1220
J. Phys. Chem. 1993,97, 1220-1223
The Scaled Particle Theory and the Tait Equation of State J. V. Leyendekkers Department of PhysicallTheoretical Chemistry, University of Sydney, Sydney 2006, Australia Received: May 18, 1992; In Final Form: October 6, 1992
The pressure coefficient of the Tait equation of state for water has been compared to the network stress or pressure derived from the application of the scaled particle theory (SPT). The two pressures are equivalent when the mean molecular distance of closest approach, u, is around 2.8 A, dropping to 2.7 A as the boiling point is approached. The effect of applied pressure, to 1000 bar, was considered also and a found to be a linear function of density. The values of a are consistent with an equation derived by P. W. Bridgman, giving the thermal conductivity as a function of the sound velocity and u. Using equations derived from the scaled particle and fluctuation theories, the second parameter of the Tait equation was interpreted as the cell volume/sites ratio of a quasilattice. The equations are applicable to any liquid, and preliminary results for ethanol show equivalence of the Tait and SPT pressures for this liquid.
I. Introduction The following statement by Sti1linger:I “the scaled particle theory (SPT) offers a powerful conceptual and computational framework within which to examine molecular order and thermodynamic properties” is still valid today.2 Although a geometric intensive theory, the SPT has been remarkably successful and has generated many exact results.2 On the other hand, the Tait equation of state was originallyderived empirically over a hundred years ag0.~9~Around the time the SPT was introduced the Tait equation was derived theoretically from association theorySand from thevan der Waals equation of state.6 The Tait equation has been applied to a wide variety of liquids, including ~ a t e r ,and ~ , an ~ extended form has been successfully used for solutions.’ An important feature of the Tait equation is that it also covers the dielectric constant, D, as a function of temperature and pressure and provides a direct, simple relationship between the density and the dielectric. In the present paper it will be shown that for water the Tait equation parameters can be related to functions derived from SP theory. That is, the pressure-volume-temperature (PVT) and PDT characteristics of a liquid can be described in essentially geometric terms; such geometry, of course, arising from the inherent molecular structure and interaction patterns of the system, which dictate the orientations and required volumes, averaged over time. The elegant simplicity of the Tait equation results in compact derivatives so that the derived quantities are less susceptible to cumulative errors than the corresponding values calculated from more complex equations of state (section 11). In section 111, the relevant equations of the SPT are outlined as proposed by Stillinger.] Numerical results are given for water (based on Pierotti’s equationss as developed by Stillinger’). This enables the “network stretch” pressure,pspt,to be estimated and compared with the resultant of the cohesive and expansive pressures, as given by the Tait equation (section IV). The averaged distance of closest approach of the molecules is considered in section V, the isothermal compressibility in section VI, and the effect of applied pressure in section VII. 11. The Tait Equation of State
BT is the difference between cohesive and expansive pressures of the liquid; it is a function of temperature but independent of pressure. j3 is the isothermal compressibility. CTis constant for a given liquid.
The integrated form from pressure 1 to p is
P) - v’” = -c,v(’) log ( ( B , + p ) / ( B , + 1))
The derivatives of j3 with respect to temperature and pressure are, from eq 1,
+
(d/3@’/dT), = -/3@)(a@) - a(l) (dB,/dT)/(B,
(dfl@)/dp),= /32( 1 - c y ) with c’ = (In 10/C~)and V’ = W ) / V ) , (a2/3@)/ap2), = (2//3)(aa/W: which, when p = 1 becomes
+ b3CY’
(4)
(5)
(a2/3/ap2)T= /33(c’+ 2(1 - c y ) Similarly, for the dielectricconstant, D, the following equations app1y:IO (1 - ( D ( ’ ) / D @ ) = ) ) A , log ((BT+PI/@, + 1)) (6) Combining eq 2 and 6 gives the relationship between D and V: (1 - ( D ( ’ ) / D @ ) = ) ) (A,/CT)( 1 - (@’/ v’”))
(7)
For water, AD = A b exp(CDT)and is independent of pressure.10
Over the range 0-100 “ C ,A b = 0.108 and CD= 3.8 X lO-3T in degrees Kelvin. As for the volume, the derivatives are simple:
(a In ~ @ ) / a p =) ,(A,/ln
~o)(D@)/D(’))/+ ( Bp, ) (8)
Thus,
(a In D@)/ap)T= k p
(1) where Vis the volume,p is the pressure, and Tis the temperature.
+ PI)
(3) where a is the coefficient of thermal expansion, (l/V)(dV/dT),
This is given by3-4.9
- ( a V + ) / a ~=) ~/3@’V’J” = ((C,/ln I O ) ~ ’ ” / ( B ,+ p ) )
(2)
(9)
with
k = ( A D /C,) (D@)v @ ) ) / D ( ’v’ ) ’)
0022-3654/93/2097-1220SO4.00/0 0 1993 American Chemical Societv
(10)
The Journal of Physical Chemistry. Vol. 97, No. 6,1993 1221
The Squeezing Pressure of Liquids
and
so that, at 1 unit pressure
k = (AD/CT)
(1 1)
III. The &led
Particle Theory The equations given here have been explained in more detail by Stillinger.1 The probability P(A) that a position in the liquid (chosen at random) lies outside of any exclusion sphere is given by
P(X) = exp(-WX)/kT) (12) where W(A)is the amount of reversible, isothermal work needed to create an empty spherical cavity of radius Xa in the liquid; a is the molecular contact distance (collision diameter) and varies in the range 0 IA < =, k is the Boltzmann constant, and T i s the absolute temperature. The density of molecule centers at the surface of the cavity is denoted by pG(A) where
= N/V
(13) N is the number of molecules in the volume V (e.g., Avogadro's number N A in the molal volume V,,,). For a given temperature and pressure, the work expended in creating the cavity (increasing the radius from 0 to Aa) is given by p
W(X)/kT= 4 1 r p a ~ t ( A ) ~ G ( AdA)
t 14)
TheprobabilityP(A)maybeexpressedin termsofthemolecular correlation functions g(n) for molecular centers in the liquid:
with SArepresentingthe spherical cavity of radius Aa. The series terms vanish for orders of n exceeding the maximum number of molecule centers that can fit into sphere SA. Ineq 15,witha thedistanceofclosestapproachoftwomolecules in the liquid, all terms beyond n = 1 vanish when 0 IA Il / 2 . Thus, since gC1) = 11,
P(A) = 1 - (4?rpa3/3)X3
(16a)
w(x)/~ = -In T (1 - (4?rpa3/3)h3)
( 16b)
and G(X) = (1 - (4?rpa3/3)X3)-' (1 6 4 When A becomes greater than 0.5 the n = 2 term in eq 15 contributes and G(A) becomes G(X) = ( p / p k T )
+ (2ym/pakT)/A - (4ym6/pa2kT)/A2+ ...
(17) where 7.. represents the surface tension in the planar boundary limit. As pointed out by Reiss and Schaaf? a boundary tension is distinct from an interfacial tension, which has one less thermodynamicdegree of freedom because of the coexistence of two phases. Thus, with the boundary tension approaching 7.. when Aa approaches infinityl*2 y = y m ( l- 26/Aa)
(18)
The quantity 6 provides the leading curvature dependence for y. Here p is the pressure corresponding to the experimental
conditions. Since the A > 1/2 approximation to G (first three terms in eq 17) must continuously and differentially connect to the exact expressions eq 16 at A = 1/2, then1 Ym
= (3ykT/*a2)(1/(l - Y )
+ 1.5Y/(l - Y ) 2 - P / P k T )
(19)
+
+
b (a/8)(1 3 ~ / ( 2 Y - 2s)) (20) where y = rpa3/6 and s = (1 - ~ ) ~ p / p k T . For water, when G(A) is plotted against A a maximum occurs when Aa 2 A, and A > 0.5. From eq 17, when dG(A)/dA = 0,
-
A = 46/a
and G(X)max = ~ = / ( 4 p k T b )+ P / P ~ T (22) The inward stress, or pressure, exerted by the water molecules at the surface of the cavity is given by1 ~sp= t pkTG(X)max
= ( ~ m / 4 6+ P)
(23)
from eq 22. Using eqs 19 and 20, p s p t = p k T ( l + 0 S y - ~ ) ~ / ( ( +l 2 y - s ) ( l - ~ ) ~ ) + (24) p
IV. Comparison of Aptand BT The quantity (BT + p), at any pressure p, has long been identified empirically as the difference between the thermal and attractive pressures of a liquid or is equal to the "network" stretch p r e s ~ u r e . ~This , ~ assumption is supported by the fact that the increaseof ( B T + ~with ) temperature, at constantvolume,closely follows the decrease of (dU/dv), the thermodynamic internal pressure, with temperature at constant volume, U being the thermodynamic total energy. This is shown by the fact that the attractive pressure, represented by ( T ( d p / d T ) ~ +BT),is thesame function of the volume for a range of temperatures and for pressures up to 1OOO bar.3 The thermal pressure coefficient (dp/ dT)v equals ((dU/dv)~+ p ) / T . The theoretical interpretations of B8v6 identify this quantity as a pressure of the type deduced empirically. Sincep,,, is an estimate of the network stretch pressure using a quite different approach, it is of interest to compare these two pressures. Another method of comparison would be to differentiate the SPT pressure equation of state with respect to volume at constant temperature, as Ginell did5 using associationtheory. However, since the pair correlation function is ill-defined for water in regard to effects of pressure, this is not feasible at present, or at least could not be expected to give accurate enough results. If a is taken as 2.82 A (the van der Waals and the collision diameter of the water molecule1I ) thevaluesofp,,, areverysimilar to the Tait pressure, BT, for the range 0-45 OC. When BT and pspt are equated,then a becoma a function of temperature,ranging from 2.627 A at 100 OC to 2.831 A at 23.8 OC and 2.801 A at 0 OC (Table I). With these values of a, A ranges from 0.698 at 100 OC to 0.748 at 24 OC for G(A)max, eqs 20 and 21. V. Mean Distance between Centers of Adjacent Molecuks in a Liquid The values of a in Table I are consistent with the equation a = (2kjU/X')1/2 (25) where k is the Boltzmann gas constant, Uis thevelocity of sound, and A' the thermal conductivity,and f is a structural parameter. Equation 25 was derived by BridgmanI2J3from considerations of the molecular energy transfers in the liquid. He took f as unity. As can be seen from Table 11, the value off here ranges from 1.12 at 0 OC to a maximum of 1.19 at 46.7 OC. In eq 25, differences in heat transfer for rotation and translation modes and the distribution of molecular energies have not been taken into account. As for the theoretical equations for gases, these can be covered by introducing a numerical factor, Le., f.
1222 The Journal of Physicul Chemistry, Vol. 97, No. 6,1993
TABLE I: SPT aod Tait Parameters for Liquid Water at its Saturated Vapor Pressure 0 4 10 20 25 30 40 50 60 70 80 90 100
1.OOO 160 1.OOOOOO 1.000300 1.001797 1.002961 1.004369 1.007840 1.012103 1.017081 1.022729 1.029017 1.035931 1.043470
3.3423 3.3428 3.3419 3.3369 3.3330 3.3283 3.3169 3.3029 3.2867 3.2686 3.2486 3.2269 3.2036
0.48455 0.63527 0.93868 1.7288 2.3060 3.0429 5.1405 8.3686 13.175 20.123 29.902 43.336 61.397
2672 2745 2840 2961 3004 3039 3080 3091 3075 3036 2978 2903 2814
2.801 2.810 2.821 2.829 2.829 2.827 2.818 2.803 2.781 2.753 2.719 2.677 2.627
a Experimental data of Gildseth, W.; Hasbcnschuss, A.; Spedding, F. H. J . Chem. Eng. Data 1972, 17, 402.bCalculated from eq 13. Experimental data for p are from ref 13,Table 242. Reference 4;ET = Eia,t', with a0 = 26713,2740.9;0 1 = 19.454,15.568;02 = -0.27028, -0.19437;104a3= 9.798,4.599,for 0 Q t Q 45 OC and 45 < t Q 100 OC, respectively. Calculated from eq 24 withpSpt= ET. Units: u in cm3g-I, p in lO22cm-3, ET in bar, and a in A.
TABLE Ik Effect of Pressure on a
_____
d
t(OC) 0 20 40 60 80 100
(A) at pressures of
1 bar 500bar 2.801 2.741 2.829 2.777 2.818 2.767 2.781 2.729 2.719 2.662 2.627 2.564
1OOObar 2.683 2.725 2.717 2.677 2.607 2.502
at pressures of 1 bar
1.122 1.170 1.185 1.181 1.156 1.102
500bar 1.042 1.099 1.117 1.112 1.088 1.028
1OOObar 0.964 1.030 1.052 1.047 1.015 0.950
Estimated from eq 24;experimental density from ref 4. From eq 25,experimental data for Ufrom Chen; Chen-Tung; Millero, F. J.Acousr. Soc. Am. 1976, 60, 1270. k' data from McLaughlin, E. Chem. Reu. 1964,61, 389 Table VI, and Lawson, A. W.; Lowell, R.; Jain, A. L. J . Chem. Phys. 1959,30,643.Values offfor 0-100 OC, at IOo intervals, are given (to +0.001) b y f = E,$ziti;00 = 1.1223, 103al = 2.835, 10s~ = -3.309 (1 bar). a
For example, the thermal conductivityof water vapor is related to the viscosity, I], via the equation13 where C, is the specific heat at constant volumeI4 and f , is the numerical factor. The value off, is 1.25 for water. In terms of the deviation from the ideal this is close to the fmax value for eq 25.
VI. The Isothermal Compressibility, B When pspt is substituted for BT in the Tait equation (eq l), using eq 23 for pSp,,
= V,(l/ln 10 G(X),.J/(cffZ?T) where cff = 1/CT = 3.175 for water. Equation 27 is similar in form to one derived by O'Reilly and Peterson,ls viz, where W is the work as defined above. However, unlike eq 27, eq 28 gives only qualitative agreement with the experimental compressibility. As can be seen from eq 16, when 0 I h I l/2, exp(-W/kT) = I/G(X) but this is not so for higher A. The value of c in eq 28 was taken as 3.25 for an fcc quasilattice. This is close to the value of c", the inverse of the Tait parameter CT. O'Reilly et aI.I5 arrived at eq 28 by considering the
Leyendekkers relationship of the compressibility with the mean square fluctuation of the number of particles, N, in a volume, V, of a unit cell of the lattice. The value 3.25 comes from the ratio of V/Nf for an fcc lattice, N'being the number of lattice sites, equal to 13, with thevolumeequal to4Vm. Thevalueof 3.175 wouldgive a volume of 4.O9Vm. If P i s taken as (In 1o/cT) or 7.310 then Vis about 5 V, for the same ".
VII. The Effect of Pressure on plpt Since BT is independent of pressure, it might be assumed that psptis independent of pressure also. It is of interest therefore to estimate the collision diameter (the mean distance between the centers of molecules at closest approach, i.e., a) on this basis and compare the values with those from eq 25. Values of u and f at 500 and IO00 bars were estimated on the assumption that pSpt= BT (Table 11). On this basis, an increase of pressure reduces the value of u which decreases as well with density at constant temperature, giving a family of straight lines (average deviation *0.001). The changes in u are associated with changesin the spatial and dynamic behavior of the molecules as the density increases, Le., changes in O*&*O angles, H-bond lengths, bonded and unbonded nearest neighbor distances and the relative concentrations of these different neighbor species, collision frequency and character, the collisional transfer of momentum and energy, etc. The value offbecomes closer to unity as the pressureincreases, but the temperature trends remain the same, which is consistent with the structural sensitivity of water found in other studies.49'6
Wr. Concluding Remarks The approach adopted here, of using the Tait equation as an external reference for the SPT equations, is new, and the results for water are interesting. Preliminary results for ethanol indicate that this equivalence of pressures holds for this liquid as well. As noted above, complex dynamic and static factors contribute to the averaged distance u so that its value could be expected to bea function of temperature and pressure. However, most studies using the SPT have taken u as a fixed value, which must surely limit the validity of the conclusions. The accepted value for the distance parameter of the Lennard-Jones potential for neon is 2.82 A, so that values of this order seem reasonable for water in view of the water molecule and neon atom being isoelectronic closed shell s y ~ t e m s . ~Published ~J~ estimates of u for the lower temperatures have a rangeof 2.501*-2.93A19. Gas-phasestudies for H20 give values of 2.65-2.71 A, which are consistent with the u values near the boiling point, at 1 bar. From 3&100 O C , the u values in Table I are very similar (0.05 A lower) to those estimated by Mayer19 from surface tension data, using the rigid-sphere fluid model. Of course, only a comparison with a wide range of molecular and other data will indicate whether the values of u are generally consistent with the theoretical interpretations of such data. The results here at least offer a new approach, and the temperature and pressure effects on u are consistent with those on many structural characteristics of ~ a t e r . ~ . ' ~ Obviously there are structural differences between liquid water and the hard-sphere liquid. Pohorille and Pratt (PP)20 showed that the scaled particle model will predict cavity works below the numerically exact results for liquid water. The differences are a function of cavity size and are negligible for sufficiently small cavities. More recently, PP2' have tested several SPT versions by comparing the predicted G(h)with computer simulation results. An interesting conclusion from these studies is that water applies more force per unit area of cavity surface than do the hydrocarbon liquids n-hexane and n-dodecane. This "squeezing" pressure reaches a maximum near a cavity radius of 2.4 A. Here, G(X) is a maximum for a cavity radius near 2.1 A. Nevertheless, pIDt
The Squeezing Pressure of Liquids and this squeezing pressure can be approximately equated, and the PP results are just what one would expect in view of the relative values of BTfor water and organic liquids. For example, at 25 O C , the values of BT for water, benzene, methanol, chloroethane,and acetone are, respectively, 3005,970,914,569, and 821 bar. All in all, the conclusion is that even though the hard-sphere liquid does not have the inbuilt structure of real water, the consequences of the individual structures of the two liquids are essentially the same in regard to the density and its change with temperature and pressure. In recent studies where a central force model was used22 (this model incorporates the local tetrahedral ordering characteristic of water) the role of molecular flexibility on thermodynamic and dynamic properties of water was shown to be small, which indicates that in some studies rigid models are not inappropriate for water, despite its complex structure. Leez3 also accords with this point of view.
References and Notes (1) Stillinger, F. H. The Physical Chemistry of AqueousSystems; Kay, R. L., Ed.;Plenum: New York, 1973; pp 43-60. (2) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. J . Chem. Phys. 1959, 31, 369. Reiss, H. Statistical Mechanics andStatistica1 Methods in Theory and Application; Landman, U., Ed.;Plenum: New York, 1977. Cotter, M. A.;
The Journal of Physical Chemistry, Vol. 97, No. 6, 1993 1223 Stillinger, F. H. J. Chem. Phys. 1972, 57, 3356. Reiss, H.; Schaaf, P. J . Chem. Phys. 1989, 91,2514. (3) Harned. H. S.;Owen, B. B. The Physical Chemistry of Electrolytic Solutions; Reinhold: New York, 1958. Tait, P. G. Phys. Chem. 1888,2, 1. Gibson, R. E.; Lotffler. 0. H. J . Am. Chem. Soc. 1939,61,2515. 14) Levendekkers. J. V. Thermodvnamicsofseawater as a Multicompone”; Ele&olyte Solution; Marcel tkkker: New York, 1976. (5) Ginell, R. J . Chem. Phys. 1961, 34, 1249. (6) Dayantis, J. J . Chim. Phys. 1972. 69, 94. (7) Leyendekkers. J. V. J . Chem.Soc., Faraday Trans. I 1981,77, 529; 1982,78,3383;J . Phys. Chem. 1986,90.5449;Aust. J . Chem. 1991,44, 195. ( 8 ) Pierrotti, R. A. J . Phys. Chem. 1965, 69, 281. (9) Leyendekkers, J. V.; Hunter, R. J. J . Phys. Chem. 1977,81, 657. (IO) Owen, B. B.; Brinkley, S. R., Jr. Phys. Rev. 1943,64, 32. ( 1 1) Rahman. A.: Stillinner. F. H. J . Chem. Phvs. 1972. 57. 128 (l2j Bridgman, P.W. P;oc.’Am. Acad. Arts. S h . 1923,’59.’141. (13) Dorsey, N. E. Properties of Ordinary Water-substance; ACS Monograph Series 81; Reinhold: New York, 1940. (14) Kell, G. S. In Water a Comprehensiue Treatise; Franks, F., Ed.; Plenum: New York, 1972; Vol. 1, Chapter 10. (15) OReilly, D. E.; Peterson, E. M. J . Chem. Phys. 1971, 55, 2155. OReilly, D. E. J. Chem. Phys. 1974, 60, 1607. (16) Leyendekkers, J. V. J. Phys. Chem. 1979.83.347. (17) Rahman, A,; Stillinger, F. H. J . Chem. Phys. 1971, 55, 3336. (18) van der Waals, J. H.; Platteu, G. Adu. Chem. Phys. 1959, 2, I . (19) Mayer, S. W. J . Phys. Chem. 1963,67, 2160. (20) Pohorille, A.; Pratt, L. R. J . Am. Chem. Soc. 1990, 112, 5066. (21) Pratt, L. R.; Pohorille, A. Proc. Natl. Acad. Sci. U S A . 1992, 89, 2995. (22) Smith, D. E.; Haymet, A. D. J. J. Chem. Phys. 1992, 96,8450. (23) Lee, B. Biopolymers 1991. 31, 993.
