I N T E R M O L E C U L A R F O R C E S IN W A T E R V A P O R J.
P. O ' C O N N E L L ' A N D J . M . P R A U S N I T Z
Department of Chemical Engineering, University of California, Berkeley, Calif. 94Y20 Vapor-phase thermodynamic and transport properties at low to moderate densities were analyzed to determine intermolecular forces and hydrogen bonding of pure water vapor. Hydrogen-bond energies up to about - 6 kcal. per mole for an open-chair dimer model are consistent with experimental second virial coefficient and heat capacity data. In addition, Stockmayer potential parameters fitted to the data indicate that the energy parameter should be less than 300" K., while the distance parameter is between 2.68 and 3.00 A. These values appear more consistent with the structure of water than previous ones which ignored hydrogen-bonding and induction forces. However, the pure component data are not sufficiently sensitive to the potential to be able to establish the parameters more closely. Analysis of the anomalous effect of pressure on the viscosity of water vapor indicates that further theoretical work may be needed to explain the experimental temperature dependence, although hydrogen bonding is apparently not inconsistent with the observed effect.
ALTHOUGHmuch effort has been expended in interpreting
and correlating the wealth of experimental thermodynamic and transport property data for steam, relatively little success has been achieved. Characterization of molecular interactions in water vapor is a particularly important subject, since an understanding of aqueous, vapor-phase properties is valuable for interpreting liquid-phase properties (particularly in biological systems), and solid-phase properties, both in ice and in hydrates. In addition, it is often necessary to predict the properties of aqueous vapor mixtures for practical applications. In this work, we discuss a model for dilute water vapor which includes dimerization of water molecules, and show how our model is consistent with experimental compressibility, heat capacity, and viscosity data a t low to moderate densities. We first postulate the structure of an open-chain, linear dimer of hydrogen-bonded water molecules. For various assumed energies of dimer formation, the potential functions are delineated which describe the dipolar, induction, and dispersion forces between the monomers consistent with experimental second virial coefficient and heat capacity data. The effect of chemical dimer formation is included along with the contributions of physically bound monomers (bound states) for interpreting the pressure dependence of the viscosity of water vapor. Dimer Model
There appear to be three models of chemically bonded dimers (Pimentel and McClellan, 1962; Van Thiel et al., 1957) (see Figure 1 ) : open chain, bifurcated, and cyclic. Based on the work of Schneider (1955), the open-chain dimer was chosen because it appears to be the most stable. While Van Thiel, Becker, and Pimentel (1957) argue persuasively that, for dimers trapped in inert gas matrices a t 20' K., the spectroscopically determined frequencies indicate a cyclic form, the additional entropy of an open chain due to molecular rotation and internal rotation about the hydrogen bond probably makes it the more stable species a t room temperature and above. This conclusion was also reached Present address, Department of Chemical Engineering, University of Florida, Gainesville, Fla. 32601.
Open Chain Dimer
Bifurcated Dimer
... ...
H-0,
Yi Cyclic Dimer
Figure 1.
Possible configurations of water dimer --Out
of plane of drawing
by Schneider (1955) on the basis of approximate electrostatic arguments alone. Further, although the permanent dipoles are in the most attractive position in the bifurcated structure, ab initio molecular orbital calculations (Morokuma and Pedersen, 1968) show that its bent hydrogen bond does not have the stability of the linear bond of the open-chain dimer. For an open-chain dimer, the length of the 0 . * e 0 distance can be varied over a fairly wide range consistent with most data. For instance, Schneider (1955) used the distance 2.76 A., while Viktorova (1964) used 2.85 A. Since the amount of covalency of the bond, and thus the hydrogen-bond energy, is extremely sensitive to this distance (Coulson and Danielsson, 1954), we have chosen to investigate the two different hydrogen-bond lengths occurring in ice (2.76 A.) and in methanol crystals (2.66 A . ) . While the length of the single bond in the dimer may be somewhat different from that in the doubly or quadruply hydrogen-bonded molecules in the crystals, since the electrostatic energy and moments VOL.
8
NO. 3 A U G U S T
1969
453
of inertia change relatively little between these two configurations, it is likely that the equilibrium constant of the dimer formation is not extremely sensitive to this distance. Table I shows the lengths and angles of the model. In structure A the hydrogen-bond distance is 2.66 A ; in structure B, 2.76 A. Free rotation of the two end groups about the hydrogen bond has been assumed, since the distance betn een potentially overlapping atoms is large and calculations of the rotational barrier show it t o be small or negligible (Xorokuma and Pedersen, 1968; Viktorova, 1964).
We have assumed that the molecules are rigid rotors with harmonic vibrations. These approximations are probably sufficiently accurate over a temperature range of a few hundred degrees. The expressions for the partition functions are then
V 2amkT
jtrans =
e
(7)(3) 3/2
where V is the volume, and e is the base of the natural logarithms. Also, for the dimer,
Equilibrium Constant
For the gaseous reaction 2h@ A 2
where
IA,
the equilibrium constant, K,, is expressed (11oelwyn-Hughes, 1957) as a function of the partition functions, j , for translation, rotation, and vibration, and the zero-point energy difference -
5=
P-
n
(ftransfrotfvib)Aq
12 T ( ftrans frot
PA2
IB,IC = principal moments of inertia of the molecule Zm,I’, = moments of inertia of internally rotating groups cr = symmetry number
x exP
fYib
n [1 - exp (-h~i/’kT)]-l
(51
1=1
fvib l2A
[
=
- (UA,” - 2b’~”
+AH)]
kT
(11
where C0 is the ground-state vibrational energy, Lro = c $ h v t , and A H is the configurational hydrogen-bond energy.
where v is the frequency of vibration. (Electronic partition functions have not been indicated, since they are assumed to be unity.) I t can also be shown that
I
A H , as wed in this work, is defined as the enthalpy of formation of the dimer from infinitely separated monomers a t zero of their potential energies. Since the zero-point vibration energy difference of about 2.8 kcal. per mole has been removed from the enthalpy of formation, A H is the same as that of Coulson (1957), which is the sum of the four contributions to the energy of dimer formation: electrostatic attraction, dispersion, repulsion, and delocalization. Coulson gives a theoretical value of -8.6 kcal. per mole for ice, but Eisenberg’s revised value of the electrostatic contribution makes it -6.8 kcal. per mole. The uncertainty in this number is large. AH differs from the standard-state enthalpy change, which is obtained by differentiating the equilibrium constant
a In K ,
A H o = R-
a I/T
The standard-state enthalpy, AH’, is a function of temperature and is 1 to 2 kcal. more positive than A H in the temperature range 273’ to 1173’ K. Structural Parameters for Open Chain Water Dimers
Table 1.
H
where GAO is the spectroscopically determined ideal-gas Gibbs function a t 1 atm and p~ is the number density. The equilibrium constant is then given by
(GO- U o ) A n A 2 hvj - j-12kT kT
nA
hvi
AH
c-+ x---] hT kT i=l
(7)
where for the open-chain water dimer C = 85.5 for structure A and C = 94.0 for structure B. The three vibrational frequencies of the monomer (%A = 3) are well established from spectroscopic measurements (Coulson, 1957). In the dimer, however, several more vibrational modes are established. Since there are 3N degrees of freedom (where N = 6, the number of atoms in the dimer) and six are used in molecular translation and rotation, and one more in internal rotation, there are 11 independent vibrations in the dimer (RA, = 11). These may reasonably be assigned to one stretching mode for each of the five bonds and six bending modes, including two librations along the hydrogen bond. Table I1 shows the various modes and estimates of the frequencies with which these modes have been assumed Table II. Vibrational Frequencies for Water Dimers S o . of Struc- StrucModes in ture ture Monomer Mode Dimer A B
Structure 0-0
distance, A.
11, A. 12, A. 1 3 , A. el, degrees e2, degrees 03, degrees
454
lhEC
A
B
2.66 1.64 1.02
2.76 1.78 0.98
0.98 105 108 108
0.98 105 105 105
FUNDAMENTALS
0-H stretch H-0-H bend 0-H.. stretch -H. - 0 stretch H-0-H. * 0
-
bend
H-0-H.
libration
*
*0
3 2 1 1 2
3650 1640 2700 230 1640
3700 1610 3200 190 1610
2
900
700
3700 (2 modes) 1600 (1 mode)
... ... ... ...
to vibrate. The 0-H stretch frequency for the bonds not along the hydrogen bond is largely unaffected by the dimerization, while the other 0-H stretch frequency is given approximately by the correlation of Pimentel and Sederholm (Pimentel and McClellan, 1962). The frequencies of the H-0-H bending modes increase slightly upon hydrogenbond formation (Pimentel and McClellan, 1962). The libration and hydrogen-bond stretching frequencies are taken from the spectra of ice (Fox and Martin, 1940; Gross, 1959; Zimmerman and Pimentel, 1962). Because of compensating effects, a 15y0 variation in these estimates causes only slight variations in the equilibrium constant. Anharmonic vibrations about the O..*O bond may be important a t high temperatures, but there is no simple way to account for them. The ideal-gas thermodynamic functions for water were taken from McRride and Gordon (1961). Contributions of Dimers to Physical Properties
Virial Coefficients. Schafer and Foz (1942) and Lambert et al. (Lambert, 1951; Lambert et al., 1947) have shown how the compressibility factor, z z=
PV nRT
is affected by dimer formation in the gas. If the pressureseries virial equation is used,
in the limit of zero pressure, it can be shown that the second virial coefficient is composed of two parts: B p h y s from the physical interactions of the monomers, and &hem from dimer formation. The experimental second virial coefficient obtained from PVT measurements is then &xp
= Bphls
&hem
=
Bphys
- RTKp
with a dipolar term included : u= a
forr
< 2a
f o r r > 2a
(11)
where f(e,~$)is a known function of the angles between the dipole vectors (Stockmayer, 1941). The dipole is assumed to be ideal and located in the center of the molecule. In addition, neglecting the effects of the core and assuming axial symmetry, the contributions to the second virial coefficient from inductive, dipole-quadrupole, quadrupole-quadrupole, and off-center dipole interactions are obtained from the expansions of Pople (1954) and Buckingham and Pople (1955), of Kielich (1961), and of Lawley and Smith (1963). Unfortunately, the magnitude of the third-order terms in these expansions indicates that higher-order terms may be significant for the range of parameters used. However, by ignoring the (positive) third-order quadrupole and off-center dipole terms, the contribution of unaccounted-for (negative) induction terms may be partially taken into account. We estimate that the calculated values are within f 1 0 cc. per gram mole of the values given by a more complete formulation. The molecular constants were adopted from Eisenberg (1964) and Coulson and Eisenberg (1966); the axially symmetric quadrupole moment is assumed to be 1.0 X esu from the calculations of Glaeser and Coulson (1965). While the uncertainty in this number is probably large, the contributions determined by the quadrupole moment are sufficiently small that relatively little error can be expected. Figure 2 summarizes the contributions of the dimer to the second virial coefficient for a few values of the hydrogenbond energies for structure A. The equivalent values for
(10)
Although B p h y s has usually been estimated from the Berthelot equation using critical properties (Foz and Vidal, 194i; Lambert, 1951; Lambert et al., 1947; Rowlinson, 1951; Schafer and Foz, 1942), it seems likely that dimerization would also affect the critical properties and lead to erroneous values of B p h g s . Furthermore, since there are strong attractive forces between monomers, previously assumed constant values (Barrow, 1952; Hirschfelder et al., 1942; Kreschmer and Pitzer, 1951; McCullough et al., 1951; Weltner and Pitzer, 1951) of B p h y s are even less realistic. Since expressions for B p h y s are available in terms of potential parameters (Buckingham and Pople. 1955; Cheh et al., 1966; O’Connell and Prausnitz, 1965; Pople, 1954; Stockmayer, 1941), we have attempted to fit second virial coefficient data for water with various assumed values of the hydrogen-bond energy and potential parameters. This is an approximation, since the orientation of the monomers corresponding to dimerization are included again in the integrations over phase space to obtain B p h y s . We believe that this contributes a negligible error to our model, bince we assume only a single orientation appropriate for the dimer. The data have been obtained from Goff and Gratch (1949) a t low temperatures and from a refitting of the data reported by Keyes (1949) a t high temperatures. The calculational procedure is essentially that given previously (Cheh et a l , 1966; O’Connell and Prausnitz, 1965). Our potential function is a spherical-core Kihara function
Temperature , O K
Figure 2. Contribution of dimerization to second virial coefficient of water for various hydrogen-bond energies VOL.
8
NO. 3 A U G U S T
1969
455
structure B are slightly smaller. The hydrogen-bond energy of this model appears to be limited to a maximum of -5.8 kcal., while the minimum energy, where significant contributions are made, is about -4.0 kcal. Potential Parameters from Second Virial Coefficients of Water
The ranges of potential parameters which fitted Bphys within experimental accuracy for various assumed values of A H are shown in Figure 3. Calculations are not available for large values of t* = p2/Z/sLro:oa3,and therefore the tables reported by O'Connell and Prausnitz (1965) have been extended to t* = 3.0 for this work. Using the Stockmayer potential, the second virial coefficient data were fitted to within experimental error for the temperature range 273' to 773' K. for all hydrogen-bond energies for structure A, although for values of -4.6 2 AH 2 -5.2 kcal. per gram mole, it appears that only the very low energy parameters (Uo/k < 200' K.) can be used. For structure B, for values of A H where R T K , was significantly different from that of structure A, the data could not be fitted by any parameters, indicating that the hydrogen-bond distance may be closer to that in the methanol crystal than that in ice. However, the insensitivity of the calculations t o this distance precludes any conclusions. For A H more positive than -4 kcal. and for A H between -5.2 and -5.7, the fitted energy parameter ranged from 150' to 270' K. When a finite core was used (2a > 0) in the polar Kihara potential model (O'Connell and Prausnitz, 1965), even for structure A there was a range of A H for which the data could not be fitted (-4.0 > A H 2 -5.4 kcal.). The variation of the fitted size parameter (a+ 2a) is a smooth function of hydrogen-bond energy and changes rapidly with A H more negative than -5.2 kcal. On the other hand, a hydrogen-bond energy of a t least -4.5 kcal. per gram mole is required to change the potential parameters significantly from those obtained by fitting the data without con-
sidering dimer formation. Thus, if dimerization is significant, establishing the distance parameter provides considerable information about the structure and energy of the dimer. The Stockmayer potential parameters obtained here are different from those obtained by earlier workers (Monchick and Mason, 1961; Rowlinson, 1949, 1951) partly because the effects of induction, which are significant for water, were not separated in earlier analyses. In particular, the effect of induction on the reduced dipole moment has not been taken into account previously. As Hirschfelder et aZ. (1954) have shown, the spherically symmetric portion of the (classical) induction effect can be included in the Stockmayer dispersion parameters for a pure polar gas if the values are modified by
uo'= uo
(1 + - 2zJ
For water, ap2/2Uoa6 is about 0.1 to 0.2, causing a significant change in the energy parameter and in the reduced dipole moment in either case. However, even when the previous parameters are modified according to Equation 12 to get the parameters on the same basis, there are considerable differences between the parameters obtained here and those reported previously (Table 111). Since we have shown that the parameters are in no way unique, these differences are not surprising. In any case, using the techniques described here, indications are that the energy parameter, Uo/k, for water is less than 300' K., while the center-to-center distance is between 2.68 and 3.00 A. Considering the structure and polarizability of water, these parameters appear reasonable, but on the basis of second virial coefficient data alone, it is impossible to give more exact values.
I --- Ca ICb lated
B,&
Not Available
Heat Capacity
li
aa b'
W e l l Depth, U,/k,OK
Figure 3. Effect of hydrogen-bond energy on potential parameters obtained from second virial coefficients of water 456
ILEC
FUNDAMENTALS
The effect of pressure, P, on the heat capacity a t constant pressure, C,, may be expressed in the form (Hirschfelder et aZ., 1954)
where B is the second virial coefficient and C,O is the ideal-gas value. This may also provide an extremely sensitive test for expressions for the second virial coefficient. The major sources of experimental heat capacity data are from direct measurements (McCullough et al., 1952; Knoblauch and Jakob, 1907; Knoblauch and Mollier, 1911; Knoblauch and Winkhaus, 1915) and from differentiation of saturated enthalpy data (Osborne et al., 1939). We have used the latter together with results of Goff and Gratch (1946, 1949) to calculate C, - C,O along the saturation line. Figure 4 shows a plot of all of the data reduced to (C, - C,O)/PT which, a t the limit of zero pressure, is the second derivative with respect to temperature of the second virial coefficient (Equation 13). Also shown is the sum of the analytic second derivative of RTK, and of the numerical second derivative of Bphys. Since the second term in Equation 13 is very small, this line represents essentially all of the contribution to C, - CpO due to the second virial coefficient. The
Table 111.
Stockmayer Potential Parameters for Water
ua/k, K.
Data Source Virial coefficients (Rowlinson, 1951) Viscosity (Monchick and Mason, 1961) Virial coefficients (this work) AHDimerisation = O A H ~ i ~ ~ ~= i -5.6 ~ ~ tkcal./mole i ~ .
U’o/kl.
K.
0,
A.
u’, A.a
f*
p ia
305 430
380 506
2.70 2.74
2.65 2.71
1.45 0.94
1.2 0.85
150 250 150 250
270 365
2.69 2.69 2.88 2.95
2.56 2.60 2.81 2.91
3.5 1.6 1.25 0.59
2.3 1.2
200 290
1.0
0.52
U‘O,u’, and t * ’ include flrst-order (classical) spherically-symmetric Induction effect (see Equation 12).
calculated values are consistent with the data, but it appears that contributions from the third virial coefficient are also important (Barrow, 1952; McCullough et al., 1952; Weltner and Pitzer, 1951). low-Pressure Transport Properties
Considerable viscosity (Kestin and Wang, 1961; Moszynski, 1962; Rivkin and Levin, 1966; Shifrin. 1959; Smith, 1924) and thermal conductivity (Baker and Brokaw, 1964; Bruges, 1963; Foz et al., 1948; Vargaftik and Zaitseva, 1963) data are available for water vapor. Using approximate kinetic theory expressions, such as those of Monchick and Mason (1961), potential parameters may be obtained by analysis of these data. Unfortunately, the results may not be as significant as those from the virial coefficient analysis because of the approximations of the theory and the limitations of the tables of collision integrals. Many of the sets of parameters obtained above correspond to reduced dipoles beyond the range of Monchick and Mason’s tables (1961).
0.6
--
From Correlotion of Goff 8 Gratch From Data of McCullough, et al. From Data of Knoblauch P = 2 atm. + P - 8 otm.
0.1 N
0.06-
Y
t
m
E
$ 0
Oa03-
0
u
0.006 0.01
-
0.003
270
350
430
510
Temperature,
590
670
OK
Figure 4. C a l c u l a t e d and o b s e r v e d heat c a p a c i t y of water vapor
In addition, the effects of the quadrupole moment and the rigid core cannot be taken into account, since the required calculations have not been performed. As a result, we have only tried to test sets of parameters for rough consistency with the data. For the coefficient of viscosity, the calculated values are low by a few per cent for those reduced dipole moments close to and within the range of the collision integral tabulations, Decreasing the collision diameter for a given energy parameter, in an attempt to increase the calculated values, is often more than compensated by the increased collision integral of the correspondingly higher reduced dipole moment. However, the trends indicate that low values of Uo/k (less than 200’ K.), where t* is beyond the range of the table, may yield good agreement with experiment. The thermal conductivity analysis showed similar results. Effect of Pressure oh Transport Properties
Low-temperature experimental data for the pressure dependence of the viscosity of steam (Kestin and Wang, 1961; Moszynski, 1962; Rivkin and Levin, 1966) show anomalous behavior: A t a constant temperature the viscosity decreases with increasing pressure. Barua and das Gupta (1963), using a free-volume model, indicated that the qualitative trends could be explained by consideration of the formation of bound pairs. Their concept of bound pairs is somewhat different from the present hydrogen-bonded dimers, and a distinction should be made between these two types of molecular associations which are important for transport properties. From the dynamics of collisions between molecules with attractive and repulsive forces, three distinct regions of potential energy for a pair of molecules result (Hirschfelder et al., 1954). The effective potential energy for a typical low-energy collision is shown in Figure 5. If the energy of interaction is greater than the maximum, the molecules separate after collision; the molecules are considered free in this case. However, a bound pair results if a nonadiabatic collision occurs and the resulting translational energy is lass than the maximum, while the distance between molecules is smaller than that at the maximum. When the resulting potential energy is positive, the pair is “metastably bound.” Each region contributes to the total second virial coefficient (Stogryn and Hirschfelder, 1959).
Physically bound pairs occur because of dispersion and attractive polar forces which are assumed to be independent, or a continuous function, of orientation. On the other hand, the interaction of monomers in the single hydrogen-bonded dimer orientation must be described by a potential function, which takes into account the delocalization energy of the VOL
8
NO.
3 AUGUST
1969
457
to the interaction between a monomer and a dimer. The first term on the right-hand side may be estimated (Stogryn and Hirschfelder, 1959) from a modification of Enskog’s dense-gas theory :
1 ~
Free
Fraction of Bound Pairs
~~
Distance Between Molecular Centers Figure 5.
Regions of potential energy for molecular pairs
hydrogen bond (Coulson and Danielsson, 1954). This is consistent with our earlier approximation for the second virial coefficients and leads to the two basically different associations which we do not consider to be interconvertible. The mole fraction of the bound pairs has been shown by Stogryn and Hirschfelder (1959) to be related to the bound second virial coefficients x2R5 -
Bbound
+ Bmetsstable RT
(15)
Since the lifetime of metastable pairs is comparable to that of several consecutive collisions, these pairs can be treated as stable species. Contributions to the second virial coefficients of the three possible pairs (free, metastably bound, and bound) are available for a number of potential models, including the Lennard-Jones (12-6) and the Stockmayer potentials (Barua et al., 1965; Saran et al., 1967; Singh et al., 1967). In the latter case, the contributions were averaged over all angles of orientation. Stogryn and Hirschfelder (1959) present equations for the initial pressure dependence of the viscosity for a substance obeying the Lennard-Jones (or Kihara) potential. [Although we consider only viscosity, it is also possible to treat thermal conductivity in much the same manner. On the other hand, the uncertainties in the prediction of this property due to internal degrees of freedom (Baker and Brokaw, 1964; Mason and Illonchick, 1962) lead us to believe that a detailed analysis of the data would not yield meaningful results.]
where q c is the “collisional viscosity,” P is the pressure, is the mole fraction of the dimer, and subscript 12 refers
x2
458
l&EC
FUNDAMENTALS
We tried several methods to determine a set of monomer parameters which could be used to calculate the concentration of bound pairs in water vapor and to determine the pressure dependence of viscosity from Equations 17 and 18. It was impossible to fit adequately the zero pressure viscosity data for water vapor using either the nonpolar Lennard-Jones collision integrals (Hirschfelder et d.,1954; hlonchick and Xason, 1961; O’Connell and Prausnitz, 1965), or the formulas for temperature-dependent parameters used by Bae and Reed (1967) with parameters inside the range of the tables for bound second virial coefficients (Barua et al., 1965; Singh et al., 1967; Stogryn and Hirschfelder, 1959). I t was possible to fit the viscosity data well with parameters from Krieger’s polential (Itean et al., 1961; Krieger, 1951) and the Stockmayer potential, provided the dipole moment was an additional floating parameter, but the calculated bound-pair second virial coefficients obtained from these parameters and the tables of Barua et al. (1965) and Singh et al. (1967) were considerably more negative than the experimental ones. an unrealistic result at the reduced temperatures of interest. In view of the inapplicability of the available tables for bound pairs, another method was tried. The second virial coefficient, Bphys, is in part an equilibrium constant for bound pairs, and the numerical values of Stogryn and Hirschfelder (1959) for the Lennard-Jones potential indicate that the contribution from free pairs is relatively important only a t temperatures above the Boyle point. Further, the reduced second virial coefficient for free pairs of polar gases should not be markedly different from that for nonpolar gases; the major difference is in the boundpair contribution. Thus, it seems reasonable to assume that most, if not nearly all, of B p h y s is due to bound pairs. In the temperature range of interest, this is likely to make only a small error in the dimer concentration, and some semiquantitative results, limited in the manner discussed above, might be obtained through the use of Equation 17 and the collision integrals of Monchick and R‘Iason (1961) for the Stockmayer potential. Since the resultant dipoles of the dimer and the bound pairs are probably different, there are two terms like the last one on the right-hand side of Equation 17. However the values within the parentheses are relatively uncertain, since the dipole of the bound pair is open to question. Barua et al. (1965) assert that the Krieger model is realistic in this case because the most favorable alignment of dipoles is assumed. However, it is likely that oscillations about this alignment due to the kinetic energy of the individual molecules will reduce the effective dipole moment. Another uncertainty in the calculation is the ratio of size parameters for monomer-bound pair interactions and for monomer-monomer interactions, ulz/a11. Although Stogryn and Hirschfelder found that al~/crll= 1.04 was required for nonpolar substances, using the collision integrals for the largest reduced dipole from Monchick and Mason (1961), this value yielded a positive pressure effect, not the observed
negative effect. (The pressure effect due to the hydrogenbonded dimers is positive.) Higher values of the reduced dipole moment, as indicated by the analysis of second virial coefficient data, would lower the size ratio required to fit the data. Comparisons with Experimental Viscosity Data
Figure 6 shows the pressure dependence of water vapor viscosity with the zero-pressure viscosities, q (0), obtained by the best straight-line fit of the experimental data. The ordinate { [q (0) - 17 ( P ) ] / q(0) ) (l/P) has been chosen, since the initial pressure dependence of the viscosity is linear and the coefficient is a function only of temperature. The parameters of Monchick and Mason’s “free fit” (1961), elk = 260’ K., u = 2.825 A., t* = 1.78, were used along with the parameter ratios e12/ell= 1.32, and u12/u11 = 1.16, and with the bound-pair reduced dipole moment, t* = 1.78, to predict the viscosity of water vapor as a function of temperature and pressure. The solid line on Figure 6 shows the results. If the hydrogen-bond energy of dimer formation is assumed to be -5.6 kcal. per gram mole, then from the reduced dipole for the dimer-monomer interaction, t* = 1.04, and the parameter ratios given above, the dashed line results. A higher value of the hydrogen-bond energy yields an even smaller negative variation of viscosity with pressure. Using other sets of parameters more consistent with the second virial coefficient calculations (up to the maximum tabulated value of t* = 1.78) yielded similar results. The discrepancies in Figure 6 suggest several possibilities for further work, since the temperature dependence of the function { [q (0) - q (P)]/17(0)) (l/P) is much stronger than predicted by either model. In particular, extended collision integral tables are again 0.004 0
-
,(O)
‘ e E
1
0
Dimerization Ignored Dimerization Included Data of Kestin 8 Moszynski Data of Rivkin 8 Levin From Best Fit of Each lsothei Experimental Variation
suggested, because if a larger reduced dipole moment for the monomer-bound pair interaction were used with the same size ratio, the pressure effect would be more negative. Then, for the expected values of the dipole moment, the collision integral values would be consistent with a parameter ratio u12/(~11 somewhat lower than 1.16, as found by Stogryn and Hirschfelder, and the results would also be consistent with a larger hydrogen-bond energy, as the thermodynamic data indicate. U‘hile the temperature effect would not be in agreement with the data even if the parameters were adjusted, perhaps the Enskog approximation for the collisional term, qc, is not valid for highly polar molecules a t low temperatures. Another possible explanation is that the dipole moment of the bound pairs decreases markedly with temperature because of oscillation about the head-to-tail position of the molecular dipoles. In any case, our model is again not inconsistent with the data on the pressure effect on viscosity, but further calculations and better experimental results are necessary to be more certain about its significance. Conclusions
Upon postulating chemical dimer formation in water vapor, we have shown that an open-chain model for the dimer is consistent with the experimental data on second virial coefficients, and heat capacities and limits can be placed on significant parameters of the model. The configurational hydrogen-bond energy which is obtained must be more positive than - 6 kcal. per gram mole, but is probably more negative than -4 kcal. per gram mole. The potential parameters obtained from fitting second virial coefficient data indicate that the energy parameter is less than 300’ K., a value lower than previously found, and the size parameter is between 2.68 and 3.0 A. These values appear more consistent with the structure of water than earlier values. An analysis of the zero-pressure and low-to-moderate pressure dependence of viscosity indicates that present calculations are inadequate for the interactions of water monomers, but that the potential parameters and assumed dimerization are probably consistent with the pure-component experimental data. To establish more closely the parameters describing the important dispersion forces in water, it will be necessary to consider aqueous mixture data as discussed in the following paper. Acknowledgment
The authors are grateful to John Dahler for helpful discussions, to the Office of Saline Water, G. S. Department of the Interior, for financial support and to the Computer Center, University of California, Berkeley, for the use of its facilities. literature Cited
Bae, J. H., Reed, T. M.,IND.ENG.CHEM.FUNDAMEKTALS 6, 67 (1967).
Baker, C. E., Brokaw, R. S., J . Chem. Phys. 40, 1523 (1964). Barrow, G. M., J . Chem. Phys. 20, 1739 (1952). Barua, A. K., Chakraborti, P., Saran, A,, Mol. Phys. 9, 8 (1965) Barua, A. F., das Gupta, A,, Trans. Faraday Soc. 69,2243 (1963). Bruges, E. A., Univ. Glasgow Mech. Eng. Dept., Tech. Rept. 9 (B) (1963).
0 400
Buckingham, A. D., Pople, J. A,, Trans. Faraday SOC. 61,
1 450
500
550
Temperature, OK F i g u r e 6. E x p e r i m e n t a l and c a l c u l a t e d p e n d e n c e of viscosity of water vapor
pressure
de-
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RECEIVED for review January 29, 1968 A C C E P T E D December 5, 1968
INTERMOLECULAR FORCES IN AQUEOUS VAPOR M I X T U R E S M . R I G B Y , ’ J . P. O ’ C O N N E L L , 2 A N D J . M . P R A U S N I T Z Department of Chemical Engineering, University of California, Berkeley, Calif.94720 Experimental data for second virial cross coefficients and for vapor-phase diffusivities of binary aqueous mixtures were reduced with a potential function of the Kihara type. The second component in the aqueous mixtures was argon, nitrogen, methane, or oxygen. A good fit of all the data can be obtained using the following parameters for woter: &/k = 170’ K., u (core-to-core distance) = 2.65 A., and 20 (core diameter) = 0.265 A. These parameters are consistent with those obtained from a study of vapor-phase properties of pure water.
OLECULAR interpretation of equilibrium and transport properties of water vapor at low and moderate pressures requires a reasonable potential function for water-water interactions. The large dipole moment of water and the tendency of water to dimerize by hydrogen bonding complicate the analysis of experimental data. I n an earlier paper (O’Connell and Prausnitz, 1969) an attempt was made to interpret the available data using a Stockmayer-Kihara potential,
r > 2a =
QD,
r
(1)
7 2a and all e, cp
Present address, Queen Elizabeth College, London, England. Present address, University of Florida, Gainesville, Fla. 460
l&EC
FUNDAMENTALS
+
where u 2a is the center-to-center separation for which the angle-independent potential is zero, U,is the maximum energy of attraction due to nonpolar forces, and 2a is the diameter of the spherical, hard core (a* = 2 a / u ) . The final term represents the dipole-dipole interaction, where f(0, c p ) is a known function of the angles between the dipole vectors. For water dimer, an open-rhain linear model was assumed (O’Connell and Prausnitz, 1969) and it was possible to determine the range of possible parameters for the Kihara potential which was compatible with various values for the heat of formation of the dimer. It was concluded that the value of U,/k is less than 300°K., and that the distance parameter, u 2a, is between 2.68 and 3.00 A. I n a n attempt to determine these values more precisely, recently obtained experimental data leading to information about the interactions of water with simple nonpolar molecules were studied. Interpretation of these interactions is
+
