2O and D2O and within the 3-4 decimal point precision of the isothermal compressibility4 of H2O. It is also known from wholly independent work5 that this approach, with close to the same parameters, is capable of explaining the temperature dependence of the Raman spectrum of liquid water from -24 to +95 °C. In addition, differential X-ray scattering data6 have indicated that radial distribution functions might also be reproduced7 by this approach. In the discussion section of the paper3 on D2O, following the lead of Angell,8 it was stated that “when the temperatures are scaled according to the temperatures of maximum density, the bonding fractions in liquid D2O are not very different from those in liquid H2O”. In other words, according to this concept, shifting the temperature scale for H2O upward by the difference between the temperatures of maximum density9 of H2O (3.984 °C, where F ) 0.999 972 g/cm3) and D2O (11.185 °C, where F ) 1.106 00 g/cm3), or about 7.2 °C, would cause the structural properties of these two liquids to be nearly identical over a fairly wide range of temperatures. If true, this idea should be relevant not only to the densities of the two liquids but also to the radial distribution functions10 and the dynamics,11 or any other property that depends solely on the structure of the liquid, once any effect of the mass difference, 18.010 565 for H2O and 20.022 92 for D2O, has been accounted for.
V(P,T,H) ) V(P,T′,D) f(P,T,H) VI(P,T,H) + [1 - f(P,T,H)]VII(P,T,H) f(P,T′,D) VI(P,T′,D) + [1 - f(P,T′,D)]VII(P,T′,D)
for the ratio of the specific volumes of H2O and D2O at pressure P and Kelvin temperatures T and T′, where VI and VII are the specific volumes of the assumed two bonding types in the mixture-model picture, and f is the bonding fraction of the type I component. The temperature-shift conjecture states that the structure, thus the bonding fraction f, in H2O is the same as that in D2O when T′ ) T + δ, where δ ) 7.2 K. We can now assume that, when the 7.2 K temperature shift is applied over a range of temperatures, the structural properties of type I are the same for both isotopic molecules, so that their densities at the pairs of temperatures T and T′ are related by merely a constant factor,
FI(P,T′,D) ) RsFI(P,T,H)
FII(P,T′,D) ) RsFII(P,T,H)
Department of Chemistry. Department of Physics. X Abstract published in AdVance ACS Abstracts, February 1, 1996. ‡
0022-3654/96/20100-3825$12.00/0
(3)
where the ratio Rs refer to the shifted temperature scale, T′ ) T + δ. The constant Rs should be close to the ratio of the molecular weights, 20.02292/18.010565 ) 1.11173. However, because of slightly different hydrogen-bond distances and atomic root-mean-square displacements within the same structural constraints, the ratio would not be expected to be exactly this value.12 The relationships in eqs 1-3, together with the bonding fraction equivalency, then allow one to write, very simply, for the liquid densities,
F(P,T′,D) ) Rs F(P,T,H)
†
(2)
Assume also that exactly the same relationship with the same constant factor holds for the type II bonding structures,
II. Density Relationship The simplest property of water for testing such a relationship is the density. In fact, the mixture model for the density provides an explicit expression,2,3
(1)
(4)
III. Results Densities of H2O and D2O. In Table 1 we present the densities13 of liquid H2O at a series of temperatures, the densities3,9,14 of D2O at these same temperatures, and the densities, obtained from Kell’s fitting equation,9 of D2O at the shifted temperatures [t(H2O) + 7.2] °C. While the density of © 1996 American Chemical Society
3826 J. Phys. Chem., Vol. 100, No. 9, 1996
Vedamuthu et al.
TABLE 1: Scaled and Unscaled Densities of D2O t(H2O), °C F(H2O), t F(D2O),t -30 -25 -20 -15 -10 -5 0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50
0.983 85 0.989 59 0.993 55 0.996 28 0.998 12 0.999 26 0.999 84 0.999 96 0.999 70 0.999 10 0.998 20 0.997 04 0.995 65 0.994 03 0.992 22 0.990 21 0.988 04
1.0738 1.0845 1.0918 1.0970 1.1006 1.1031 1.1047 1.1056 1.1060 1.1059 1.1053 1.1045 1.1032 1.1017 1.1000 1.0979 1.0957
Ru 1.0914 1.0959 1.0989 1.1011 1.1027 1.1039 1.1049 1.1057 1.1063 1.1069 1.1073 1.1077 1.1081 1.1083 1.1086 1.1088 1.1090
t′ ) t(H2O) + 7.2 °C F(D2O),t′ -22.8 -17.8 -12.8 -7.8 -2.8 +2.2 +7.2 +12.2 +17.2 +22.2 +27.2 +32.2 +37.2 +42.2 +47.2 +52.2 +57.2
1.0880 1.0943 1.0987 1.1018 1.1039 1.1052 1.1059 1.1060 1.1057 1.1050 1.1040 1.1026 1.1010 1.0991 1.0970 1.0947 1.0921
Rs 1.1059 1.1058 1.1059 1.1059 1.1060 1.1060 1.1060 1.1060 1.1060 1.1060 1.1059 1.1059 1.1058 1.1057 1.1056 1.1055 1.1053
H2O is very likely known to around five significant figures, that of D2O is known only to about four. It is seen from Table 1 that the ratio Rs of these densities for temperatures between -30 and +30 °C (on the H2O scale) is 1.1059 ( 0.0001, the small variation very probably being caused by the slight D2O density uncertainty, particularly in the lower temperature range of the supercooled regime.14 This constant ratio on the shifted temperature scale is in contrast to the density ratio Ru shown in Table 1 when no temperature shift is applied. The Ru ratio increases from 1.0914 at -30 °C to 1.1081 at +30 °C, a variation that is over 2 orders of magnitude larger than that of Rs. Above about +30 °C, the ratio Rs begins to deviate more from its low-temperature value. This is caused, as discussed in earlier work,2-4 by different thermal expansivities of the I and II components, a change in the structure of the type II component to more varied bonding forms and eventually to a breakdown of the entire two-state mixture-model concept with increasing temperature. Application to the Density of T2O. This scaling procedure should also apply to T2O, where density measurements, as discussed by Goldblatt,15 are much more demanding. In the case of T2O, the temperature of maximum density15 is near +13.4 °C and the density at this temperature is reported to be 1.215 02 g/cm3. Thus, for T2O, temperatures must be shifted by about 9.4 °C, and the density ratio should be 1.215 05, giving T2O densities of 1.1954 g/cm3 at -20.6 °C, 1.2128 g/cm3 at -0.6 °C, 1.2140 g/cm3 at +24.4 °C, and 1.2056 g/cm3 at +49.4 °C. These values are close to (actually about 0.0001-0.001 g/cm3 higher than) those given by Goldblatt,15 but are probably more accurate, not only because of the high precision of the H2O data used in our assessment but also because of Goldblatt’s method for correcting to 100% from his 99.3% T2O sample, which took no account of the temperature-shift phenomenon discussed here. Furthermore, it can be presumed that Goldblatt’s density determination at the temperature of maximum density, on which our calculated densities for T2O are based, except for the slight error caused by his correction method, is among his most accurate. IV. Other Properties A number of other properties of isotopic water can be compared using this temperature-shift concept. However, the melting and boiling points of H2O and D2O do not behave similarly because they depend on the thermodynamics of phases other than the liquid, rather than simply structural considerations of the liquid itself.
Shear Viscosities. Here, we briefly demonstrate how this sort of scaling applies to other quantities such as the shear viscosity. For instance, according to Osipov et al.,16 the shear viscosity of H2O for 26 temperature points between -4 and -35 °C is essentially equal, usually within a few tenths of a centipoise, to that of D2O at t(H2O) + 7 °C. This is somewhat remarkable because of the rather wide variation of viscosity in this temperature range, 2-19 cP. More interesting perhaps is the fact that temperature scaling of the more precise H2O and D2O viscosity data at not such low temperatures gives an isotope ratio η(D2O)/η(H2O) of about 1.05, which is close to what the square-root-mass law would predict. Since there has been confusion in the past17 concerning these isotope effects, it would be of value in future work to be aware of the structural effect of temperature when eliciting an isotope effect for any static or dynamic property of water. V. Discussion The very precise temperature-scaling phenomenon discussed in this paper for the densities is not simply a curiosity. As seen for the shear viscosity data, it should be extremely useful in correlating other properties of the isotopic forms of water with those of H2O, thus providing a validity check at the much higher precision usually available for H2O data. These correlations also lend support to our modern mixturemodel approach,1-4 since no parameter variations have been necessary here. This mixture model for liquid water assumes that hydrogen bonds can be bent rather than broken,18 as in the moderately dense ice polymorphs, II, III, V, and VI. The model is currently leading to a much better molecular-level understanding of the “anomalous” properties of liquid water, and in fact it is the only model that seems capable of providing a precise, yet simple, quantitative understanding of virtually all the properties of water over a wide range of not only T but also P. This model differs from others, such as the random network model,19 which concentrate almost entirely on the bonding in ice Ih as a starting point, then analyze deviations from this structure in terms of ring formation20 or broken bonds, without specifically taking into account any of the other ice polymorph structures. In our model, the number of nearest-neighbor hydrogen bonds is always close to four, as in all the ice forms as well as the liquid, but so-called “fifth neighbors”21 (having a second-neighbor O‚‚‚O distance in the range of about 3.4 Å, as in the ice polymorphs of moderate density22,23) increase in number with increasing temperature, replacing some of the ∼4.5 Å structure found in ice Ih. This picture of the liquid is in complete agreement with the experimental isochoric X-ray data of Bosio et al.6 It has also recently been found possible24 computationally to create with this model a realistic density maximum in liquid water, which other computational models have really not been able to do very well.25,26 Acknowledgment. Financial support was provided by the Robert A. Welch Foundation (D-0005 and D-1094) and by private donations from one of the authors (G.W.R.) and the G. Wilse Robinson, Jr., Trust, Commerce Bank, Kansas City, MO. References and Notes (1) Bassez, M.-P.; Lee, J.; Robinson, G. W. J. Phys. Chem. 1987, 91, 5818. (2) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Phys. Chem. 1994, 98, 2222. (3) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Phys. Chem. 1994, 98, 8591. (4) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Phys. Chem. 1995, 99, 9263.
Relationship between the Properties of Isotopic Water (5) d’Arrigo, G.; Maisano, G.; Mallamace, F.; Migliardo, P.; Wanderlingh, F. J. Chem. Phys. 1981, 75, 4264. (6) Bosio, L.; Chen, S.-H.; Teixeira, J. Phys. ReV. A 1983, 27, 1468. (7) Urquidi, J.; Singh, S.; Robinson, G. W. Work in progress. (8) Angell, C. A. Ann. ReV. Phys. Chem. 1983, 34, 593. (9) Kell, G. S. J. Chem. Eng. Data 1967, 12, 66. (10) Soper, A. K.; Phillips, M. G. Chem. Phys. 1986, 107, 47. (11) Lang, E. W.; Lu¨demann, H. D. In NMR Basic Principles and Progress; Jonas, J., Ed.; Springer-Verlag: Berlin, 1990; Vol. 24, pp 129187. (12) Thiessen, W. E.; Narten, A. M. J. Chem. Phys. 1982, 77, 2656. (13) Kell, G. S. J. Chem. Eng. Data 1975, 20, 97. (14) Hare, D. E.; Sorensen, C. M. J. Chem. Phys. 1986, 84, 5085. (15) Goldblatt, M. J. Phys. Chem. 1964, 68, 147. (16) Osipov, Yu. A.; Zheleznyi, B. V.; Fondarenko, N. F. Russ. J. Phys. Chem. (Engl. Transl.) 1977, 51, 748. (17) DeFries, T.; Jonas, J. J. Chem. Phys. 1977, 66, 5393. (18) Kamb, B. In Structural Chemistry and Molecular Biology; Rich, A., Davidson, N., Eds.; W. H. Freeman and Co.: San Francisco, 1968; pp 507-542.
J. Phys. Chem., Vol. 100, No. 9, 1996 3827 (19) Sceats, M. G.; Rice, S. A. In Water-A ComprehensiVe Treatise; Franks, F., Ed.; Plenum: New York, 1982; Vol. 7, Chapter 2. (20) Belch, A. C.; Rice, S. A. J. Chem. Phys. 1987, 86, 5676. (21) Sciortino, F.; Geiger, A.; Stanley, H. E. Phys. ReV. Lett. 1990, 65, 3452. (22) Eisenberg, D.; Kauzmann, W. The Structure and Properties of Water; Oxford Univ. Press: London, 1969. See Table 3.4, p 85. (23) Robinson, G. W.; Zhu, S.-B.; Singh, S.; Evans, M. W. Water in Biology, Chemistry and Physics: Experimental OVerViews and Computational Methodologies; World Scientific: Singapore, 1996; Chapter 4. (24) Cho, C. H.; Singh, S.; Robinson, G. W. To be published. (25) Billeter, S. R.; King, P. M.; van Gunsteren, W. F. J. Chem. Phys. 1994, 100, 6692. (26) Wallqvist, A.; A° strand, P.-O. J. Chem. Phys. 1995, 102, 6559.
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