J. Chem. Soc., Faraday Trans. 1, 1989, 85(6), 1315-1326 A Cubic Equation of State for Mixtures Containing Steam Christopher J. Wormald* and Neil M. Lancaster Department of Physical Chemistry, University of Bristol, Bristol BS8 ITS A cubic equation of state and combining rules for mixtures containing steam has been developed. The equation is based on the earlier equations of Clausius, Martin and Kubic, and contains two temperature-dependent parameters, which give it enough flexibility to fit the residual properties of both steam and non-polar fluids with sufficient accuracy. Mixture properties are calculated using pseudo-critical parameters for steam and a single temperature-independent interaction parameter { in the combining rule Tl2 = c(ql q,):. For mixtures such as steamxarbon dioxide, where there is a specific interaction between the unlike components, { must be obtained from cross-term second virial coefficients.Where there is no specific interaction is given by < = 2(K1 V,,): Kl2-l(I1 12)~(Il+12)-1. The equation of state fits measurements of the excess molar enthalpy H: of mixtures containing steam at pressures of 0.1-10 MPa, at temperatures up to at least 700 K, and allows the calculation of ( p , V,, T ) properties. Attempts to measure (p, Vm, T ) properties of mixtures containing steam using a compression apparatus have generally yielded results of low accuracy. Large adsorption errors spoil the measurements, particularly at low densities. An alternative to ( p , Vm, T ) is direct measurement of the excess molar enthalpy HZ using a flow mixing calorimeter.The calorimeter is sensitive only to changes in the enthalpy of the fluid passing through it; adsorbed molecules become part of the apparatus and have no effect on the measured enthalpy change. A low pressure differential flow-mixing calorimeter for HE measurements on dilute gases has been described.' The calorimeter has been used to make measurements on binary mixtures of steam-hydrogen,2 -nitrogeq3 -argon,4 -methane,2-C2-C8 n-alkane~,~~' -ethene,5 -propene,6 -benzene,' -cyclohexane,* -carbon mon~xide,~, -carbon d i ~ x i d e , ~ -chloromethane, lo -chloroethane, lo and -trichloro- methane. l1 Analysis of the measurements yielded cross-term second virial coefficients B12. The only other B,, values for mixtures of water-gaseous substance have been obtained from measurements of the solubility of water in compressed gas.l2?l3 For the few mixtures on which measurements have been made, agreement with B12s from the Hz measurements is within the combined experimental error. As the enthalpy of mixing work progressed and experience was gained, better analysis of the HE measurements became possible.As this paper draws heavily on conclusions reached during the course of the low-pressure measurements it is useful to summarise the relevant points before proceeding. For low density gaseous mixtures it has been shown1 that where #" = B - T(dB/dT), A$") is a function of B and $" for the mixture, and fly) is a function of third virial coefficient terms. If the pair potentials of components 1 and 2 are known B,, and &'2 can be obtained by adjusting the parameter in the combining rule E l 2 = &11&22>~ (2) 13151316 Equation of State for Steam Mixtures until the right-hand side of (1) agrees with the experimental HE.When the Stockmayer parameters ElkB = 233 K, CT = 0.312 nm and t* = 1.238 are used for water it was found’’ that 5 was given by where Z is the ionisation energy. Eqn (3) fits all the HZ measurements except those for steam-carbon dioxide, -ethene, -propene and -benzene. For these mixtures there are specific interactions between the water lone-pair electrons and unfilled orbitals of the other component. Eqn (3) has been shown to give a better fit to cross-term B12s for mixtures of n-alkanes than six other similar formulae.“ 3 l5 Experimental and calculated values of 5 for mixtures containing steam are listed in ref. (10). Where there are specific interactions the experimental values of 5 are bigger than those calculated from eqn (3). An alternative to pair potentials is to use a corresponding states correlation to obtain second virial coefficients and their temperature derivatives. For mixtures of n-alkanes it has been shown14* l5 that the experimental Hg values can be fitted using the correlation of McGlashan and Potter16 together with the combining rule where (5) As pair-potential parameters fitted to second virial coefficients vary according to the quality of the data, values of calculated from eqn (5) are slightly different from those calculated from eqn (3). For the 20 substances for which steam mixture measurements have been made, the average value of 5 calculated from eqn (3) is only 1.5 O/O greater than that calculated from eqn (5).For mixtures of water-Cl-C, n-alkanes the experimental HE values can be fitted using eqn (4) and (5) together with the McGlashan-Potter correlation and the pseudo-critical temperature T, = 230 K for water. V , for water was set equal to the true critical volume 55.5 cm3 mol-’ and the effective chain length was N = 1 . For other choices of pseudo- critical temperature, and N fit the Hgs equally well, our choice of 230 K for T, is not unique. In the analysis of the HZ measurements on water-argon it was noted4 that the HE values could be fitted to within experimental error using fluoromethane as a homomorph for water.The second virial coefficient of fluoromethane is close to that calculated from the McGlashan-Potter correlation using = 230 K for water, and that calculated using the Stockmayer potential with the parameter ElkB = 233 K, CT = 0.312 nm and t* = 1.238 for water. Fluoromethane has the same dipole moment 1.85 Dt as water, although a smaller reduced dipole moment t* = 1.044 and stronger dispersion forces. By using the fluoromethane parameters (T, = 318 K, pc = 5.87 MPa, = 124 cm3 mol-l, w = 0.19) as pseudo-critical parameters for water in the third virial coefficient correlation of Orbey and Vera17 it was shown’* that high-pressure HZ values for water-n-pentane could be fitted to within experimental error. The function fly) in eqn (1) was calculated in the same way.The use of fluoromethane as a homomorph for water was also a key step in the development of our first modell’ to fit the high-pressure enthalpy of mixing of steam-hydrocarbons. Enthalpies of mixing of several binary mixtures containing steam at high pressure have been measured using a flow-mixing calorimeter enclosed in a pressure vessel. Most of the measurements extend from 448.2 to 698.2 K at pressures up to ca. 12 MPa. The mixtures studied include steam-hydrogen,20 -nitrogen,21 -methane,22 -ethene,23 -pro- pane,24 -butane,24 -n-pentane,’, -n-he~ane,~~ -n-he~tane,~~ -carbon monoxide,26 xarbon dioxide.26 Small corrections to some of the measurements listed in ref. (20)-(24) t 1 D = 3.33564 x C m.C. J. Wormald and N . M. Lancaster 1317 and (26) have been published." Most of the measurements were made at x = 0.5, though some measurements of the composition dependence of HE were made under selected conditions for each mixture.At pressures up to ca. 12 MPa the HE values for mixtures containing n-alkanes up to n-pentane can be fitted using the virial equation of state." For steam-n-hexaneZ5 the virial equation fits the HE values only up to ca. 5 MPa and for ~team-n-heptane'~ it fits only up to ca. 2.5 MPa. Cubic equations of state, such as that of Peng and Robinson, are worse than the virial equation, failing to fit HZ measurements at both low and high den~ities.'~ It is clear that for steam mixtures another approach to the problem is needed. Previous Equations for Mixtures Containing Water Attempts to construct cubic equations of stzLc to fit the properties of mixtures containing water are aimed at fitting either liquid/vapour equilibria or at fitting vapour phase properties only.Robinson et a1." applied the Peng-Robinson equation to phase equilibria of binary mixtures containing water and methanol. When their equation is used to calculate HE values for steam-hydrocarbon mixtures poor agreement with experiment is found. Baumgartner et al. proposed an association model" based on the cubic equation of Schmidt and W e n ~ e l . ~ ~ It requires the simultaneous solution of equations for the chemical equilibria of water oligomers containing up to 14 molecules, and is almost intractable for mixture calculations. Ghemling et aL3' extended the perturbed hard-chain equation of Donohue and Prausnitz3' to include dimerisation equilibria for polar molecules.Their equation is used for industrial calculations of liquid/vapour equilibria at high pressures. The equation is not cubic and calculations are complicated. In common with the other equations described above, the adjustable parameter kij must be fitted to experimental data. Cubic equations of state constructed specifically for water vapour have been proposed by de Santis et al.,33 Nakamura et al.,34 Oellrich et Nishida et al.,36 and Wormald." The perturbed hard-sphere equations of Nakamura et al., Oellrich et al., and Nishida et al. give a good representation of the ( p , Vm, T ) properties of steam at high temperature and pressure by using the Carnahan-Starling expression for the hard-sphere compressibility factor and making the covolume b temperature dependent.The first two of this set of equations of state give a poor fit to B(T) for steam; the latter gives B(T) accurately but requires three temperature-dependent parameters. Only a few fluids have been correlated with the equations and they have not been used for calculating mixture properties. The equation of de Santis et al. gives a reasonable fit to B(T) and to steam densities at pressures up to 220 MPa, but the form of a ( T ) is non-analytic. Enthalpy calculations require a( T ) to be a continuous function. Their equation can be used for the calculation of mixture volumes and has the advantage that, given the parameters for steam, no further information is needed to determine the mixture volume, except when there is a specific interaction.Our previous equation of statel' for steam mixture properties was a modification of the Peng-Robinson equation in which the a(T) term was separated into a part due to London and Keesom forces and a part due to hydrogen bonding. This gave a reasonable fit to the residual enthalpy of steam at moderate densities, and provided a way of calculating cross-terms similar to the association treatment of Woolley3' and Lambe~t.~' To improve the fit to both low-pressure and high-pressure HE measurements a temperature-dependent interaction parameter calculated from the second virial coefficient of the mixture was used. To fit the properties of mixtures in which there are specific interactions an additional parameter was needed.The equation fits the H:s of water-C,-C, n-alkanes up to 10 MPa quite well, but fails at higher pressures. We have now developed an improved equation of state for the vapour phase properties of steam mixtures which largely overcomes these limitations.1318 Equation of State for Steam Mixtures A Cubic Equation for Mixtures Containing Steam Any equation of state designed to fit the properties of steam mixtures must first fit the residual properties of the pure components adequately. As the volumetric properties of steam are very different from those of say n-hexane, much is demanded of an equation if it is to fit the properties of both fluids. The Soave-Redlich-Kwong, Peng-Robinson and Patel-Teja equations of state were developed primarily for liquid/vapour equilibrium calculations, and do not give as good a representation of volumetric properties of the vapour as does the Martin3’ equation which was developed specifically for this purpose.Kubic4’ has modified the Martin equation for the calculation of liquid/vapour equilibria. After much experimentation with recent cubic equations we have also taken the Martin equation as a starting point, and have used some of Kubic’s modifications in the development of our equation for steam mixture properties. Martin made a detailed analysis of the ability of two-term cubic equations to predict densities of liquids and gases. He concluded that the Clausius41 form was both the best and the simplest, and modified it by making the parameter a temperature dependent: RT a(T) P=(V-b)-(V+c)2. The virial expansion of eqn (6) is 2ca(T) 1 3c2a(T) 1 Z = l + ( b - - s)) ++ ( b2 +-&) v2+ ( b3 -T) v3.(7) Kubic’s modification was to make c temperature dependent and to equate the second virial coefficient [b - a( T)/ RT] to that given by the Tsonopoulos c~rrelation,~~ which gives adequate values of B(T) for non-polar and slightly polar fluids and so makes the parameter a temperature dependent. For mixtures, the term [b - a( T)/ RT] is equated to the second virial coefficient of the mixture where cross-terms are calculated from combining rules for critical parameters. Kubic’s formulae for the coefficients of eqn (6) are RT, b = - (0.082 - 0.07 1 3 ~ ’ ) Pc (9) The coefficients a’, al, yo and yl are polynomials in powers of the reduced temperature a’= -0.1514T,+0.7895+0.3314~1+0.029~2+0.0015~7 (1 1) ul= -0.237T,-0.7846c1 + 1.0026c2 +0.019c7 (12) yo = 4.275 - 8.879c’ + 8.509c2 - 3.481 c3 + 0.576c4 (1 3) yl= 12.86-34.74c1+ 37.43c2- 18.06r3+3.51C4.(14) (15) T* The parameter m’ is related to Pitzer’s acentric factor o through W’ = 0.000756 + 0.90980 +O. 16230~ +0.1455w3. The residual molar enthalpy H: is given by [a- T(da/dT)] aT(dc/dT) H: = pV- RT- - (V+C) (V+ c)2 -C. J. Wormald and N. M. Lancaster 1319 2 * E X -1 21 1 I + € X - 0 4 8 12 16 20 0 4 8 12 plMPa p l W a Fig. 1. (a) The fit to the residual molar enthalpy HE of steam obtained using eqn (18) and (16). (-), HZ from the 1984 HGK43 Steam Tables. (----) calculated from eqn (1 8) and ( I 6). (b) The fit to the residual molar enthalpy HZ of n-hexane obtained using eqn (10) and (16).0, experimental measurements of Wonnald and Yerlett.45 (-) HZ from eqn (10) and (16). (----) calculated from the BWRS equations of state. (- .---) calculated from the Patel-Teja46 equation of state. Making the a and c terms of eqn (6) temperature dependent gives the equation considerable flexibility. To fit the properties of steam we calculated a( T ) from the second virial coefficient B(T) of steam a(T) = RT[b-B(T)]. (17) B(T) was calculated from a polynomial in powers of T1 fitted to second virial coefficients calculated by Gallagher'' which are consistent with the Haar-Gallagher-Kell (HGK) equation of These virial coefficients agree very closely with those given by the correlation of L e F e ~ r e . ~ ~ The temperature dependence of c(T) was chosen to make saturated vapour densities calculated from eqn (6) agree with those obtained from the HGK equation for steam from the normal boiling temperature to the critical temperature : With these modifications, eqn (6) fits the molar volume of steam with good accuracy up to 20 MPa.It is not possible to fit the molar volume and the saturated vapour pressure of steam simultaneously. When c( T ) was fitted using saturated vapour pressures, poor values of Vm and Wm were obtained. The fit to the residual enthalpy HE of steam obtained using (1 7) and (18) is shown in fig. 1 (a). The fit to H z for n - h e ~ a n e ~ ~ given by Kubic's formulae is shown in fig. 1 (b). At low densities Kubic's equations are superior to the Patel-TeJa46 equation and the multi-parameter BWRS equation. c(T) = 1220.7- 3656.5T1 +4043.8T2- 1847.5T3+252.3T4.(18) Combining Rules for Mixtures We use combining rules for a( T ) and b that are consistent with the statistical-mechanical expression for the second virial coefficient of a mixture of components 1 and 2: a( T ) = xt a,, + 2x, x, al, + xi a,, b = X; b,, + 2x, X, b,, + xi b,,. (19) (20)1320 To calculate c for the mixture we used Equation of State for Steam Mixtures c = x , c, + x, c,. (21) Eqn (21) was recommended by Peneloux and Rauzy4’ for three-parameter cubic equations of state based on the concept of volume transition. Kubic proposed a linear rule for b though this is apparently inconsistent with the virial interpretation of his equation. The difference between linear and quadratic rules for b is, however, unimportant in the density region of interest to this work.Cross-term parameters a12 and b,, were calculated from eqn (8) and (9) using cross- term critical constants. q12 was obtained from equations (4) and (5). Other cross-terms were calculated from the equations (22) CU,, = 0.5 (CO, -t 0,) yC12 = 0.125 ( VEl + V!2)3 ZCl2 = 0.291 - O.08Ul2 (24) Pc12 = z c 1 2 R L , / v c l 2 . Pseudo-critical Constants for Steam From their analysis of cross-term 4 , s for mixtures of water-non-polar-fluid, Smith et al.’ concluded that water in its interaction with a non-polar substance behaved like a small molecule intermediate between methane and argon. Cross-term 4 , s could be fitted by the Potter-McGlashan correlation using a pseudo-critical temperature T, = 230 K for water together with the true critical volume = 55.5 cm3 mol -l.It was also suggested4 that fluoromethane might be a useful homomorph for water in its interaction with a non-polar component. However when fluoromethane critical constants were used as pseudo-critical constants for water in the above equations, agreement with high- pressure experimental HE values was not as good as that obtained by choosing the constants differently. For many non-polar and slightly polar fluids the critical constants are related by the equation (26) Eqn (26) is not exact, but is a good approximation. Using the pseudo-critical temperature T, = 230 K for steam, yC = 55.5 cm3 mol-l, and o = 0.19 (the value for fluoromethane), we obtain a pseudo-critical pressure p , = 9.5 MPa.Many other sets of pseudo-critical constants for water are possible, but this set is adequate for present purposes. Substitution into eqn (4) and (9, (22)-(25), and finally (8) and (9) yields cross-terms a12 and b12, and hence the residual molar enthalpy H: of the mixture. The residual molar enthalpy H:, of the non-polar component was calculated from Kubic’s equations. The residual molar enthalpy H:, of steam was calculated using eqn (17) and (1 8). The excess molar enthalpy HE is given by (27) pc = (0.29 1 - 0.08~) RT,/ K. HEW, P , x ) = H 3 T , P , x ) - Xlll*,,(T, P ) - X,H$,(T’ PI. Comparison with Experiment Excess molar enthalpies H i ( x = 0.5) calculated from the above equations are compared with experimental results for steam-n-alkane mixtures in fig.2. The fit to the HE ( x = 0.5) measurements on mixtures containing ethane, propane, butane and n-pentane is to within experimental error at all temperatures and at pressures up to 14 MPa. The fit to the measurements on (0.5H,0+0.5C,H14) is to within experimental error up to 10 MPa, but above this pressure some of the experimental HEs lie below theC. J . Wormald and I?. M. Lancaster 1321 "0 4 8 12 p l W a Fig. 2. Comparison of calculated and experimental excess molar enthalpies HZ of (0.5H20 + 0.5C,H2,+,) for n = 2-7. Solid curves were calculated from eqn (27) as described in the text. Experimental measurements are listed in ref. (18) and (23H25). (a) (0.5H20 + 0.5C2H,),23 (b) (0.5H2 + 0.5C3H,),24 (c) (0.5H20 + 0.5C4H,,),24 (d) (0.5H20 + O.5C5Hl2),ls (e) (0.5H20 + 0.5C,H,4),25 (f) (0.5H20 + 0.5C,H1,).25 Measurements on all mixtures were made at 448.2,473.2,498.2, 523.2, 548.2, 573.2, 598.2, 648.2 and 698.2 K.Additional measurements on (0.5H20 + 0.5C6H,,) were made at 623.2 and 673.2 K.0 Equation of State for Steam Mixtures 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.0 1 0 0.2 0.4 0.6 0.8 1 Fig. 3. Comparison of calculated and experimental excess molar enthalpies HZ of [xH,O + (1 - x ) C ~ H ~ ~ + J for n = 5-7. Solid curves were calculated from eqn (27) as described in the text. (a) [xH,O + (1 - x)C,H,,],l* all measurements at 4.50 MPa, (b) [xH,O + (1 - x)C,H,,],~~ measurements at 548.2 K and 4.93 MPa, 598.2 K and 9.41 MPa, 648.2 K and 11.48 MPa, (c) [xH,O + (1 --x)C,H,,],~~ measurements at 548.2 K and 4.58 MPa, 573.2 K and 6.00 MPa, 598.2 K and 7.68 MPa.X calculated line by an amount greater than the uncertainty on the measurements. The (0.5H20 +0.5C,H1,) measurements at 573.2 and 598.2 K at pressures above 6 MPa also lie below the calculated line. The vertical sections of the isotherms at 448.2, 473.2 and 523.2 K shown in fig. 2(f) are at the saturated vapour pressure of n-heptane. Above the saturation pressure n-heptane enters the calorimeter as a liquid which evaporates to form a vapour mixture. The height of the vertical sections corresponds to theC. J . Wormald and N . M. Lancaster 1323 p l W a Fig. 4. Comparison of calculated and experimental excess molar enthalpies H z of (a) (0.5H20 + 0.5C,H4) and (b) (0.5H20 + OSCO,).Experimental measurements are listed in ref. (23) and (26). Solid curves were calculated from eqn (27) as described in the text using < = 1.21 for (0.5H20 + 0.5C,H4) and < = 1.33 for (0.5H20 + 0.5C0,). evaporation of n-heptane. Agreement with the calculated HE values at these temperatures is good. For the six n-alkane mixtures shown in fig. 2, < was calculated from eqn (5). The fit to HZ measurements on mixtures of steam-hydrogen, -nitrogen, -methane and -carbon monoxide obtained using 5 from eqn ( 5 ) is as good as that for the C,-C, n-alkanes. The composition dependence of HZ(x) for steam-n-pentane, n-hexane and n-heptane is shown in fig. 3. The solid curves calculated from our equations fit the measurements on [xH,O + (1 - x)C5H12] to within experimental error.The calculated curves are slightly above the experimental points for [xH,O + (1 - x)C6H14] and slightly below for Where there is a specific interaction between water and the other component of the mixture, is greater than that given by eqn (3) or (5) and must be obtained by fitting either cross-term second virial coefficients or excess enthalpies at low pressures. Measurements of HZ 0, = 101.325 kPa) for (0.5H20 + 0.5C,H4) and (OSH,O + OSCO,) yielded values of < of 1.21 and 1.33, respectively. Eqn (5) gives 0.97 and 0.99. High- pressure H z s calculated using these experimental 5 values are shown in fig. 4. For (0.5H20 + 0.5C,H4) the HZ values calculated from our equations agree with experiment at pressures up to 5 MPa. At higher pressures the calculated curves lie slightly below the experimental points.Fig. 4(b) shows the fit to measurements on (0.5H20 + 0.5C0,) *at pressures up to 14 MPa. The fit is very good up to 648.2 K. Only the measurements at 698.2 K are not fitted well. Our equations are a good fit to the H: measurements on all mixtures up to (0.5H20 + 0.5C,H1,). For (0.5H20 + 0.5C6H14) and (0.5H20 +0.5C7H16) at high pres- sures some of the calculated HE values are smaller than those obtained experimentally. The fit to the HZ values for these mixtures is, however, to within the uncertainty on the measurements at pressures up to p , for the hydrocarbon. The lack of agreement at higher pressures is probably due to the inability of our simple equation of state to fit the residual enthalpy of the hydrocarbons at temperatures just above T,. At temperatures well above T, the fit is better, and better agreement with the isotherms at 648.2 and 698.2 K is observed, as shown in fig.2(e) and (f). We expect our equations to work well at [XH,O 4- (1 - X)C7HI6].1324 Equation of State for Steam Mixtures 1 1 1 I I I 360 3 80 400 420 TIK Fig. 5. Comparison of calculated and experimental excess molar enthalpies H: of (0.5H20 + 0.5CflH,,+,) for n = 1-7 at p = 101.325 kPa. Solid curves were calculated from eqn (27) as described in the text using values of 5 calculated from eqn (5). The low pressure H: measurements are listed in ref. (5)-(7). temperatures above 698.2 K, and above this temperature it is likely that they will give good thermodynamic properties of mixtures at pressures up to 20 MPa.The success of our equations in fitting mixture properties over a wide range of pressure is highlighted by the fit to the HEs at p = 101.325 kPa for steam-C,-C, n-alkanes shown in fig. 5 . These HE measurements were made using our low-pressure flow-mixing Agreement with experiment is to within the uncertainty on the measurements at all pressures. The fit to low-pressure HE measurements on mixtures containing ethene and carbon dioxide is just as good. In the comparisons with experiment shown in fig. 2-5 HZ, for steam was calculated using eqn (16)-(18), which were fitted to HGK residual enthalpie~.~~ In the calculation of HE from eqn (27) an alternative procedure is to calculate HZl for steam directly from the HGK equation of state. For the pressure range covered by our measurements there is little point in doing the extra computation this involves, but for the calculation of mixture properties at high pressures it is essential to have an accurate value of the residual properties of steam.Many mixtures of interest contain steam and simple gases such as oxygen, nitrogen or methane. For such mixtures the largest term in eqn (27) is Pml. H:, for the gas is smaller, and as the gas is usually well above its critical temperature its residual enthalpy will be given with good accuracy by eqn (16). Under these conditions the residual enthalpy of the mixture calculated using eqn (19H21) will also be smaller than H:l, and the value calculated from eqn (16) should not be too much in error. The above remarks about residual enthalpies apply equally well to residual volumes. In fig.6 (a) we show the fit to the molar volume Vm of [xH,O + (1 - x)A,] at 673.2 K and at 50 and 100 MPa. The measurements were made by Lentz and Fran~k.~* The curves calculated from our equations fit the measurements well. Fig. 6(b) shows the fit to excess molar volumes V'g of (0.35H2O+0.65N2) at 673.2 K and at pressures up to 250 MPa obtained from the measurements of Japas and F~anck.~' In the 50-250 MPa region our equations are a good fit to the measurements. At 32 MPa our equation has a maximum in VE. Such maxima in supercritical region excess functions are to be expected50 and have been seen before.51 We have shown that experimental H t s for mixtures of steam with fluids as diverse as hydrogen, carbon dioxide and n-heptane can be well fitted using a simple two-termC.J. Wormald and N. M. Lancaster 1325 X plMPa Fig. 6. Comparison of calculated and experimental molar volumes at high pressures. Solid curves were calculated from our cubic equation of state together with HGK molar volumes of steam as described in the text. (a) molar volumes V, of [xH,O+ (1 -x)Ar] at 673.2 K measured by Lentz and F r a n ~ k , ~ ~ (b) excess molar volumes V: of (0.35H2O+0.65N2) at 673.2K from the measurements of Japas and F ~ a n c k . ~ ~ equation of state and a single temperature-independent parameter c. The calculation of densities, compressibilities and other vapour phase thermodynamic functions is now straightforward. References 1 C. J. Wormald, J. Chem. Thermodyn., 1977, 9, 901.2 G. R. Smith, A. 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