Examination of the Excess Thermodynamic Properties of n-Alkane

J. Phys. Chem. B 2005, 109, 12145-12153

12145

Examination of the Excess Thermodynamic Properties of n-Alkane Binary Mixtures: A Molecular Approach Marıá Carolina dos Ramos and Felipe J. Blas* Departamento de Fı´sica Aplicada, Facultad de Ciencias Experimentales, UniVersidad de HuelVa, 21071 HuelVa, Spain ReceiVed: February 9, 2005; In Final Form: April 19, 2005

A modification of the statistical associating fluid theory, the so-called Soft-SAFT equation of state, is proposed to predict the excess thermodynamic properties of binary mixtures of n-alkanes. n-Alkane molecules are modeled as fully flexible Lennard-Jones chains. This molecular model accounts for the most important microscopic features of real chainlike molecules: attractive and repulsive interactions between different chemical groups and the connectivity of the segments that form the molecules. In this work we consider an additional microscopic effect that can profoundly affect certain thermodynamic properties, namely, the conformational changes when two different n-alkane molecules are mixed. We propose, following the work of Vega and co-workers [J. Chem. Phys. 1999, 111, 3192], a simple model to account for the conformational changes in molecules. The resulting free energy is combined with the SAFT free energy to describe the excess thermodynamic properties of binary mixtures of n-alkanes. Predictions from the theory are compared with experimental data taken from the literature. The agreement between the experiments and the theoretical predictions is excellent in all cases. This work shows that although minor microscopic effects, such as the conformational changes in the molecules that form the mixtures, have only a very small effect on the usual thermodynamic properties, such as pressure, chemical potential, phase equilibria, and excess volumes, they can contribute significantly to other thermodynamic properties. In fact, one of the main conclusions of this work is that it is essential that conformational effects be taken into account in molecular-based theories if an accurate description of certain excess properties (excess enthalpy for instance) is desired.

1. Introduction The determination of the thermodynamic properties of complex systems, such as hydrocarbons and their mixtures, is currently an active research field not only because of the importance of these properties within the chemical and petrochemical industries but also because of the wide range of phenomena that these systems exhibit. In addition, there is a strong interest in predicting the thermodynamic behavior of these complex systems, including phase equilibria, interfacial properties, pressure-volume-temperature (PVT) and calorimetric data, and excess thermodynamic properties, from a theoretical point of view. Most of the interest in recent years has focused on the phase equilibria behavior of hydrocarbons and their mixtures partially because of the importance of this chemical process design in various fields, including the traditional chemical and oil industries, supercritical fluid extraction, chromatography, surfactancy, and the development of separation and extraction processes.1-3 However, an accurate description of other properties, particularly the excess thermodynamic properties of mixtures, has been given less attention until recently. Usually, the phase equilibria behavior of a system is used to test the accuracy of a theory in predicting its thermodynamic behavior. Although this approach is valid and adequate in most cases, some situations require strong tests to truly determine whether a theoretical approach accurately predicts a given * Corresponding author: F. J. Blas (Tel.: + 34 959219796, FAX: +34 959219777, E-mail: [email protected]).

system. In particular, excess thermodynamic properties serve as a valuable tool for this purpose because they are generally more sensitive to the molecular details than are phase equilibria and can therefore be used to check whether theories are suitable for accurately describing the thermodynamic behavior of a given system.4-6 In this work we examine the excess thermodynamic properties of binary mixtures of n-alkanes using a molecular approach. The n-alkanes constitute one of the most studied chemical families from both experimental and theoretical points of view. The distinct molecular architecture of the n-alkane molecules has presented an interesting challenge for statistical mechanics since the beginning of the development of liquid theories. Different authors have contributed to the understanding of these systems,7 including Prigogine and co-workers8 (on the basis of the previous works of Lennard-Jones and Devonshire9); Flory and co-workers;10-14 Barker and Henderson;15,16 and Weeks, Chandler, and Andersen17 among many others. Although all of these contributions have provided an excellent picture of the liquid state from a theoretical point of view, the most important theoretical developments in the field of flexible molecules in the liquid state have been achieved in the last 15 years since the formulation of the so-called first-order thermodynamic perturbation theory (TPT1) of Wertheim.18-21 The original Wertheim’s theory is a statistical mechanics approach to account for the thermodynamic properties of hard spherical associating fluids. One of the major extensions, proposed by Wertheim,22,23 Jackson et al.,24 and Chapman et al.,25 allows for the use of this theoretical approach to analyze

10.1021/jp0507142 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/26/2005

12146 J. Phys. Chem. B, Vol. 109, No. 24, 2005 chainlike molecules. The statistical associating fluid theory (SAFT) is a formalism based on Wertheim’s theory that, in general, combines a chain reference contribution with an associating perturbation term (for systems with specific interactions, i.e., hydrogen bonding, reactive systems, etc.) to analyze complex-chain and associating fluids. The original SAFT was proposed by Chapman et al.26,27 and has since been extended and used to predict the phase behavior, among other thermodynamic properties, of a wide variety of pure components and their mixtures.28-66 For a detailed description of the different versions of SAFT existing in the literature, we strongly recommend the excellent reviews by Mu¨ller and Gubbins.67,68 More recently, there have been new developments related to the formalism of Wertheim (see refs 69-78 for further details). The goal of this work is to apply the Soft-SAFT equation of state to predict the excess thermodynamic properties of binary mixtures of n-alkanes. The major excess functions of n-alkanes became very well-known in the 1960s and the 1970s.4 The experimental and theoretical interest in these properties stems from the importance of alkanes in the oil industry, from the desire to test the theoretical predictions of the treatment developed by Flory and co-workers10-14 and other authors in the 1960s, and from interest in testing the principle of congruence of Bronstead and Koefoed.4,79-81 In a previous work, hereinafter referred to as paper I,82 one of us used the Soft-SAFT equation of state and computer simulation to determine the major excess functions of binary mixtures of Lennard-Jones chains. Theoretical predictions, in excellent agreement with simulation data, accounted for the most important features of excess volumes and heats of these mixtures. Although results indicated that the theory could be used to predict the excess properties of real n-alkanes, we only concentrated on model systems in that previous work. Independently, Vega and co-workers83 have also examined the excess properties of n-alkane binary mixtures using a perturbation theory based on the work of Wertheim. This work is particularly interesting because they show how the conformational population of a mixture of n-alkanes plays a determinant role in the behavior of the excess heats of n-alkanes. Theoretical predictions of the resulting theory indicate that the excess properties of n-alkanes can be predicted accurately and in excellent agreement with those found experimentally. Here we follow an approach similar to that used by Vega and co-workers and modify the Soft-SAFT theory to investigate the effect of the conformational population on excess properties. This work can be considered as the natural extension of paper I. The rest of the paper is organized as follows: in section 2 we present the most relevant features of the Lennard-Jones chain model and the Soft-SAFT equation of state, including the theoretical modeling of the intramolecular interactions (conformational changes); the results and discussion are presented in section 3, and the conclusions are presented in section 4. 2. Model and Theory In this work we consider n-alkane molecules, which constitute the simpliest homologous series of real molecules that exhibit chainlike molecular behavior. To model these substances, we use the so-called Lennard-Jones pearl necklace model, in which molecules are represented by m Lennard-Jones spherical units, equal in diameter and dispersive energy, tangentially bonded to form the chains. Intermolecular and intramolecular interactions between monomers i and j of different and/or the same

dos Ramos and Blas chain(s) are taken into account through the Lennard-Jones potential model

[( ) ( ) ]

φ(r) ) 4ij

σij r

12

-

σij r

6

(1)

in which σij and ij are the segment size and the dispersive energy between segments i and j, respectively. 2.1 The Soft-SAFT Approach. Because the SAFT approach has been widely used in the literature, here we will just explain the most important features of the Soft-SAFT equation of state for binary mixtures of Lennard-Jones chains. The Soft-SAFT theory, along with other versions of SAFT, is written in terms of the Helmholtz free energy, which can be expressed as a sum of different microscopic effects:56

A Aideal ALJ Achain ) + NckBT NckBT NckBT NckBT

(2)

in which Nc is the number of chain molecules in the system, kB is the Boltzmann constant, T is the temperature, and Aideal is the Helmholtz free energy of an ideal mixture of n chains.4,84 The reference term accounts for both the repulsive and the attractive interactions of the segments forming the chains. ALJ is the Helmholtz free energy of a mixture of spherical LennardJones sites. In this work we use the Lennard-Jones equation of state proposed by Johnson et al.85 The contribution to the chain formation is accounted for in Achain. This term was independently introduced by Wertheim22 (as well as by Jackson et al.24 and Chapman et al.25) from the first-order perturbation theory for associating spherical molecules.18-21 For mixtures of Lennard-Jones chains with bond lengths equal to σii, the diameter of the Lennard-Jones segments in species i, the final expression takes the form n

Achain ) NckBT

xi(1 - mi) ln y(ii) ∑ LJ (σii) i)1

(3)

in which y(ii) LJ (σii) is the contact value of the cavity correlation function for spherical segments of species i in the LennardJones reference fluid.84 y(ii) LJ (σii) is easily related to the pair radial distribution function of the Lennard-Jones fluid, 84 g(ii) LJ (σii). Because we are also interested in mixtures, we use the van der Waals one-fluid theory (vdW-1f) to describe the ALJ term of the mixture, as well as the pair correlation function of the Lennard-Jones mixture.84 The unlike segment size and dispersive energy of the Lennard-Jones reference fluid are given here by the Lorentz-Berthelot combining rules.84 According to the model presented here, the n-alkanes are characterized by three molecular parameters, m, σ, and . Pa`mies and Vega86 have proposed a set of transferable parameters for describing n-alkanes. They obtained these parameters by fitting the experimental saturated liquid density and vapor pressure of the first eight members of the n-alkane series and appropriately selecting the thermodynamic range of the experimental data available. The transferable parameters proposed by Pa`mies and Vega are not only able to describe the phase equilibria of longer n-alkanes outside of the range in which they were fitted, but they are also able to predict the phase behavior of mixtures without any further fitting. 2.2 Excess Thermodynamic Properties. Excess thermodynamic properties are calculated, via standard relationships, as

Thermodynamic Properties of n-Alkane Mixtures

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TABLE 1: VE and HE values for Several Equimolar Mixtures of n-Alkanesa as Obtained from Experiment and the Original Soft-SAFT Approachb mixture

T (K)

VEexp (cm3/mol)

VESAFT (cm3/mol)

HEexp (J/mol)

HESAFT (J/mol)

c6/c10 c6/c16 c8/c16 c8/c24

293.15 293.15 293.15 379.15

-0.2087 -0.4988 -0.1988 -1.3380

-0.29 -0.76 -0.31 -1.56

16.787 12989-91 88.887 -25.181

-25.7 -59.4 -21.9 -177

a c : n-hexane, c : n-octane, c : n-decane, c : n-hexadecane, and 6 8 10 16 c24: n-tetracosane. b The superscripts of the experimental data indicate the references from which the particular data was taken.

the difference between the actual thermodynamic function of mixing and the corresponding value for the ideal mixture.82 Because the goal of this work is to predict the excess thermodynamic properties of binary mixtures of n-alkanes, we have performed preliminary calculations to obtain the excess volume (VE) and excess heat (HE) of several mixtures of n-alkanes at selected thermodynamic conditions. The results, presented in Table 1, indicate that the Soft-SAFT approach provides a reasonable description of the excess volumes of n-alkanes. However, the results given in Table 1 unfortunately suggest that the theory is unable to predict positive values for the excess heat at low temperatures. This point is consistent with the results from Vega and co-workers.83 2.3 The Effect of Conformational Changes. The theory presented in the previous sections, along with other versions of SAFT, neglects the effect the conformational changes in molecules have on the thermodynamic properties, including excess functions. Small conformational changes exist when two different alkanes are mixed.92,93 On the basis of the work of Vega et al.,83 we assume that positive values for the excess enthalpies of n-alkanes at low temperatures are due to conformational changes in the mixture. According to Vega and collaborators,83 the free energy due to the intramolecular interactions of a pure fluid is given by

{

}

Egt Aintra ) (nc - 3) 2xg+ + xt ln xt + 2xg+ ln xg+ NckBT kBT

(4)

in which nc is the number of carbon atoms forming the molecules of the system; xt, xg+, and xg- are the fractions of torsional degrees of freedom in the trans (t), gauche+ (g+), and gauche- (g-) states, respectively; and Egt is the energy difference between the gauche and trans states. In the rest of the work, Egt will equal 3302.2 J/mol. This simple model can be extended to analyze mixtures and can therefore account for the conformational effects. On the basis of the work of Vega and collaborators,83 the free energy contribution due to the conformations can be readily obtained assuming that the conformational population of the components of the mixture is the same:

{

Egt Aintra ) [x1(nc1 - 3) + x2(nc2 - 3)] 2xg+ + xt ln xt + NckBT kBT

}

2xg+ ln xg+

(5)

in which nc1 and nc2 are now the number of carbon atoms of components 1 and 2, respectively, and x1 and x2 are the molar fractions of the mixture. Notice, however, that the fractions of the torsional degrees of freedom in the mixture change with respect to the pure components. According to Vega and coworkers, the conformational population in the mixture is

Figure 1. Vapor-liquid phase equilibria of n-hexane(1)/n-alkane(2) binary mixtures at T ) 308 K. Symbols correspond to experimental data taken from the literature for different n-alkanes, and curves are the predictions from the original theory (Aintra ) 0) and the modified Soft-SAFT approach (including the intramolecular interactions) for the same systems: n-decane (circles94 and solid curves), n-undecane (squares95 and dashed curves), and n-dodecane (triangles96 and dotdashed curves).

expressed in terms of the change with respect to the ideal gas population according to

xg+ ) xgig+ + F(T)Fcx1x2(n2 - n1)1.6

(6)

in which Fc is the chain density of the mixture, and F(T) is an empirical parameter that depends only on temperature (see the work of Vega et al.83 for further details). It is important to mention here that the contribution due to intramolecular interactions affects the pressure and other properties, such as the chemical potential and the free energy, only by a very small degree. In particular, the excess Volumes are not affected by the conformational changes at the level of approximation used here. To show this particular aspect, we have calculated the vapor-liquid phase equilibria of selected mixtures of n-alkanes using both the original Soft-SAFT (Aintra ) 0) and the modified theory (including the intramolecular term presented in this section). Figure 1 shows the phase behavior of several n-alkane mixtures. As can be seen, the results from both versions of the theory are nearly indistinguishable, as expected from the work of Vega and collaborators, and the agreement between the theoretical predictions and the experimental data is excellent in all cases. Values of parameter F(T) have been obtained by fitting the experimental excess heat of the n-hexane/n-hexadecane mixture at different temperatures (from 293 to 373 K). This function can be accurately described by the expression F(T) ) 0.03813028 - 0.005609567 ln T, in which T is expressed in kelvins. 3. Results We have applied the theoretical approach described in the previous sections to determine the excess thermodynamic properties of selected binary mixtures of n-alkanes. In particular, we are interested in the excess volumes and excess heats of different mixtures. As previously mentioned in the introduction, our main interest focuses on the behavior of these two properties when different thermodynamic conditions are varied, such as temperature and composition, as well as when the number of carbon atoms (or chainlength) of one of the components or both of them is changed. In the first part of this section, we consider the excess volume of several binary mixtures of n-alkanes. In the second part of

12148 J. Phys. Chem. B, Vol. 109, No. 24, 2005

dos Ramos and Blas

Figure 2. VE of an n-hexane(1)/n-hexadecane(2) binary mixture at p ) 101325 Pa. Symbols correspond to experimental data taken from the literature at different temperatures: 293.15 K (squares),87 298.15 K (circle),97 303.15 K (diamond),98 313.15 K (up triangle),98 324.15 K (crosses),80 and 333.15 K (left triangle).98 Solid curves are the predictions from the Soft-SAFT equation of state at the same temperatures (the upper curve corresponds to 293.15 K and the lower one to 333.15 K).

Figure 3. VE of n-hexane(1)/n-alkane(2) binary mixtures at p ) 101325 Pa and T ) 298.15 K. Symbols correspond to experimental data taken from the literature for different n-alkanes (the second component of the mixture): n-decane (circles),94 n-undecane (squares),95 n-dodecane (diamonds),97,96 and n-hexadecane (plus).97 Solid curves are the predictions from the theory for the same systems [the upper curve corresponds to the n-hexane(1)/n-decane(2) mixture and the lower one corresponds to the n-hexane(1)/n-hexadecane(2) binary mixture].

this section, we discuss in detail the thermodynamic behavior of the excess heat. 3.1 Excess Volumes. The excess volumes (VE) of different binary mixtures of n-alkanes are presented in this section, with special emphasis on their dependence on temperature and composition. In particular, we are interested in the effect changing the molecular weight of one or both components has on VE. To assess the accuracy of the theoretical approach in predicting the VE of these mixtures, we have also included experimental data taken from the literature for selected mixtures. First, we study the influence temperature has on the VE of a given mixture, namely, n-hexane/n-hexadecane. We chose this mixture because it is probably the one n-alkane homologous mixture for which more experimental data exists in the literature, including VE and HE data, at different thermodynamic conditions. In addition, this mixture was selected by Flory and coworkers10-14 in their seminal work on the thermodynamic properties of chainlike mixtures. Figure 2 shows the VE of this system in a wide range of temperatures (from 293 to 373 K) versus the composition of component 1 (n-hexane). We have also included experimental data taken from the literature to assess the accuracy of the theory in predicting VE properties. As can be seen, VE is negative at all temperatures and nearly quadratic, with the minimum displaced toward the more volatile compound (n-hexane in this case). This property increases in magnitude (becomes more negative) as the temperature is increased. The Soft-SAFT predictions qualitatively describe the behavior observed for the VE of the n-hexane/n-hexadecane binary mixture as it accounts for the most important features of the VE functions. Unfortunately, the theory underestimates the VE values at all temperatures. At this point, it is important to recall that in our calculations we have not used any adjustable parameters; that is, the results from the theory are pure predictions. In fact, the only experimental information is derived from the phase equilibria data of the first members of the n-alkane series (from methane to n-octane), from which the molecular parameters (segment size, dispersive energy, and chainlength) were obtained. In addition, the mixture behavior is obtained from the original Lorentz-Berthelot combining rules without any further fitting procedure.

Because we are interested in the behavior of VE when the number of carbon atoms of one of the components is changed, we first consider the effect of increasing the chainlength of the less volatile compound, which in this case is n-hexadecane. Figure 3 shows the results corresponding to mixtures of n-hexane with longer molecules (but shorter than n-hexadecane). As shown in the figure, VE is negative for all mixtures, showing the same qualitative shape as that of the previous systems studied. This remarkable effect, which seems to be rather singular, is a direct consequence of the liquid structure and packing arrangements in the mixture. In fact, in paper I, one of us demonstrated that mixtures with the same molecular parameters (i.e., segment size and dispersive energy parameters) but different chainlengths, exhibit negative VE values, which can only be ascribed to a packing effect in the liquid mixture structure. As the chainlength of the second component is decreased, VE becomes less negative, and the minimum is displaced toward lower compositions of component 1, which is as expected because the components become more similar. The theory captures the essential features exhibited by these mixtures, although theoretical predictions overestimate the experimental data for all the systems considered here. It is important to recall here that no single adjustable parameter has been used in the theory, and the results from Soft-SAFT are pure predictions. We now consider mixtures in which the second component is fixed and the chainlength of the less volatile component increases with respect to that of the n-hexane/n-hexadecane mixture. In all cases we only consider components lighter than n-hexadecane, as shown in Figure 4. Obviously, the behavior observed is similar to that shown in the previous figure because the main effect of changing the chainlength is to make both components similar. As can be seen, VE is again negative for all compositions, having a nearly quadratic shape. As expected, VE becomes less negative as the molecular weight of the first component is increased. Although the agreement between the experimental data and the theoretical predictions is only qualitative, the theory provides a good picture of the main trends of the VE of n-alkane mixtures of different molecular weights. Finally, we consider very asymmetric mixtures of n-alkanes in which one of the components is fixed (n-octane), and the second one is much heavier. Figure 5 shows the VE of three


Figure 4. VE of n-alkane(1)/n-hexadecane(2) binary mixtures at p ) 101325 Pa and T ) 293.15 K. Symbols correspond to experimental data taken from the literature for different n-alkanes (the first component of the mixture): n-hexane (squares),87,88 n-heptane (circles),87,88 n-octane (diamonds),87,88 and n-decane (triangles).87,88 Solid curves are the predictions from the theory for the same systems [the upper curve corresponds to the n-decane(1)/n-hexadecane(2) mixture and the lower one to the n-hexane(1)/n-hexadecane(2) binary mixture].

Figure 5. VE of n-octane(1)/n-alkane(2) binary mixtures at p ) 101325 Pa and T ) 379.15 K. Symbols correspond to experimental data taken from the literature for different n-alkanes (the second component of the mixture): n-tetracosane (diamonds),80 n-dotriacontane (circles),80 and n-hexatriacontane (squares).80 Solid curves are the predictions from the theory for the same systems [the upper curve corresponds to the n-octane(1)/n-tetracosane(2) mixture and the lower one corresponds to the n-octane(1)/n-hexatriacontane(2) binary mixture].

mixtures of n-octane with longer molecules (up to n-hexatriacontane). Although the same qualitative behavior is observed (as seen in the previous systems), the VE values are larger in magnitude, and the shape of the curves when plotted against the composition is more asymmetric, with the minimum shifted up to x1 ≈ 0.7. As can be seen, the agreement between the experimental data and the results from the theory is remarkable if we take into account the fact that the theoretical calculations are pure predictions. In addition, the only (experimental) input in the theory is the set of molecular parameters optimized for describing the vapor-liquid phase equilibria of short n-alkanes (from methane to n-octane). In summary, the Soft-SAFT equation of state is able to provide an excellent qualitative picture of the behavior of VE in n-alkane binary mixtures under a wide range of thermodynamic conditions. The only experimental information needed is the phase equilibria data of pure short n-alkanes. Then, using a set of transferable rules for the molecular parameters, namely, segment size, dispersive energy, and chainlength, one can obtain

J. Phys. Chem. B, Vol. 109, No. 24, 2005 12149 the thermodynamic properties of longer n-alkanes in a straightforward manner. This allows one to obtain, in a truly predictive manner, the VE of different binary mixtures, including very asymmetric mixtures at different thermodynamic conditions for which parameters have been fitted. The poor agreement between the theoretical predictions and the experimental data for some of the mixtures considered could be somewhat surprising, especially when an accurate molecular approach such as SAFT is used. However, it is important to take into account two points: First, although it has been demonstrated that SAFT is an excellent and accurate theoretical framework in predicting the phase equilibria properties of complex systems, excess thermodynamic functions are extremely difficult to predict because of their low values and their sensitivity to molecular details. In this respect, we think that the theoretical predictions are consistent with the experimental data. Second, our main interest in this work was not to provide the best description of these properties at any price, but to obtain the general trends of the VE of n-alkane mixtures with a minimum amount of experimental information. We think this goal has been achieved because predictions from Soft-SAFT provide a qualitative and correct description of the most important features of n-alkane binary mixtures. 3.2 Excess Enthalpies. Once the theory had proven that it could describe the VE of n-alkane binary mixtures, we applied the Soft-SAFT approach to examine the excess heats (HE) of selected mixtures. According to the discussion of sections 2.3 and 2.4, the original theory is not able to predict, even at a qualitative level, the behavior of the HE of n-alkanes for varying temperature and molecular weight. Hence, we follow the approach developed in section 2.4, previously proposed by Vega and co-workers.83 According to this, a new contribution to the free energy, which accounts for the conformational changes in the mixture, is included in the original free energy expression of the Soft-SAFT theory. As previously mentioned in section 2.4, a new empirical parameter should be introduced into the theory, which essentially accounts for the temperature dependence of the conformational population in the mixture. To determine the F(T) function, we have fit it to the experimental HE of the n-hexane/n-hexadecane binary mixture in a wide range of temperatures (from 293 to 373 K). Once the function F(T) is optimized, it can be used in a transferable way to predict the HE of different n-alkane mixtures without any further adjustment. Figure 6 shows the HE of the n-hexane/n-hexadecane binary mixture at different temperatures. We have also included the corresponding experimental data taken from the literature to assess the accuracy of the fitting procedure of the function F(T) in the whole range of temperatures. The introduction of the new contribution in the free energy, which accounts for conformational changes in the mixture population, allows us to accurately predict the most important features of HE as the temperature of the mixture is varied. As can be seen, HE exhibits a different qualitative behavior as the temperature is increased. At low temperatures, HE is positive and nearly quadratic. At high temperatures, HE becomes negative and more asymmetric, with the minimum displaced toward the rich compositions of the more volatile component. At intermediate temperatures (between 323 and 333 K), HE behaves as a sigmoidal curve, with HE < 0 for mixtures rich in n-hexane and HE > 0 for those rich in the less volatile compound (in this case n-hexadecane). We now follow an approach similar to that used for examining the VE of selected mixtures. We first consider the effect of increasing the chainlength of the less volatile compo-

12150 J. Phys. Chem. B, Vol. 109, No. 24, 2005

Figure 6. HE values of an n-hexane(1)/n-hexadecane(2) binary mixture at p ) 101325 Pa. Symbols correspond to experimental data taken from the literature at different temperatures: 293.15 K (squares),87-91 298.15 K (circle),99 303.15 K (diamond),89 313.15 K (up triangles),81,90 324.15 K (right triangles),81 333.15 K (down triangles),81 349.15 K (pluses),81 and 373.15 K (cross).100 Solid curves are the predictions from the SoftSAFT equation of state (including the intramolecular Helmholtz free energy) at the same temperatures (the upper curve corresponds to 293.15 K and the lower one corresponds to 373.15 K). Dashed curves correspond to the predictions from the original Soft-SAFT (Aintra ) 0). From top to bottom, dashed curves are the predictions from low to high temperatures, respectively.

Figure 7. HE values of n-hexane(1)/n-alkane(2) binary mixtures at p ) 101325 Pa and T ) 349.15 K. Symbols correspond to experimental data taken from the literature for different n-alkanes (the second component of the mixture): n-hexadecane (circles),81 n-tetracosane (squares),81 and n-hexatriacontane (diamonds).81 Solid curves are the predictions from the theory for the same systems [the upper curve corresponds to the n-hexane(1)/n-hexatriacontane(2) mixture and the lower one to the n-hexane(1)/n-hexadecane(2) binary mixture]. Dashed curves correspond to the predictions from the original Soft-SAFT (Aintra ) 0). From top to bottom, dashed curves are the predictions for the n-hexane(1)-n-hexadecane(2), -n-tetradecane(2), and -n-hexatriacontane(2) systems, respectively.

nent at a constant temperature. In Figure 7 we present the HE of mixtures of n-hexane with heavier n-alkanes. As can be seen, the main effect of increasing the molecular weight of the second component is an important increase of the HE: the HE goes from negative values for the n-hexane/n-hexadecane mixture to high positive values for the n-hexane/n-hexatriacontane binary mixture. For the intermediate mixture n-hexane/n-tetradodecane, the HE behaves as a sigmoidal curve, as previously observed in Figure 6. The agreement between the theoretical predictions and the experimental data is quite good if we take into account the fact that no single parameter has been fitted for these mixtures. The F(T) function was fitted for the n-hexane/n-hexadecane

dos Ramos and Blas

Figure 8. HE values of n-alkane(1)/n-hexadecane(2) binary mixtures at p ) 101325 Pa and T ) 293.15 K. Symbols correspond to experimental data taken from the literature for different n-alkanes (the first component of the mixture): n-hexane (squares),87-91 n-heptane (circles),87,91 n-octane (diamonds),87 and n-decane (triangles).87,91 Solid curves are the predictions from the theory for the same systems [the upper curve corresponds to the n-hexane(1)/n-hexadecane(2) mixture and the lower one corresponds to the n-decane(1)/n-hexadecane(2) binary mixture]. Dashed curves correspond to the predictions from the original Soft-SAFT (Aintra ) 0). From bottom to top, dashed curves are the predictions for n-hexane(1)-, n-heptane(1)-, n-octane(1)-, and n-decane(1)-n-hexadecane(2) systems, respectively.

binary mixtures and then used in a transferable way for the rest of mixtures considered in this work. In a similar way, we now consider binary mixtures of n-hexadecane with another component less volatile but heavier than n-hexane to investigate the effect increasing the chainlength of the first component has on HE. As shown in Figure 8, the HE of these mixtures exhibits positive values in the whole range of compositions, with the maximum values of HE centered at x1 ≈ 0.6. The agreement between the experimental data and the theoretical predictions is excellent in all cases, although the Soft-SAFT theory overestimates the excess values of mixtures with longer components. Obviously, as the chainlength of the second component increases, HE decreases because the two n-alkanes become more similar. One can consider the effect of temperature on HE for different binary mixtures from a complementary but similar analysis by calculating the HE of equimolar mixtures of n-alkanes as a function of temperature. We have obtained the values of HE(x1 ) 0.5) for a number of mixtures at different temperatures, and the results are shown in Figure 9a. As can be seen, the HE of n-alkane mixtures, at equimolar compositions, is positive at low temperatures and negative at high temperatures, as expected. The theory also predicts a linear decrease of HE with temperature, independent of the particular mixture considered, as indicated by the experimental results taken from the literature. It is also interesting to compare the behavior of HE versus T when one of the components is fixed and the chainlength of the second one is varied. In particular, we have obtained the HE for the mixtures of n-hexane with different longer n-alkanes. As can be seen, the effect of increasing the molecular weight of the second component is 3-fold: first, the range of temperatures at which HE goes from positive to negative values increases as the chainlength of the second component is increased; second, the slope of HE versus T becomes more prounounced as the molecular weight increases, which is as expected because the compounds of the mixtures become more


J. Phys. Chem. B, Vol. 109, No. 24, 2005 12151

Figure 9. HE values of n-alkane binary mixtures at equimolar compositions in which p ) 101325 Pa. Panel (a) represents HE vs T and panel (b) represents HE vs Tr as defined by eq 6. Symbols correspond to experimental data taken from the literature and curves correspond to the predictions from the theory for different binary mixtures: n-hexane(1)-n-decane(2) (open circles87,94 and solid line), -n-undecane(2) (open squares95 and dotted line), -n-dodecane(2) (open diamonds96 and short-dashed line), -n-hexadecane(2) (triangles81,87,90,99 and long-dashed line), -n-tetracosane(2) (crosses81 and dot-dashed line), and n-heptane(1)- (filled circle87 and thick solid line), n-octane(1)- (filled square87 and thick dotted line), and n-decane(1)-nhexadecane(2) (filled diamond87 and thick dashed line).

different; and finally, the temperature at which HE vanishes increases as the molecular weight of the second component is increased. This complex but understandable behavior can be represented in a more uniVersal way. Recently, Martins101 has found a universal behavior of the temperature at which HE vanishes for different n-alkane binary mixtures, plotting HE against a reduced temperature, Tr, instead of the absolute temperature of the mixture. This new reduced temperature is defined as

Tr )

T (Tc1Tc2)1/2

(7)

in which T is the temperature of the mixture and Tc1 and Tc2 correspond to the critical temperatures of the pure components of the mixture. Following this approach, we have represented the HE of different equimolar mixtures of n-alkanes as a function of Tr defined by the previous equation. As can be seen in Figure 9b, Soft-SAFT predicts a similar behavior for HE(x1 ) 0.5) versus temperature when compared with the results shown in Figure 9a, but with an important difference: the temperature at which

Figure 10. T (a) and Tr (b) as functions of the molar composition of component 1, at which HE(x1) ) 0 and p ) 101325 Pa for different binary mixtures of n-alkanes. Curves correspond to the predictions from the theory (systems and the corresponding legends are the same as in the previous figure) and symbols represent the experimental data taken from the literature for the n-hexane(1)/n-hexadecane(2) (triangle)81 and n-hexane(1)/n-tetracosane(2) (cross)81 binary mixtures. Note that the scales in panels (a) and (b) are different.

HE becomes zero is an approximately universal value (Tr ≈ 0.51-0.52) for all binary mixtures considered. The agreement between the experimental data and the theoretical predictions is excellent if we take into account the fact that no further fitting procedure has been used to calculate HE. Finally, one can consider the effect of temperature on HE for different binary mixtures by calculating the temperatures at which HE(x1) ) 0 as functions of the molecular composition of the mixtures. Figure 10a shows the predictions from Soft-SAFT for several binary mixtures of n-alkanes. This figure includes a more general analysis because we are now considering mixtures with any composition. However, the temperature range is more limited because it is restricted to the temperatures at which HE exhibits a sigmoidal behavior when plotted against the composition. As the temperature of the system is increased for a given mixture (e.g., n-hexane/n-hexadecane), the composition at which HE vanishes moves toward lower values, that is, it is displaced to values corresponding to mixtures rich in the less volatile component. This is the expected behavior because the range of temperatures at which there is a value for x1 in which HE(x1) ) 0 corresponds to the temperatures for which HE behaves as a sigmoidal curve when plotted against x1. In these cases, HE < 0 for mixtures rich in the more volatile component, and HE > 0 for those rich in the less volatile component. As the temperature is increased, HE becomes more negative, and consequently, x1 f 0.

12152 J. Phys. Chem. B, Vol. 109, No. 24, 2005 This behavior is similar for all the mixtures considered here, with an important difference: as the chainlength of one of the components is increased, the temperature values at which HE exhibits a sigmoidal behavior increases. In other words, the results seem to indicate that the main effect of changing the molecular weight of the components of the mixture is to displace the scale of temperatures at which HE has a sigmoidal behavior. The results shown in Figure 9b indicate that the Tr at which HE vanishes exhibits a universal behavior. Here we have generalized the behavior presented in Figure 9a,b by considering the temperatures at which HE(x1) ) 0, expressed in reduced units, as functions of the molecular composition of several mixtures. As can be seen, the Soft-SAFT approach provides an excellent description of the system behavior compared with the experimental data. The results presented in Figure 10b show that all of the curves collapse in a unique and approximately universal curve when the temperature is reduced to the critical temperatures of the pure components. This behavior, which generalizes the results obtained in Figure 9b, should be related to the principle of congruence of Bronstead and Koefoed.79 We are currently investigating the relationship between this behavior and the principle of congruence.

dos Ramos and Blas enthalpic and entropic properties, such as HE, can be profoundly affected by small variations in the conformational properties of a given mixture. This hypothesis is consistent with the results obtained by Vega and co-workers.83 It will be interesting to see whether our main hypothesis could be confirmed via a different strategy. Molecular simulation will be probably the most direct and conclusive approach to clarify this point, demonstrating from a fully microscopic point of view that conformational changes can profoundly affect certain excess thermodynamic properties. Acknowledgment. M.C.dR. would like to thank the Programme Alβan, European Union Programme of High Level Scholarships for Latin America, Identification No. E03D21773VE for funding a studentship. We wish to thank Carlos Vega and Eduardo J. M. Filipe for helpful discussions. We acknowledge further support from Project No. FIS2004-06227-C02-01 of the Spanish DGICYT (Direccioń General de Investigacioń Cientı´fica y Tećnica), as well as additional financial support from X Plan Propio de Investigacioń de la Universidad de Huelva and III Plan Andaluz de Investigacioń de la Junta de Andalucıá. References and Notes

4. Conclusion In this work, we have used a version of SAFT, the so-called Soft-SAFT equation of state, to predict the excess thermodynamic properties of binary mixtures of n-alkane molecules. The molecules were modeled as fully flexible Lennard-Jones chains. Molecular parameters were obtained from a set of transferable parameters that allowed us to predict the thermodynamic properties of any n-alkane. We described the binary mixtures using the vdW-1f theory, and the unlike parameters were obtained from the Lorentz-Berthelot combining rules. The Soft-SAFT approach enabled us to predict the major features of the VE of n-alkanes: negative VE values and nearly quadratic functions with the minimum displaced toward the more volatile compound. As the temperature was increased, this excess function became more negative and more asymmetric. In addition, the main effect of increasing the molecular weight of one or both of the components was to increase the absolute values of VE (i.e., VE became more negative). To accurately account for the HE of n-alkanes, we assumed that positive values of HE at low temperatures were due to conformational changes in the molecules when they are mixed. Using simple assumptions and plausible approximations, we have modified the Soft-SAFT to account for the HE of n-alkanes. The new version of the Soft-SAFT enabled us to predict the most important features of this property: positive values at low temperatures, negative values at high temperatures, and sigmoidal behavior at intermediate temperatures. The agreement between the theoretical predictions and the experimental data was excellent in all cases. We also found that the HE of n-alkanes can be represented in a universal form, defining Tr in terms of the critical temperatures of the pure components. Using Tr, we found that HE vanishes at the same value (Tr ≈ 0.510.52) for all mixtures considered in this work, which is in excellent agreement with the experimental data. The Soft-SAFT equation of state, along with other versions of SAFT, is based on the assumption that intramolecular interactions, particularly the conformational populations of the components of a given mixture, are functions of temperature only and depend neither on density nor on composition. This is a good assumption when analyzing volumetric properties, such as PVT data, VE, and phase equilibria, among others. However,

(1) McGlashan, M. L. Pure Appl. Chem. 1985, 57, 89. (2) McGlashan, M. L. J. Chem. Thermodyn. 1985, 17, 301. (3) Schneider, G. M. Pure Appl. Chem. 1991, 63, 1313. (4) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworth Scientific: London, 1982. (5) Blas, F. J. Mol. Phys. 2002, 100, 2221. (6) Blas, F. J.; Fujihara, I. Mol. Phys. 2002, 100, 2823. (7) Equations of State for Fluids and Fluid Mixtures; Sengers, J. V.; Kayser, R. F.; Peters, C. J.; White , H. J., Jr., Eds.; Elsevier: Amsterdam, 2000. (8) Prigogine, I. The Molecular Theory of Solutions; North-Holland: Amsterdam, 1957. (9) Lennard-Jones, J. E.; Devonshire, A. F. Proc. R. Soc. London, Ser. A 1937, 168, 53. (10) Flory, P. J.; Orwoll, R. A.; Vrij, A. J. Am. Chem. Soc. 1964, 86, 3507. (11) Flory, P. J.; Orwoll, R. A.; Vrij, A. J. Am. Chem. Soc. 1964, 86, 3515. (12) Flory, P. J. J. Am. Chem. Soc. 1965, 87, 1833. (13) Orwoll, R. A.; Flory, P. J. J. Am. Chem. Soc. 1967, 89, 6814. (14) Orwoll, R. A.; Flory, P. J. J. Am. Chem. Soc. 1967, 89, 6822. (15) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 4714. (16) Barker, J. A.; Henderson, D. ReV. Mod. Phys. 1976, 48, 587. (17) Weeks, J. D.; Chandler, D.; Anderson, H. C. J. Chem. Phys. 1971, 54, 5237. (18) Wertheim, M. S. J. Stat. Phys. 1984, 35, 19. (19) Wertheim, M. S. J. Stat. Phys. 1984, 35, 35. (20) Wertheim, M. S. J. Stat. Phys. 1986, 42, 459. (21) Wertheim, M. S. J. Stat. Phys. 1986, 42, 477. (22) Wertheim, M. S. J. Chem. Phys. 1986, 85, 2929. (23) Wertheim, M. S. J. Chem. Phys. 1987, 87, 7323. (24) Jackson, G.; Chapman, W. G.; Gubbins, K. E. Mol. Phys. 1988, 65, 1. (25) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Mol. Phys. 1988, 65, 1057. (26) Chapman, W. G.; Gubbins, K. E.; Jackson G.; Radosz, M. Fluid Phase Equilib. 1989, 52, 31. (27) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29, 1709. (28) Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29, 2284. (29) Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1991, 30, 1994. (30) Galindo, A.; Whitehead, P. J.; Jackson, G.; Burgess, A. N. J. Phys. Chem. 1996, 100, 6781. (31) Galindo, A.; Whitehead, P. J.; Jackson, G.; Burgess, A. N. J. Phys. Chem. B 1997, 101, 2082. (32) Garcıá-Lisbona, M. N.; Galindo, A.; Jackson, G.; Burgess, A. N. Mol. Phys. 1998, 93, 57. (33) Garcıá-Lisbona, M. N.; Galindo, A.; Jackson, G.; Burgess, A. N. J. Am. Chem. Soc. 1998, 120, 4191. (34) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. J. Chem. Phys. 1997, 104, 4168. (35) Galindo, A.; Davies, L. A.; Gil-Villegas, A.; Jackson, G. Mol. Phys. 1998, 93, 241.

Thermodynamic Properties of n-Alkane Mixtures (36) McCabe, C.; Galindo, A.; Gil-Villegas, A.; Jackson, G. Int. J. Thermophys. 1998, 19, 1511. (37) McCabe, C.; Gil-Villegas, A.; Jackson, G. J. Phys. Chem. B 1998, 102, 4183. (38) McCabe, C.; Galindo, A.; Gil-Villegas, A.; Jackson, G. J. Phys. Chem. B 1998, 102, 8060. (39) Galindo, A.; Gil-Villegas, A.; Whitehead, P. J.; Jackson, G. J. Phys. Chem. B 1998, 102, 7632. (40) Galindo, A.; Florusse, L. J.; Peters, C. J. Fluid Phase Equilib. 1999, 158, 123. (41) Galindo, A.; Gil-Villegas, A.; Jackson, G.; Burgess, A. N. J. Phys. Chem. B 1999, 103, 10272. (42) Davies, L. A.; Gil-Villegas, A.; Jackson, G. Int. J. Thermophys. 1998, 19, 675. (43) Davies, L. A.; Gil-Villegas, A.; Jackson, G. J. Chem. Phys. 1999, 111, 8659. (44) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2001, 40, 1244. (45) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2002, 41, 1084. (46) Chapman, W. G. J. Chem. Phys. 1990, 93, 4299. (47) Ghonasgi, D.; Chapman, W. G. Mol. Phys. 1993, 80, 161. (48) Johnson, J. K.; Gubbins, K. E. Mol. Phys. 1992, 77, 1033. (49) Ghonasgi, D.; Llano-Restrepo, M.; Chapman, W. G. J. Chem. Phys. 1993, 98, 5662. (50) Ghonasgi, D.; Chapman, W. G. AIChE J. 1994, 40, 878. (51) Johnson, J. K.; Mu¨ller, E. A.; Gubbins, K. E. J. Phys. Chem. 1994, 98, 6413. (52) Kraska, T.; Gubbins, K. E. Ind. Eng. Chem. Res. 1996, 35, 4727. (53) Kraska, T.; Gubbins, K. E. Ind. Eng. Chem. Res. 1996, 35, 4738. (54) Tang, Y.; Lu, B. C.-Y. Fluid Phase Equilib. 2000, 171, 27. (55) Tang, Y.; Wang, Z.; Lu, B. C.-Y. Mol. Phys. 2001, 99, 65. (56) Blas, F. J.; Vega, L. F. Mol. Phys. 1997, 92, 135. (57) Blas, F. J.; Vega, L. F. Ind. Eng. Chem. Res. 1998, 37, 660. (58) Blas, F. J.; Vega, L. F. J. Chem. Phys. 1998, 109, 7405. (59) Blas, F. J.; Vega, L. F. Fluid Phase Equilib. 2000, 171, 91. (60) Blas, F. J.; Vega, L. F. J. Chem. Phys. 2001, 115, 3906. (61) Blas, F. J.; Vega, L. F. J. Chem. Phys. 2001, 115, 4355. (62) Florusse, L. J.; Peters, C. J.; Pa`mies, J. C.; Vega, L. F.; Meijer, H. AIChE J. 2003, 49, 3260. (63) Dias, A. M. A.; Pa`mies, J. C.; Coutinho, J. A. P.; Marrucho, I. M.; Vega, L. F. J. Phys. Chem. B 2004, 108, 1450. (64) Herdes, C.; Pa`mies, J. C.; Marcos, R. M.; Vega, L. F. J. Chem. Phys. 2004, 120, 9822. (65) Filipe, E. J. M.; Gomes de Azebedo, E. J. S.; Martins, L. F. G.; Soares, V. A. M.; Calado, J. C. G.; McCabe, C.; Jackson, G. J. Phys. Chem. B 2000, 104, 1315. (66) Filipe, E. J. M.; Martins, L. F. G.; Calado, J. C. G.; McCabe, C.; Jackson, G. J. Phys. Chem. B 2000, 104, 1322. (67) Mu¨ller, E. A.; Gubbins, K. E. In Equations of State for Fluid and Fluid Mixtures; Sengers, J. V., Kayser, R. F., Peters, C. J., White, H. J., Eds.; Elsevier: Amsterdam, 2000; Chapter 12. (68) Mu¨ller, E. A.; Gubbins, K. E. Ind. Eng. Chem. Res. 2001, 40, 2193. (69) Llovell, F.; Pa`mies, J. C.; Vega, L. F. J. Chem. Phys. 2004, 121, 10715.

J. Phys. Chem. B, Vol. 109, No. 24, 2005 12153 (70) Jiang, J. W.; Prausnitz, J. M. J. Chem. Phys. 1999, 111, 5964. (71) Jiang, J. W.; Prausnitz, J. M. AIChE J. 2000, 46, 2525. (72) McCabe, C.; Kiselev, S. B. Fluid Phase Equilib. 2004, 219, 3. (73) McCabe, C.; Kiselev, S. B. Ind. Eng. Chem. Res. 2004, 43, 2839. (74) Duque, D.; Vega, L. F. J. Chem. Phys. 2004, 121, 8611. (75) Duque, D.; Pa`mies, J. C.; Vega, L. F. J. Chem. Phys. 2004, 121, 11395. (76) Blas, F. J.; Martıń del Rıó, E.; de Miguel, E.; Jackson, G. Mol. Phys. 2001, 99, 1851. (77) Gloor, G. J.; Blas, F. J.; Martıń del Rıó, E.; de Miguel, E.; Jackson, G. Fluid Phase Equilib. 2002, 194-197, 521. (78) Gloor, G. J.; Jackson, G.; Blas, F. J.; Martıń del Rıó, E.; de Miguel, E. J. Chem. Phys. 2004, 121, 12740. (79) Bronstead, J. H.; Koefoed, J. K. K. Dan. Vidensk. Selsk., Mat.Fys. Skr. 1946, 22, 17. (80) Holleman, Th. Physica 1963, 29, 585. (81) Holleman, Th. Physica 1965, 31, 49. (82) Blas, F. J. J. Phys. Chem. B 2000, 104, 9239. (83) Vega, C.; MacDowell, L. G.; Lo´pez-Rodrıguez, A. J. Chem. Phys. 1999, 111, 3192. (84) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1990. (85) Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993, 78, 591. (86) Pa`mies, J. C.; Vega, L. F. Ind. Eng. Chem. Res. 2001, 40, 2532. (87) McGlashan, M. L.; Morcom, K. W. Trans. Faraday Soc. 1961, 57, 907. (88) Desmyter, A.; van der Waals, J. H. Recl. TraV. Chim. Pays-Bas 1958, 77, 53. (89) Stoeckli, H. F.; Fernańdez-Garcıá, J. G.; Boissonn, C. G. Trans. Faraday Soc. 1966, 62, 3044. (90) McGlashan, M. L.; Morcom, K. W. Trans. Faraday Soc. 1961, 57, 581. (91) van der Waals, J. H.; Hermans, J. J. Recl. TraV. Chim. Pays-Bas 1950, 69, 949. (92) Vega, C.; Lago, S.; Garzoń, B. J. Chem. Phys. 1994, 100, 2182. (93) Padilla, P.; Vega, C. Mol. Phys. 1995, 84, 435. (94) Marsh, K. N.; Ott, J. B.; Costigan, M. J. J. Chem. Thermodyn. 1980, 12, 343. (95) Marsh, K. N.; Ott, J. B.; Richards, A. E. J. Chem. Thermodyn. 1980, 12, 891. (96) Ott, J. B.; Marsh, K. N.; Stokes, R. H. J. Chem. Thermodyn. 1981, 13, 371. (97) Go´mez-Ibań˜ez, J. D.; Liu, C. T. J. Phys. Chem. 1963, 67, 1388. (98) Dıáz Penã, M.; Benı´tez de Soto, M. An. R. Soc. Esp. Fis. Quim., Ser. B 1965, 61, 1163. (99) Larkin, J. A.; Fenby, D. V.; Gilman, T. S.; Scott, R. L. J. Phys. Chem. 1966, 70, 1959. (100) Friend, J. A.; Larkin, J. A.; Maroudas, A.; McGlashan, M. L. Nature 1963, 198, 683. (101) Martins, L. F. G. Ph.D Dissertation, Universidade Tećnica de Lisboa, Instituto Superior Tećnico, Lisboa, Portugal, 1999.

Examination of the Excess Thermodynamic Properties of n-Alkane

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