8856
J. Phys. Chem. 1995,99, 8856-8862
Equation of State for Representing the Thermodynamic Properties of Liquids at High Pressure Mercedes Taravillo, Valentin G. Baonza," Mercedes Chceres, and Javier Ndiiez Departamento de Quimica Flsica, Facultad de Ciencias Qulmicas, Universidad Complutense de Madrid, 28040-Madrid, Spain Received: July 11, 1994; In Final Form: January 23, 1995@
A new equation of state intended to be used for representing the thermodynamic properties of liquids at high pressure has been developed. The complete expression is based on an isothermal equation suggested by our group; the temperature dependencies of the characteristic parameters are obtained from their limiting behavior at a pseudospinodal curve using accurate experimental data from literature. The equation of state has been subsequently applied to liquid toluene, and the following thermodynamic properties have been calculated as functions of pressure and temperature: isothermal compressibility, thermal expansion coefficient, thermal pressure coefficient, and isobaric heat capacity. The comparison with experimental results from literature appears to be very satisfactory, and the possibility of using the equation of state for extrapolation is discussed.
Introduction and Background
A fundamental problem in high-pressure research is to find an equation of state (EOS) to describe liquid systems in general. A reliable EOS should be able to give not only a good description of the pVT surface of a liquid but also accurate values of its derived properties: isothermal compressibility, KT, thermal expansion coefficient, a,, and isobaric heat capacity, C,. However, most of the equations of state proposed in the literature to represent pVT data of liquids are based on polynomials depending on a large number of coefficients, which are very frequently meaningless from a fundamental point of view. This implies that the derived properties are also represented by polynomials, so their predictive power out of the experimental range is obviously restricted, making them unpractical for most purposes. Other kinds of EOSs are those derived from isothermal equations based on simple phenomenological relations.' Maybe the most widely known examples are those of Tait and of Mumaghan, indistinctly used for solids and liquids through the years. Their derivation is based on the assumption that a linear relation between the bulk modulus B and the pressure holds at a given temperature. However, both theory2 and experiment3 show that a certain curvature is expected for B at very high pressures. In this work we derive a rather simple EOS, based on an isothermal expression, which incorporates the curvature of B(p) by means of a simple power law! The complete EOS is applicable to liquid systems up to very high pressures in the normal liquid range, e.g., subcritical temperatures except in the vicinity of the critical point. As an example, liquid toluene will be studied in detail. Using only pVT data, the EOS reproduces experimental KT, a,, and C, results for this liquid up to very high pressures. The extrapolation capabilities of the EOS have also been tested. Fundamentals In recent works we confirmed that a, and KT follow power laws in the pressure, p , along isothermal path^.^^^ According to this, a,@) and KZ@) can be represented by the following phenomenological expressions: t
* To whom correspondence should be addressed. @Abstractpublished in Advnnce ACS Absrracrs, May 1, 1995.
K&I)
= K*@ -p , > p
where p , is the divergence pressure at the pseudospinodal curve at this temperature, a* and K* are proportionality constants for each temperature, and y' and y are the pseudocritical exponents, which are independent on temperature and characterize the divergence of these quantities as the pseudospinodal curve is approached. While the mean-field pseudocritical exponent, y' = '/2? remains almost constant for a,@) in both metastable8 and stable619J0liquid ranges, K&I) diverges with a pseudocritical exponent y = 0.85 in the stable liquid range.4.5.9 A valuable characteristic of eqs 1 and 2, particularly for the problem we are dealing with, is their simplicity. In addition, eq 2 gives the proper curvature expected for B at high pressures! However, from a general point of view, it is probably more important that the same p s should work in correlating both experimental a, and KT data at a given temperature. This issue, first suggested by Pruzan9 and recently c o n f i i e d by our group,6 supports the idea of the mutual relation of the characteristic divergence exponents in the normal liquid range. The integration of eq 2 yields the following isothermal expression for the molar density as a function of press~re:~."
where esis the density of the liquid branch of the pseudospinodal curve at this temperature. Equation 3 with y = 0.85 represents experimental densities of many liquids within their estimated In addition, it is noticeable that the divergence pressures, obtained from the fit of experimental densities, lie very close to those calculated from maximum superheat measurements4 and to those obtained from the correlation of experimental a, and KT data.5 An additional feature of eqs 1-3 is that they proved useful for extrapolation.'* Thus, it would be desirable to obtain a complete EOS based on eq 3 by looking for universal functions to represent the temperature dependence of its characteristic parameters. Development of the EOS A common problem to any isothermal EOS is to find the appropriate temperature dependence of the characteristic pa-
0022-3654/95/2099-8856$09.00/0 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 21, 1995 8857
Thermodynamic Properties of Liquids at High Pressure rameters, being combinations of polynomial series usually unsati~factory.~~ The actual problem is therefore to obtain a reliable EOS based on the form given by eq 3, which requires adequate expressions be found for ps(T), K*(T), and es(T). A. Temperature Dependence of ps. We have found5 that one of the most reliable and simple expressions for representing p,(T) is that proposed by Alba et al.:1°
20
15
(4) where pc is the critical pressure, t = 1 - T/T, is a reduced temperature, Tc being the critical temperature, and U I and a2 are characteristic parameters for each substance. When eq 4 is incorporated into a given EOS, the divergence pressures obtained lie very close to other estimations of a pseudospinodal curve.5 B. Temperature Dependence of ps. Critical data are commonly represented by the following function in liquidgas and binary fluids systems
where e is the density of the liquid under orthobaric conditions, ec is the critical density, B, is the critical amplitude, and /? is the critical exponent. We have found that eq 5 is successful in describing the coexistence curve in the whole temperature range also. Osman and SorensenI4tested the so-called pseudospinodd assumption and pointed out the possible existence of a universal pseudospinodal curve that describes several thermodynamic and transport properties for various substances. Under the pseudospinodal assumption, the pseudospinodal curve has a form similar to that of eq 5 :
e m = e c ( l + B,!')
(6)
where B, and /3t may be called the pseudocritical amplitude and exponent, respectively. These authors found that both exponents /? and /?twere quite close, although a certain tendency of being /?tslightly greater than /? was observed. In any case, these results suggested to us that eq 6 should be useful in representing es(T)of eq 3. We have chosen CF4I5 and n-hexaneI6 to study the temperature dependence of eSin eq 3. First, we fitted the liquid branch of the coexistence curves of both liquids to eq 5. We thereafter fitted some experimental density isotherms of both substances to eq 3 with y = 0.85. The esresults were fitted to eq 6 using an exponent /?tequal to /? obtained from the coexistence data. The comparison of the different pseudospinodals are plotted in Figures l a and 2a. Unfortunately, although eq 6 represents relatively well es(T),we have found that it does not work properly for representing the derived properties of liquids over the whole temperature range when it is included into our EOS, so we analyze es(T)from other arguments. Without making any assumption about the functional form of e,(T) and K*(T), the following expression for a,(p,T) can be derived from eq 3 through the standard thermodynamic relation
a,
=
-e-I(ae/aT),:
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TIK Figure 1. (a) Temperature variation of
e,
for liquid CF4: (A) Correlation of experimental density data to eq 3 with y = 0.85; (- * ) liquid-vapor coexistence curve; (- -) fit of the values of es (A) in terms of eq 6; (-) eq 9 with coefficients of Table 3. (b) Temperature variation of a, = -es-'(QJd7) for liquid CF4: (H) fit of experimental a, data to eq 7; (- -) calculated from fit in terms of eq 6 of eS(A); (-) eq 8 with coefficients of Table 3. The critical point is denoted as C.P. Experimental values taken from ref 15.
(a) First, using experimental KT data and supposing that y = 0.85, the isotherms of KT are fitted to eq 2 separately. (b) Then, the divergence pressures, ps, obtained from the fit are correlated in terms of eq 4. (c) Finally, the isotherms of KT are fitted again to eq 2 with y = 0.85 and with K* as the only adjustable parameter; then, using the improved values of K*, its temperature derivative (dK*/ dT) is evaluated numerically and subsequently included in eq 7. With these results for K* and (dK*/dT), and with y = 0.85, a,(T)can be obtained from experimental a&) data as the only adjustable parameter in eq 7. The results of a,(T)obtained for CF4 and n-hexane are plotted in Figures l b and 2b, respectively. Notice the near linearity exhibited by both curves over the whole temperature range. Similar results were obtained for other liquids such as argon, xenon, and 2,3-dimethylbutane, so it seems reasonable to use a simple linear function to represent this quantity. Therefore, the following linear function to represent a,(q is proposed:
+
a,(T)= u3/Tc (2a4/T:)T
where a,(T)= -es-'(QJdT). Since it is assumed that ps(T)and its temperature derivative can be evaluated from eq 4,we have obtained the temperature dependence of a,(T) as follows:
-l
c
(8)
where a3 and a4 are characteristic parameters for each substance. Thus, the exponential form which follows for es(T),supposing eq 8 for a,(T),is
'
Taravillo et al.
8858 J. Phys. Chem., Vol. 99, No. 21, 1995 '
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100
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500
400
300
p I MPa
Figure 3. Comparison of calculated isotherms from eq 7 and experimentalI6 a, isotherms for n-hexane: (-) by using eq 9 to represent es;(- -) by using eq 6 to represent es. Experimental results: (0)T = 273.15 K; (0)T = 473.15 K.
F
1 00
200
250
300
350
400
450
m-xylene
500
TIK
es for liquid of n-hexane. Symbols as in Figure 1. (b) Temperature variation of a,for liquid n-hexane. Experimental values taken from ref 16.
Figure 2. (a) Temperature variation of
In Figures l b and 2b, we also compare the results of a,(T) obtained from eq 8 with the coefficients of Table 3 for CFd and n-hexane, respectively. The comparison shows that these variations are related by a proportionality constant which arises from using y = 0.85 as a fixed value. Figures l a and 2a show the pseudospinodal curve given by eq 9 using the coefficients given in Table 3. The agreement with es obtained above from the correlation of density isotherms is now much more satisfactory than from use of eq 6. In order to confirm that eq 9, once it is introduced into our EOS, gives reliable results for the derived properties of liquids, we compare in Figure 3 several experimental a,,isotherms of n-hexane with those calculated from eq 7, using both eqs 6 and 9 to represent e,. The comparison with the results obtained using eq 9 to represent g, is rather satisfactory. On the other hand, it can be observed that the results obtained using eq 6 do not reproduce satisfactorily the overcrossings experimentally found for the a, isotherms.I6 However, it seems that eq 6 could succeed in correlating experimental data at low temperatures. Figure 4 compares a, results obtained from eq 7, using eq 6 to represent @, for m-xylene. The comparison with results reported in ref 17 is very satisfactory. C. Temperature Dependence of K*. If one correlates isothermal data of KT in terms of eq 2, one finds that K* is a smooth increasing function of temperature. A similar procedure using isothermal a, data and eq 1 shows that a* exhibits a smooth decrease with tem~erature.~ Both temperature variations are shown in Figures 5a,b and 6a,b for CFq and n-hexane, respectively. At this point, one can speculate about the existence of a general relation between both a* and K* valid for any liquid.
0.85
1 0
10
20
30
40
'
50
60
70
80
90
100
p MPa
Figure 4. Comparison of a, isotherms for m-xylene:
(-) results obtained in ref 20; (- -) calculated results from eq 7 using eq 6 to represent es;(0)T = 248.15 K; (0)T = 293.15 K.
This can be an appealing procedure because successful expressions for representing a*(T)are available in the l i t e r a t ~ r e . ' ~ . ' ~ . ' ~ In order to obtain the temperature dependence of K*, we now take into account the relation between a, and KT by means of the thermal pressure coefficient yV.*O Following Speedy,21a, and KT are asymptotically proportional to each other near the line p,(T), along any path. This requires that y y' as p ps(T), which allows us to obtain a limiting relation between K*(T) and a*(T)at a spinodal curve
-
-
-a * ( ' - (dp/dT), E (dpJd7') K*(T)
This argument is equivalent to that of Skripov,22who stated that though a, and KT are infinity at the spinodal, their ratio is completely defined by the shape of the spinodal curve in @,T) variables. Equation 11 is an identity in the vicinity of the pseudospinodal curve because both pseudocritical exponents are identical and equal to however, since we obtain the characteristic coefficients of the EOS from measurements of the stable liquid range, where the pseudocritical exponents are quite different,
Thermodynamic Properties of Liquids at High Pressure
J. Phys. Chem., Vol. 99, No. 21, 1995 8859
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n-hexane
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TI K Figure 5. (a) Temperature variation of K* for liquid CFd: (- -0--) fit of experimental K~ data to eq 2 with y = 0.85; (- -0--) the same with pr fixed of previous fit; (-) calculated from eqs 13 and 14 with coefficients of Table 3. (b) Temperature variation of a* for liquid from fit of experimental a, data to eq CF4: (- 4--) a* (MPao.5K1) 1 with y' = 0.5; (-) a*+(MPay-K-I) from eq 13 with coefficients of Table 3. (c) Temperature variation of y* for liquid CF4: (- -0--) y* (MPa0.65-K-1) from fit of experimental y. data to eq 10 with exponent equal to 0.35; (-0-) (dp/dvS(MPa-K-') obtained from the correlation of experimental KT to eq 2 using y = 0.85; (-) (dp/dn, (MPaK') obtained from eq 4 with coefficients of Table 3. eq 11 is not an identity but only a proportionality, so an additional constant arises, and we must write
This can be confirmed from the analysis of experimental y y data in terms of eq 10 (please see ref 20) with an exponent equal to 0.35. We compare in Figures 5c and 6c the temperature variation of y* (MPao,65*K-')with that of (dp/dnS(MPa-K-') obtained from the correlation of experimental KT to eq 2 using y = 0.85 (the same used to study a, in eq 7, see previous section) and with that of the (dp/dnSobtained from eq 4 with coefficients of Table 3. Both curves of y*(n and (dp/dns exhibit similar behavior and are related each other by a constant, C (MPa0.35),within the uncertainties of both quantities; this confirms empirically the relation expressed by eq 12. We shall use the following expression to represent a * ( T ) : I 8 a*(n =
-P,(T)>~
(13)
where @ andpo are characteristic parameters for each substance. An important advantage of using eq 13 is that the proportionality constant may be included in the parameter without increasing the number of characteristic parameters of the equation of state, so we can write
250
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450
500
TIK Figure 6. (a) Temperature variation of K* for liquid n-hexane. (b) Temperature variation of a* for liquid n-hexane. (c) Temperature variation of y* for liquid n-hexane. Symbols as in Figure 5. K*(T) = a*(r)/y*(T) = Ca*(T)/(dpJdT) = a*'(T)/(dp,/dT)
(14) where a * ? ( n has a similar form to eq 13, but including C in Q. This can be confirmed in Figures 5b and 6b, where we compare the results of a * + (MPaYK') obtained from eq 13 with coefficients of Table 3 with those obtained for a* (MPaY'*K-') from the correlation of experimental a, data in terms of eq 1 with y' = 0.5. Both curves are related to each other by a constant, which is C also. The ratio (C = a*?(T)/ a*(n) is recorded as a function of temperature in Table 1 for CF4 and n-hexane. From Table 1 it is found that C does not vary largely with the temperature, so it can be assumed that C is approximately constant. Figures 5a and 6a show the results obtained of K*(T) from eq 14 with the coefficients of Table 3. The agreement with the results obtained from the correlation of isotherms of KT to eq 2 with y = 0.85 for both substances is very satisfactory and confirms the former expressions. In addition, this expression for K*(T) is advantageous because both ps(T) and its derivative, (dp,/dr), can be computed by means of the same equation, providing additional consistency to the complete EOS. D. Analysis of the Exponent y in Equation 3. Although eq 3 represents the general form of the complete EOS, one question needs confirmation, namely, the possibility of considering y as a constant equal to 0.85 or treating it as a free parameter. We have then fitted the experimental pVT results of CF4 and n-hexane to the complete EOS using the two prescriptions for the parameters which follow from considering the two possibilities, fixed to 0.85 or free parameter, for the exponent y .
Taravillo et al.
8860 J. Phys. Chem., Vol. 99, No. 21, 1995 TABLE 1: Ratio, C, between a*+(Z')Obtained from Equation 13 (Using the Coefficients Recorded in Table 3) and a*(Z')Obtained from the Correlation of Experimental to Equation 1 with y' = 0.5 for CF4 and n-Hexane (See Xxt) n-hexane
CF4 TIK
C/MPao.35
T/K
C/MPao.35
95.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0
1.51 1.50 1.44 1.39 1.36 1.34 1.31 1.28
243.15 273.15 298.15 323.15 348.15 373.15 398.15 423.15 448.15 473.15
2.53 2.26 2.09 1.95 1.84 1.74 1.66 1.57 1.49 1.41
TABLE 2: Average Relative Deviations, in Percentage, between Experimental Results of Isothermal Compressibility, KT, and Thermal Expansion Coefficient, q,,for CF4 and n-Hexane and Those Calculated from Equation 15 Using Different Prescriptions for the Characteristic Parameter@ CF4
y = 0.85 (fixed) y free parameter
n-hexane
AKT
AaP
AKT
AaP
3.1 3.1
2.3 2.3
7.2 3.6
11.9 5.1
The relative deviation is defined as Aq = (41 - qc)lql, where ql and qc refer to the quantity q (apor KT) found in the literature and computed from eq 15, respectively.
We have subsequently calculated KT and a, as functions of pressure and temperature and compared both quantities with experimental result^.'^^'^ Table 2 records the average relative deviations for both substances. The results show that the liquid data can be represented almost equally by the two prescriptions, when judged through the difference between experimental and calculated densities. However, when judged in terms of the accuracy and consistency of the derivatives of the pVT surface, the second possibility ( y as free parameter) is clearly much superior, as confirmed when results of n-hexane. This means that although the value y = 0.85 is good enough to represent isolated isotherms through eqs 2 and 3, once the generic form of these expressions is included into an EOS the calculated properties are rather sensitive to its exact value. Nevertheless, the resulting value is very close to 0.85, in agreement with previous observations. E. Final Form of the Equation of State. The complete EOS we propose here is the following:
where p s ( r ) , es(r),and a*'(r) are given by eqs 4, 9, and 13, respectively, and y is an adjustable parameter, which is expected to be close to 0.85: The characteristic coefficients of eq 15 used in previous sections for CF4 and n-hexane are recorded in Table 3 for the convenience of the reader. More digits than statistically significant have been included in the table to avoid roundoff errors in the numerical computation of the pVT and derived properties results. Application of the EOS to Toluene Accurate measurements of the density have been reported in the literature for toluene by many authors: Muringer et al.23 reported data up to 260 MPa at temperatures between 180 and
TABLE 3: Coefficients of Equation 15 Determined from a Weighted Least Squares Fit of the Density e of CF4, n-Hexane. and Toluene CFa p,MPa TJK ~,l(mol.dm-~) a1 a2
a3
a4 107@(MPa'r-2)K1) p a p a
Y a
4.19 227.45 7.102 -16.355 -0.8600 0.81541 0.26833 3.8203 233.37 0.84997
n-hexane
toluene
3.03 507.8 2.703 -23.080 - 1.0308 0.75235 0.15768 8.3910 107.71 0.86881
4.215 591.77 3.159 -27.924 -0.8785 0.77288 0.05778 3.0639 211.29 0.86683
The critical constants have been taken from ref 31.
320 K, M o p ~ i k *gave ~ results up to 200 MPa at temperatures between 223 and 298 K, and Kashiwagi et reported measurements up to 250 MPa from 273 to 373 K; the uncertainty of the pVT data does not exceed 0.1%. In a recent work we have also reported peT measurements for toluene using an expansion technique." Two isotherms, 223 and 303 K, were measured up to 110 MPa. The accuracy of the densities reported there was 0.003 m ~ l - d m - ~ . The overall pVT data for this substance have been correlated in terms of eq 15 by using a weighted least squares procedure; the characteristic coefficients are recorded in Table 3. The standard deviation of the fit is 0.009 mol~dm-~,a result compatible with the combined uncertainties and systematic differences found between the different sets of data. A. Isothermal Compressibility KT. From eq 15 the isothermal compressibility can be computed using the expression
The isothermal compressibility of toluene is plotted in Figure 7a for temperatures ranging from 200 to 320 K and pressures up to 260 MPa. Sun et a1.26obtained the pressure dependence of this quantity from their speed of sound measurements; the agreement with present results is very satisfactory, and discrepancies are within the accuracy of their results. The maximum difference found between the two sets of data was 0.01 GPa-I. B. Thermal Expansion Coefficient a,. Differences between values of a, calculated from eq 7 and those experimentally determined by Ter Minassian et a1.'* using a piezothermal technique between 200 and 450 K and pressures up to 400 MPa are always less than K-I. The experimental uncertainty on a, is about 0.5-1% over the whole experimental range. Selected results of a, are shown in Figure 8a for two isotherms: 200 and 320 K. A clear crossing between our calculated a, isotherms is observed at 45 MPa, while those of Ter Minassian et al. indicate a crossing at 5 1 MPa. Sun et a1.26 computed this property from their speed of sound measurements, and the intersection is found to be at 36 MPa. Similar results are plotted in Figure 8b for the isotherms of 260 and 320 K. The agreement with direct measurements is again satisfactory. In addition, the displacement with temperature observed experimentally for the intersections of the a, isotherms is adequately reproduced by our EOS. C. Thermal Pressure Coefficient yy. The thermal pressure coefficient yv can be directly calculated from eqs 7 and 16. Our results for toluene are plotted in Figure 7b together with those given by Sun et a1.26 The agreement between the two sets of data is excellent. The maximum difference found between the two sets of data is about 0.05 MPaeK-', which is found at 180 K.
Thermodynamic Properties of Liquids at High Pressure 1
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