J. Phys. Chem. 1994,98, 1993-1998
1993
Estimation of the Spinodal Curve for Liquids: Application to 2,3-Dimethylbutane Valenth Garcia Baonza,' Mercedes Chceres Alonso, and Javier Niiiiez Delgado Departamento de QuImica FIsica. Facultad de Ciencias Quimicas, Universidad Complutense de Madrid, 28040 Madrid, Spain Received: July 30, 1993; In Final Form: November I, 1993"
For a pure substance, the spinodal curve is defined as the locus where thermodynamic quantities such as the thermal expansion coefficient a,, the isothermal compressibility KT, and the isobaric heat capacity C, are expected to diverge. Its location is of particular importance from both practical and theoretical points of view, since it represents the limit beyond which a particular state of matter, in our case the liquid state, can exist or not. In this work, several predictions of the spinodal curve have been tested, compared, and discussed for 2,3-dimethylbutane. Thecomparison with results found in the literature reveals that some conclusions established here could be of general application.
Introduction It is widely known that liquids can be cooled below their freezing temperature without solidification (supercooled liquid) and that they can be heated above their boiling temperature without vaporization (superheated liquid). In both cases the liquid remains homogeneous beyond the line of phase equilibrium into the socalled metastable region. These two phenomena have been extensively described through the years in the literature.'-5 For a one-component system, the spinodal curve (first introduced by van der Waals) represents the stability boundary of the one-phase state (through this work, the liquid state only) in the two-phase region. Experimentally, it is considered that the spinodal curve can be approached quasistatically but never reached. However, the assumption of the existence of a spinodal curve has been supported by a large number of experiments.l+IO Both thermodynamic and transport properties have been used to test the existence of a universal spinodal which describes the divergence of all these properties normalized relative to the coexistence curve." For subcritical temperatures, quantities such as the thermal expansion coefficient cy,, the isothermal compressibility KT, and the isobaric heat capacity C, increase significantly as the spinodal curve is approached and, finally, diverge. It has been shown through an expansion of the Helmholtz potential that, sufficiently close to the spinodal, these quantities follow a power law with exponent equal to (-1 /2) (mean-field pseudocritical exponents) along isothermal or isobaric pathsn4 A more recent study of DAntonio and Debenedettilzconfirms these results from a general thermodynamicanalysis of loss of tensile strength in liquids. From a practical point of view, the most interesting feature to study here is the range where these power laws hold. Mean-field pseudocritical exponents should work for correlating experimental data if one considers regions far removed from an eventual spinodal singularity. However, while for c y p @ ) the exponent (-1/2) seems to remain almost constant over the full metastable and stable liquid phase? for K&) the exponent seems to be close to (-0.85) in the stable liquid range? This is mainly due to the fact that when the normal liquid range is considered (especially at low temperatures) experimental data are too far from the spinodal, so that the exponent is ill-defined. Thus, thedivergence pressure (or temperature) can beestimated from extrapolation of experimental measurements." Skripovl also pointed out that the spinodal curve could be obtained in @,T)variables as the envelope of a family of density isochores extrapolated into the metastable region. To whom correspondence should be addressed. *Abstract published in Advance ACS Abstracts, February 1, 1994.
0022-3654/94/2098-1993$04.50/0
Besides mechanical coefficients, another interesting property which can be related with the spinodal is surface tension. It has been shown that a surprisingly good estimation of the spinodal can be directly obtained from surface tension measurements by applying the elementary hole theory of Fiirth.14J5 Finally, from a general point of view, since the condition KT = m is equivalent to ( a p / a p ) ~ = 0, the spinodal curve might be estimated from any equation of state (EOS) able to reproduce both liquid and vapor phases as well as the coexistence region. In fact, the assumption of the existence of a spinodal curve was successfully used to derive an EOS for liquid-gas systems.16 Nevertheless, the estimation of the spinodal from EOS established in the literature has not been a common task. This is mainly due to the fact that most of the EOSs used to describe both liquid and vapor phases are empirical, presenting multiple maxima and minima inside the metastable region, so the estimation of the spinodal from thecondition ( a p / d p )= ~ 0 becomes uncertain. On the other hand, theoretical EOSs, usually derived from statisticalmechanics perturbation theories, involve simple models qualitatively correct but rather inapplicable for practical purposes. In this workwesuggest that successful semitheoreticalEOSs recently developed in the literature may be an appealing alternative. It is the purpose of this paper to test the reliability of the different approximations described above to estimate the spinodal curve of liquids as well as to describe their thermodynamic behavior in both stable and metastable states. We have then estimated the spinodal curve of 2,3-dimethylbutane (2,3-DMB) for which accurate high-pressure (up to 110 MPa) thermodynamic properties below room temperature (between 208 and 298 K) have been reported very recently by our laboratory.17 In the literature there also exist surface tension measurements at intermediate temperatures1*and high pressure p V T results (up to 30 MPa) in the near-critical region;lg this makes 2,3-DMB a suitable substance for our study.
Phenomenological and Theoretical Background Mechanical Coefficients. Let us first examine the expressions proposed in the literature to describe the mechanical coefficients ap(thermal expansion coefficient) and KT (isothermal compressibility), and their relation with the spinodal concept. From a theoretical point of view it has been shown4J*~zothat, when approaching the spinodal along isothermal or isobaric paths, the limiting behavior of cyp and KT follows a power law with an exponent y equal to (-1/2). This indicates that as the limit of stability is approached, what one recovers is mean-field pseudocritical exponents. Thus, along an isotherm, the limiting behavior 0 1994 American Chemical Society
1994 The Journal of Physical Chemistry, Vol. 98, No. 7, I994
of ap@)and
K&)
can be expressed as
.,(PI
= A @ - PSPY
(1)
KT(P)
= K(P - PSPY
(2)
where p is the pressure, pspis the divergence pressure at this temperature; A and K can be considered as constant parameters for each temperature (although they are slightly density dependent): and y = (-1 /2). These equations were obtained under the following conditions: (a) there exists a line of stability limits (the spinodal line) at which both properties diverge and (b) the Helmholtz potential A( V,T)can be expanded along an isotherm as a Taylor series in (V,,(T) - V), where Vsp(T) is the molar volume along the spinodal curve. The expansion breaks down as the critical point is approached4 since it is known that A(V,T) is not analytic there. In fact, it also breaks down very close to the spinodal, since there too the correlation length diverges. Equations 1 and 2 were checked against experimental results for water at atmospheric pressure,"J and they were used to give a successful unified description of the thermodynamic properties (including anomalies such as the density maxima) of superheated, stretched, supercooled, and ordinary water, under the so-called stability-limit conjecture: which has been discussed very recently.21.22 Pruzan9 found that, in the case of CYp@), the exponent y = (-1 /2) remains almost constant in the full liquid range (including metastable and stable states). In addition, he found that eq 1 reproduces the ap@)results of n-hexane over wide ranges of pressure (up to 700 MPa!) and temperature (from 240 to 473 K) with an exponent of (-0.52). However, the dependence of K&), for liquids such as argon, carbon dioxide, and n-hexane, is only reproduced when the value of the exponent y is set to about (-0.85) in eq 2. From an analytical point of view: eq 2 with y = -0.85 is not compatible with eq 1. In fact, the integration of ap@,T)9.23yields a complex functional form for (K&) - K T O ( P ) ) with y = (-1/2), where TOis a reference isotherm. However, while the latter treatment preserves mathematical consistency, the simplicity of eq 2 is lost, as well as the direct relationship between the divergence pressure and experimental compressibilities along an arbitrary isotherm. Alba et al.23 confirmed the validity of eq 1, with y = (-1/2), through a statistical analysis of the experimental apisotherms of carbon dioxide and n-butane from the triple temperature to the critical temperature and pressures up to 400 MPa. It is now interesting to bring out the experimental evidence of the existence of a crossover of the isotherms of apfor a large number of liquids of very different nature>J7,23-2* It has also been observed that when such intersections are found, they occur in a very narrow range of pressures for many l i q ~ i d s . l ~ - ~Under ~ , 2 6 the following conditions: (a) the intersection of the ap isotherms Occurs a t a single point of the @,ap)plane, (b) the spinodal curve originates at the critical point, and (c) the locus of the spinodal in (p,T) coordinates can be described by a [1/1] Pad6 approximation, Alba et al. proposed the following expression for apas a function of pressure p and temperature T 2 3
]
Po -Psp(T) (3) P - Psp(T) where po is the intersection pressure and a0 is the value of apat the intersection. The temperature dependence of the divergence pressure papalong the spinodal curve is given by =ao[
ap(P9T)
(4) wherep, is the critical pressure, t( T ) = ( I - T / Tc)is the reduced temperature, and a1 and a2 are characteristic parameters of the substance.
Baonza et al. Equation 3 takes into account the Occurrence of a crossover of the apisotherms at (a0,pO). There is no indication that this behavior could be general for any liquid, and it has not been observed for many of them in the usual experimental range (commonly about 100 MPa). However, this does not alter the fact that intersections might appear a t higher pressures, and it has been recently shown by us that eq 3 is able to reproduce ap results within experimental uncertainty, over wide ranges of pressure and temperature, by using a virtual value of (ao,po).25 In addition, when a crossover of the isotherms of apis detected in the experimental range, parameters a0 and po can be directly approximated from the experiment, and only a1 and a2 should be determined from the fit. The question about the reliability of the Pad6 approximation (eq 4) to give a realistic description of the spinodal curve still remains open and will be subsequently studied. Fiirth's Theory. Regarding the estimation of the spinodal curve from surface tension measurements, the elementary hole liquid theory of Fiirth,29 generalized by Skripov14 and Baidakov et al.,15 has turned out to be a surprisingly good approach when it is compared with results obtained from experimental measurements in both stable and metastable regionsI5 and even far from the critical temperature.9.30 In this theory, the holes are identified with bubbles of vapor spontaneously formed in the liquid and a certain size distribution of the holes corresponds to each equilibrium state of matter. The bubbles are supposed to possess ordinary surface tension u, three translational degrees of freedom, and one internal degree of freedom corresponding to the change in radius. The vapor pressure in the bubble is taken as the saturation pressure p"' at a given temperature. The mean hole size increases with superheating, and it remains very small to a certain limiting superheat; thereafter, a catastrophic growth of the bubble begins and the liquid becomes unstable.14 The spinodal in @,T) variables can be directly derived from this treatment, and the expression which relates the divergence pressure with other quantities is the following:l5
where kB is the Boltzmann constant, a n d p ' , pvsat,plsat,and u are, respectively, vapor pressure, orthobaric vapor and liquid densities, and surface tension, at the temperature T. The term (1 - pV"/ PI"') may be normally suppressed since ( p V P t / p p t )is negligible, or a rather small correction may be needed, except for temperatures higher than 0.8-0.9Tc. Equations of State. The concept of a spinodal curve was first introduced by van der Waals, and the first attempt to give a qualitative interpretation of this phenomena was made by means of his famous EOS. A simple inspectionof thes-shaped isotherms of the van der Waals EOS in a @,V) diagram allows us to locate a spinodal curve from the position of the minimum (liquid branch spinodal) and the maximum (vapor branch spinodal) exhibited into the metastable region. From the mathematical point of view, it is only necessary to find the roots (with physical significance) of the equation ( a p l a p ) = ~ 0, which implies the divergence of the isothermal compressibility KT. However, since the spinodal is found by extrapolation into the metastable region, its location will depend on the form of the EOS and its characteristic parameters. Results and Discussion
While eq 1 with y = (-1/2) has been proven to be consistent with the expansion of the Helmholtz potential about the spinodal4 and valid over wide ranges of pressure,9.23several questions remain open regarding the reliability of eqs 2 and 3 to reproduce, respectively, experimental K-&) and ap(p,T)results, as well as to yield a unique spinodal curve consistent with other predictions.
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1995
Estimation of the Spinodal Curve for Liquids
TABLE 1: Divergence hessures (AJMPa) along the Spinodal Curve of 2,3-DMB Calculated from upand KT Experimental Results of Refs 17 and 19, Fiirth s Theory, and Deiters EOS from ap T/K (w 1. Y = -0.50) 500.30 473.15 448.15 423.15 398.15 373.15 333.15 323.15 313.15 303.15 298.15 293.15 283.15 273.15 248.15 208.15
1.594 -1.54' -6.56' -1 1.59' -17.38'
-40.81b -42.9Sb -47.706 -52.91' -69.106 -1 10.33'
from KT (ea 2. Y = -0.85) 0.600 -1.700 -6.45' -1 1.23' -16.48'
-41 33' -44.87b -51 .02b -56.566 -69.94b -99.306
Alba cu, Fiirth (ea 4) (ea 5 ) 3.13 -0.18 -3.67 -7.67 -12.31 -17.75 -28.73 -32.06 -35.67 -39.60 -41.72 -43.93 -48.68 -53.93 -69.88 -108.80
3.13 0.80 -2.33 -6.25 -11.00 -16.64 -30.30 -27.93 -34.23 -31.27 -38.41 -34.85 4 2 . 8 7 -38.71 -40.75 -47.61 -42.87 -52.66 -47.37 -58.03 -52.26 -66.52 -98.58
Therefore, our experimental apisothermsgiven in ref 17 (between 208 and 298 K) and those calculated from the experimentalpw results of Kelso and Felsinglg (between 373 and 473 K) have been fitted to eq 1. The standard deviation of the fit is always less than 0.02kK-l, in agreement with theestimated experimental uncertainties.17 The divergence pressures pspobtained from the fittings are recorded in Table 1, and hereafter it is assumed that they constitute a realistic representation of the spinodal curve. Test of Pruzan Expression for KT. Regarding eq 2, to the best of our knowledge, only the work of Pruzang attempts to give a general value of y by studying the behavior with pressure of the experimental KT of three substances (argon, carbon dioxide, and n-hexane) in the stable liquid range. Therefore, the proposed value y -0.85 is still uncertain. We have found, however, that it is compatible with others obtained from the correlation of experimental K&) to eq 2 for several liquids. Table 2 records some representative results. Although the better exponent which fits the data of 2,3-DMB is close to -0.89, the value y = -0.85 has been retained for calculations; this allows us to make a qualitative comparison with the results found in ref 9 for argon without a loss of accuracy in the representation of the KTisotherms (about 0.02 GPa-l, which is within the estimated uncertainty in KT)." The divergence pressures calculated with both exponents differ by about 7 MPa on average between 208 and 298 K, always being lower than those obtained withy = -0.89. Table 1 records the results ofp,,using y = -0.85 in eq 2, including those calculated from the experimental data of ref 19 between 373 and 473 K. These results are also plotted in Figure 1 together with thosep,, resulting from individual fittings to eq 1 of the ap isotherms mentioned above. In general, the spinodal obtained from wthrough eq 2 lies very close to that obtained from q,using eq 1 and confirms the results obtained by Pruzan for the case of argon.9 Obviously, if we accept the validity of the expansion of the Helmholtz potential about the limit of stability, near the spinodal both eqs 1 and 2 should have the same exponent (-1/2), so using (-0.85) for K T should yield a slightly different spinodal curve. In any case, it seems that eqs 1 and 2, withy = (-1/2) and y = (-0.85), respectively, correlate adequately both (up@) and K&) experimental results, yielding a nearly coincident spinodal curve. Test of Alba Expression for apand Spinodal Curve. Pruzan9 suggested that the dependence with temperature T of parameter
I
I
I
Psat
C.P.
_.........
Deiters EOS
Reference 19. Reference 17.
TABLE 2
20
-20
-
le
a
h
-60-
a
-1001
f
,
;ooo;
ooooo -140 150
:;( -0.85), Furth's Theory aP (7:-0.50) KT
250
450
1 550
T3YK Figure 1. Spinodalcurvefor 2,3-DMB. pmtrepresentsthevapor pressure curve,37and C.P. denotes the critical point.&
A and the divergence pressurepsp in eq 1 could be fitted with high accuracy to polynomials in T of the form:
(7) Rubio et al.30 used these expressions to fit the spinodal curve for tetrafluoromethane between 95 K and the critical temperature (Tc = 227.6 K), although a Gaussian function was needed to fit A(T) at temperatures below 130 K, pointing to the existence of an uncommon reentrant spinodal similar to that found for other liquids such as or trifluor~methane.~~ The divergence pressures arising from apand Fiirth's model significantly differ at those temperatures at which the Gaussian was used for A ( T ) , although a satisfactory agreement was expected at temperatures above. An analysis in terms of eq 1 of recent aj,data reported by the same authord3 reveals a good agreement with Fiirth's model a t low temperatures (down to 95 K) following the normal trend of the spinodal curve in the van der Waals sense, that is, no reentrant spinodal. The importance of this analysis will be enhanced in next section. Since we are especially interested in giving a realistic representation of A ( T ) and psp(T)in terms of general and simple expressions, the use of polynomials should be avoided. From Alba's treatment,p,,( 7') is directly obtained from eq 4, while the expression for A(7') may be derived from eqs 3 and 4. The temperature dependence of A can be written in the following form:
It is now necessary to test that both eqs 4 and 8 adequately represent (1) the divergence pressures obtained from individual fittings of the experimental ap isotherms to eq 1 and (2) the temperature dependence of A resulting from these fittings. The ability of eq 3 to fit experimental apresults as well as the fitting procedure is widely described in ref 25. The fitting parameters are recorded in Table 3 for the convenience of the reader.
Average Exponent 7 of Eq 2 Obtained from Correlation of Experimental KT at High Pressure of Several Liquids substance
Kr Y BY
ref
-0.95 50
C F4 -0.81 33
CHF,
-0.80 33
cs2 -0.85 51
CH24H2 -0.85 52
Si(CHd4
cyclopentane
2,3-DMB
-0.78 53
-0.85 54
-0.89 17
1996 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 TABLE 3 Coefficients of Eqs 3 and 4 Taken from Ref 25 Determined from the Fit of Thermal Expansion Coefficient a,, of 2,3-DMB.17 pc and T,Are, Respectively, the Critical Pressure and Temperature Taken from Ref 46 0.92
45.1
-19.429
-1.2028
3.13
500.3
TABLE 4: Coefficient A of Eq 1 (Determined from Individual Fits of Experimental a Isotherms of 2,3-DMB17J9) and Eq 8 (Using the Parameters kecorded in Table 3 Obtained from the Fit of All of the a,,Data of 2,3-DMB from 208 to 298 K)17 A / ( 10-3 cm).MPal/2 A/(10-3 cm).MPa1/2 T/K 208.15 238.15 268.15 298.15 373.15 398.15 423.15 448.15 473.15
(eq 1)
(eq 8)
11.50 10.24 9.28 8.53 8.05 7.86 7.66 7.31 7.15
11.41 10.20 9.29 8.57 7.29' 6.97' 6.68' 6.43' 6.19'
,I Extrapolated. 5 ,
0
-5
6
I
I
,
I
I
Psat
...........
i
\ a XlO1
a -15
c
-25 300
I
350
400
500
1
550
T/K 450 Figure 2. Different estimations of the spinodal curve for 2,3-DMB at high temperature. put represents the vapor pressure curve?' and C.P. denotes the critical point."s The numerical results are recorded in Table 1.
As shown in Table 4, an excellent agreement is achieved for A ( T ) when it is compared with those obtained from the ap isotherms individually fitted to eq 1. The results for p,,(T) are recorded in Table 1 and plotted in Figure 1. The agreement with those pspobtained from the individual fittings is excellent, with relative differences of about 2%. These results confirm that eq 3, in addition to correlating experimental apdata over wide ranges of pressure and temperature,25 yields, simultaneously, a physically reasonable estimation of the liquid spinodal curve and an appropriate description of the temperature dependence of the parameter A in eq 1. In order to illustrate the general features of eqs 3, 4, and 8, we have compared in Figure 2 the spinodal curve obtained from the experimental results of Kelso and Felsing19 between 373 K and T, with those calculated from eq 4 with the parameters recorded in Table 3 (calculated only from our apdata between 208 and 298 K). The comparison appears quite satisfactory since extrapolations from very low temperatures are involved. The relative coincidence of the spinodals shown in Figure 2 implies that (a) the crossover of the apisotherms seems to remain invariant with temperature for 2,3-DMB, (b) A(7') exhibits a smooth variation with temperature adequately reproduced by eq 8 (see also Table 4), and (c) the coefficients of the Pad6
Baonza et al. approximant, a1 and a2, can be treated as characteristic constants for each substance and their values are not too influenced by the experimental temperature range considered in their calculation. An interesting conclusion is that eq 3 could be used to predict high-pressure results at temperatures quite apart from the available experimental range. Equation 4 follows the normal spinodal trend along the whole temperature range, i.e., the spinodal line moves to ever more negative pressures with decreasing temperature. This means that its applicability a t temperatures near the triple point is forbidden for those liquids with reentrant ~pinodals.3~36 Reentrant spinodals have been associated with those liquids that exhibit density maxima in their phase diagram, Le., water (in the stable liquid regi0n)3~.35or Si02 (in the supercooled region).36 Water is obviously the most studied example, and the reentrant spinodal concept is supported by the successful interpretation of the properties of water under the basisof thestabilitylimit conjecture given by Speedy.10 It is clear that reentrant spinodals are difficult to find from direct measurements, although some recent experimental31 and theoreticaP4 results suggest their existence. This conjecture has been recently discussed by Sorensen.2' since the simulation results of Poole et a1.22suggest that the behavior of water (including its anomalies) can be also explained under the basis of a normal spinodal curve and a new critical point (between two amorphous phases) identified in the metastable region. At present, there is still enough room to improve the description of the spinodal curve at low temperatures. In any case, for the purposes of this work, even if reentrant spinodals might be a common feature of liquids, the range where eq 3 holds is wide enough to recommend it as an excellent simultaneous representation of the spinodal curve and volumetric behavior of liquids in terms of ap from the critical point down to very low temperatures. Fiirth's Theory. The results obtained from eq 5 are recorded in Table 1 and plotted in Figure 1 together with other predictions. The experimental results of surface tension for 2,3-DMB have been taken from the compilation of Jasper18 and the liquid-vapor coexistence data from refs 17 and 37. Divergence pressures calculated from eq 5 are always slightly lower than those predicted from the analysis of ap and KT. However, the differences (about 2 MPa) are not too significant. At intermediate temperatures results for 2,3-DMB are comparable to those obtained by Pruzan9 for n-hexane at high temperatures and Baidakov and Gurina15 for superheated oxygen in the nearcritical region. The overall results confirm that eq 5 can be recommended to a first approximation for estimating the position of the liquid spinodal curve over a wide range of temperature.' The agreement which follows from our analysis of the aT results of Rubio et al." for tetrafluoromethane a t temperatures below 130 K (see previous section) provides an additional support to this idea even for temperatures relatively close to the normal melting point. At this point, one can speculate about the description of a reentrant spinodal through this theory. It is well-known that there exists a direct dependence between surface tension and density through the so-called parachor P . 3 8 At very low temperatures, the vapor density at equilibrium is negligible so u N P[pas"t]4. Asimple inspectionof eq 5 reveals that thepresence of a maximum in density implies a reversal in the spinodal curve. Unfortunately, precise measurements of density and surface tension in small intervals of temperature are rare in the literature, especially at low temperatures. Then, in most cases, the reversal in the spinodal could not be predicted through this theory. Predictions from Equations of State and PerturbationTheory. As stated before, the spinodal might be estimated directly from any equation of state able to represent accurately both liquid and vapor phases as well as their coexistence region. We have calculated the spinodal from four EOSs which cover the three different types described in the Introduction.
Estimation of the Spinodal Curve for Liquids 120
I
I
I
I
t
401
5
T
=
498 15
go
K
--_-___---
\
il
\ -60 i
i
a -40
\ I T
- 120;
.
-201
= 208 15
c
i K
\
, 2
4
p
/
1
1
6
8
10
moldm-3
Figure 3. Selectedisotherms,spinodal and vapor-liquid binodal calculated
from Deiters EOS" for 2,3-DMB.
Thevan der Waals EOS has been studied for historical reasons as an example of empirical EOSs, although it is known that it predicts a much narrower metastable region than other EOSs and that detected in superheat experiments. The theoretical approach has been introduced by means of the WCA perturbation theory39 applied to the Lennard-Jones (12,6) fluid, with the parametrization of Miyano and Masuokam which includes the optimized random phase approximation (ORPA) of the WCA41 and a correction at the low-density limit by means of the exact second virial coefficient. Two semiempirical EOSs based on the generalized van der Waals model42 have been considered, those of Deiters43 and the so-called BACK" in its new formulation (MOBACK) due to Saager et al.4s Characteristic parameters of the EOSs have been always determined from the experimental critical parameters given in ref 46. For van der Waals EOS the following well-known relations havebeenused: a = (27/64)R2Tc2/pc, b = RTc/(8pc). LennardJones parameters have been determined from recent results of Lofti et a!l7 for critical constants of the Lennard-Jones fluid (NpTplus test particle method): Tc* = 1.310 and pc* = 0.314. Finally, parameters of MOBACK and Deiters EOSs have been taken from ref 17. These two set constants were calculated following refs 48 and 49, respectively. The liquid spinodal has been calculated from the minima of thes-shaped isotherms obtained from each EOS. A representative plot of sev.era1 isotherms is given in Figure 3, which has been obtained for 2,3-DMB from Deiters EOS. The different predictions for the liquid spindoal are plotted in Figure 4 together with that obtained from eq 4 which is supposed to represent the appropriate spinodal as discussed in previous sections. As expected, the van der Waals EOS is not consistent with the other predictions even in the vicinity of the critical point. Both the MOBACK EOS and WCA perturbation theory yield analogous predictions from the critical point down to 350 K, (T/Tc = 0.7). These results are not surprising since the MOBACK EOS is derived from an WCA-type theory. At lower temperatures the Lennard-Jones fluid becomes metastable (broken line) and the WCA prediction departs from that of the MOBACK EOS, which is based on a more elaborated and realistic model. An interesting discussion of results given in Table 1 and observed in Figures 2 and 4 can be made by comparing the reduced temperature Trd = (T/Tc) at which the spinodal reaches zero pressure. Thevan der Waals EOS predicts Trd = 27/32 (=0.84), while superheat limit measurementslJs show that Trd is close to 0.9. Results given in this work agree with this value and predictions from other EOS, since they vary from 0.91 for Deiters EOS to 0.95 by extrapolating eq 4.
1
~
I
1
"
100
I/
ALBA
1
200
1 300
400
500
600
T / K Figure 4. Spinodal curve for 2,3-DMB obtained from different EOS: vdW (van der Waals), WCA: LJ( 12,6) (Weeks-chandler-Andersen
perturbationtheoryappliedtotheLennard-Jones(12,6) fluid), MOBACK (modified BACK EOS), and Deiters. pmt represents the vapor pressure curve,37and C.P. denotes the critical pointa
The most striking feature in Figure 4 is the good consistency with previous spinodals (see Figure 1) obtained from Deiters EOS, which has been also included in Figures 1 and 2 in order to give a more general view of its suitability. Even a t temperatures near critical a good consistency with other predictions is achieved from this EOS (Figure 2). Through this paper we have pointed out that only a complete EOS able to adequately reproduce both liquid and vapor phases as well as their coexistence should give an appropriate description of the spinodal. In previous works we have shown that Deiters EOS reproduces adequately the vaporliquid equilibrium of a great variety of nonassociated substances.17,54,55 In addition, we have recently found that mechanical coefficients at saturation are also well reproduced for 2,3-DMB over the whole temperature range.17 Regarding the description of metastable states, Figure 3 confirms that Deiters EOS gives a smooth variation of the isotherms (without multiple maxima and minima) inside the coexistence region of the phase diagram. At first sight, Deiters EOS seems to be a serious candidate to provide a good description of spinodal curves for real nonassociated liquids. This conjecture is confirmed by the results recorded in Table 1 and those included in Figures 1,2, and 4. Unfortunately, superheat experimental results are not available in the literature for 2,3-DMB and further comparisonsare not possible. However, experimental divergence pressures and densities have been reported for n-hexane,' and we have founds6 that Deiters EOS yields a spinodal consistent with the true13spinodal curve obtained from superheat experiments. As eq 4, Deiters EOS yields a normal spinodal trend, and on the basis of the stability-limit conjecture, discrepancies are expected at low temperatures for those liquids exhibiting density maximum. Fortunately, only a few substances seem to exhibit such a maximum and the general applicability of the EOS is guaranteed. Conclusions
It has been confirmed that both eqs 1 and 2, with exponents y = (-1/2) and y = (-OM),represent adequately both ap(p)and K&) experimental results, yielding a nearly coincident spinodal curve for 2,3-DMB. Our results confirm that eq 3, with only four adjustable and physically significant parameters, must be recommended to correlate experimental apdata over wide ranges of pressure and temperature for those substances presenting intersections of the apisotherms in a narrow range of pressures.25 This equation yields, simultaneously, a physically reasonable
1998
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
estimation of a liquid spinodal curve through eq 4, and through eq 8, an appropriate description of the temperature dependence of parameter A of eq 1. In addition, temperature invariance on the intersections of the apisotherms suggests that eq 3 could be used to predict high-pressure results at temperatures quite apart from the available experimental range. Furthermore, coefficients al and a2 can be treated as characteristic constants for each substance. It has also been confirmed that Furth’s theory (eq 5 ) can be recommended to a first approximation for estimating the locus of the spinodal curve of liquidsover a wide interval of temperature. Finally, we propose that semitheoretical EOS’s based on reasonable and significant physical models may be used to give a successful estimation of the spinodal locus for real liquids. Deiters EOS studied in this work could be a good choice for nonassociated liquids.
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