The Evolution of Multicomponent Systems at High Pressures:  I. The High-Pressure, Supercritical, Gas-Liquid Phase Transition.


J. F. Kenney

Joint Institute of the Physics of the Earth, Russian Academy of Sciences

Gas Resources Corporation,

11811 North Freeway, fl. 5, Houston, TX 77060, U.S.A.;





          The thermodynamic stability of n-octane has been investigated as a function of temperature, pressure, and degree of molecular clustering at supercritical temperatures.  At low pressures, the free enthalpy is shown to be always lowest in the unas­sociated, gas state, and the system is, in that regime, robustly resistant to clustering.  At high pressures, the free enthalpy of the unassociated, gas state exceeds that of the clustered, liquid state.  At the pressure at which the values of the free enthalpies of the gas and liquid states become equal, the system becomes abruptly unstable, and will then spon­taneously cluster into effective “cluster-poly­mers,” and undergo a phase transition to a liquid state.  This phenomenon is a geometric effect, and occurs even at super­critical temperatures.  The gas-liquid phase transition reported here is closely related to the Alder-Wainwright gas-solid phase transition, the onset of which is applied to approximate the optimal clustering parameter.  This phase transition is of the class of entropicly-driven phase transitions, characterized by an increase in spatial order accompanied by an increase in entropy, and manifests an inverted latent heat of transformation, analogous to adiabatic demagnetization.


[Keywords:  theory, chemical equilibria, equations of state, statistical mechanics.]


1.       Introduction:  The van der Waals critical state;  and the Alder-Wainwright gas-solid phase transition at supercritical temperatures and pressures.

          Approximately one hundred, forty years ago, J. D. van der Waals enunciated the first systematic analysis of the phenomenon of gas-liquid phase transitions and the equation of state which bears his name.[1]  From his analysis, and directly from the van der Waals equation, follows the prediction of a critical state at specific values of pressure and temperature at which the density of every gas experiences large fluctuations, and for greater values of temperature the gas cannot condense, regardless of how great a pressure might be applied.  Every known fluid has indeed been observed to manifest a critical gas-liquid state, as predicted by the van der Waals equation, in the thermodynamic regime of modest temperatures and pressures characteristic of the near-surface of the Earth.  Almost all fluids obey also the van der Waals equation of state modestly well in that same thermodynamic regime.  However, it deserves to be noted that the predictions of a critical state by the van der Waals equation obtain from its properties as an equation of third degree in the variable of density.  Similarly, the apparent prohibition of a phase transition at higher values of density is also a consequence of the cubic properties of that equation.  Furthermore, it must be acknowledged that the van der Waals equation is not a fundamental one, nor is it even a phenomenological equation as is, for example, the famous Boltzmann transport equation.[2]  The van der Waals equation is no more than an ad hoc statement based upon assumptions which seemed reasonable in 1870.  The developments of atomic theory and statistical mechanics, have subsequently shown that the successes of the van der Waals equation obtain by accident from its reasonable approximation to general properties of fluids in regimes of dilute densities and low pressures.  Such considerations must compel strong reservations about any suppositive prohibition of a gas-liquid phase transition in thermodynamic regimes far outside the experiences which led to the enunciation of the van der Waals equation.

          Approximately a century later, the first serious, universal failure of the van der Waals prohibition of phase transitions at temperatures higher than that of the critical state was demonstrated by Alder and Wainwright[3-6] in their analysis of the hard-sphere gas whereby they established the existence of a density-dependent phase transition at high pressure and at density approximately 65% of the “close-packed” value.  That transition had earlier been suggested by analysis of the Kirkwood equation[7] which also has no solutions at densities higher than that of the Alder-Wainwright transition.  The statistical mechanical problem of the hard-sphere gas was subsequently examined extensively,[8-19] and its analytical solution obtained in closed form independently by Wertheim[20] and Thiele,[21] using a general formalism developed by Percus and Yevick.[22]  The analysis of the hard-sphere gas applied both by Wertheim and Thiele develops slightly different equations of state depending upon whether the problem is addressed to define explicitly the pressure or the compressibility.  However, both solutions are analytic and both give accurate predictions of the Alder-Wainwright results, falling closely each on opposite sides of the observed values.  Although the Percus-Yevick procedure contains certain inconsistencies, the Wertheim-Thiele solution was later derived independently and consistently in a much simpler fashion by Reiss[11] and his coworkers[23-27] who developed scaled particle theory.  The formalism of scaled particle theory is based upon first principles and contains no adjustable parameters.  A modified equation of state was developed later by Carnahan and Starling[28] which uses a rational combination of the pressure and compressibility solutions from Wertheim and Thiele and the solution from scaled particle theory, and which fits the observed values of density with good accuracy throughout the domain of that variable to the transition value.  The analysis reported here draws upon the extensive literature of the Alder-Wainwright transition, and uses the Carnahan-Starling equation throughout.

          The high-density Alder-Wainwright transition has been interpreted as a gas-solid phase transition.  That transition depends solely upon the entropic density terms which enter the partition function, and is independent of temperature.  For the latter reason, the Alder-Wain­wright gas-solid transition may occur in the supercritical regime.  It is to be noted that neither of the Wertheim-Thiele equations (“pressure” or “compressibility”) predict a gas-liquid phase transition, nor similarly does the Carnahan-Starling equation.  A gas-liquid phase transition is characterized by a vapor-liquid equilibrium line along which the partial derivative of pressure with respect to volume at constant temperature must vanish.  Neither the Alder-Wainwright nor the Wertheim-Thiele equation admits a real solution for that constraint of the vapor-liquid equilibrium line. The principal reason why those equations do not predict a gas-liquid phase transition is because each represents the system solely as a gas phase.  Both the Wertheim-Thiele and the Carnahan-Starling equations obtain from a restricted partition function in which have been summed only those members of the ensemble which pertain to the gas state.  These aspects of the Wertheim-Thiele and the Carnahan-Starling equations are discussed in connection with the supercritical, high-pressure, gas-liquid phase transition in the last section of this article.

          In this article, the chemical potential and resulting Gibbs free enthalpy of the real fluid n-octane are examined as functions of temperature, pressure, and degree of molecular clustering.  At low or modest pressures, the effects of the entropy are proportional to the temperature which is the dominant variable and determines the evolution of the system.  At high pressures, the entropy depends upon the density of the system in a powerfully nonlinear fashion;  and in thermodynamic regimes of pressures greater than approximately 10,000 atm, pressure becomes the dominant variable, and temperature relatively unimportant.  Such results are consistent with the conclusions both of van der Waals and also of Alder and Wainwright.

          At low pressures, the entropy of the system is shown to be dominated by the contributions attributable to the molar abundance, or number of particles, and their random motion, which together may be called the entropy of disorder.  At high pressures, the entropy is shown to be dominated by contributions attributable to the hard-core repulsive terms in the inter-molecular potential, which may be called the entropy of exclusion.  For all temperatures and molar abundances the entropy of exclusion is always negative, and that of disorder always positive.  Therefore, there exists a critical bifurcation pressure below which the system evolves always toward dispersion and above which always toward concentration.

          These properties will be shown in a following article to be universal for all fluids and the causal agents of a high-pressure, supercritical, first-order, gas-liquid phase transition.  Such are responsible also for the phenomena of high-pressure polymerization and agglomeration which will also be discussed in a following article.


2.       The quantal partition function, pressure, and chemical potential of a real gas:  The formalism of the Simplified Perturbed Hard Chain Theory [SPHCT].

          In order to calculate the chemical potentials, and the entropy of a real gas or liquid in arbitrary regimes of temperature and pressure it is necessary to have an explicit, general expression for the partition function of the system.  The formalism of the Simplified Perturbed Hard Chain Theory [SPHCT] generates from first-principles, statistical mechanical argument an analytic expression for the canonical partition function, the derivation of which is independent of any specific regime of temperature or density.[29-33]  Thereby the expressions for the pressure and chemical potential of a fluid developed by the SPHCT are similarly valid in any regime of temperature or density, and hold for dense liquids equally as for dilute gases.[34]  Because the confidence with which any calculation is held depends upon an understanding of the basis for the validity of the formalism from which such was developed, that formalism is here briefly reviewed.

          The SPHCT formalism obtains from the canonical partition function for a single specie a of a real gas or liquid:



where the superscript zero indicates a single-component, pure fluid.  The first two factors in equation  (1) are recognized as the Ideal Gas canonical partition function;  and the second, W0, is the configuration integral representing the molecular interactions.  Following the procedure first developed by Prigogine for the original Hard Chain Theory, HCT,[35] the formalism of the Simplified Perturbed Hard Chain Theory, SPHCT, evaluates the configuration integral in equation (1), W0, by factoring the rotational degrees of freedom and the subset of the vibrational degrees of freedom which are volume-preserving and designated “internal,” from those which affect the translational, or kinetic degrees of freedom, such that:



In equation (2), the parameter c is the Prigogine c-factor which accounts for the volume of phase space occupied by the set of molecular vibrations and rotations which interact with the translational degrees of freedom as a modification of the dimensions of that component of the partition function of the system.

          The SPHCT applies to the configuration integral WSPHCT the decomposition procedure first enunciated by Bogolyubov,[36] later independently by Feynman,[37] and developed further by Yukhnovskii and others,[38-44] which transforms the original 3N spatial coordinates in the configuration integral into 2 sets of, respectively, coupled collective and inde­pendent-particle coordinates, each again of 3N dimensions, which are related by a statistical Jacobian which joins seamlessly those coordinates in the SPHCT partition function.  Such procedure uses an extended phase space to compute simultaneously both the powerful short-range and weaker long-range interactions, and generates a reference system:



          The SPHCT applies for the reference system of the configuration integral in equation (3) the contribution of the hard sphere gas given by the Carnahan-Starling model which has already been discussed in the previous section:



          The second factor in the SPHCT partition function, (3), accounts for both the effects of the long-range van der Waals potential and the quantal statistical Jacobian which joins seamlessly the collective and inde­pendent-particle coordinates in the SPHCT partition function:



The apparent simplicity of the expression for the contribution to the partition function from the van der Waals forces given by equation (5) belies the subtlety of its content and the sophistication of its derivation.[45-51]  To evaluate that contribution to the canonical partition function, the formalism of the SPHCT employs the “shape-inde­pen­dent” approximation to the quantal scattering phase shift first enunciated by Fermi[52] and further developed by Weisskopf,[53] Yang,[54] and others, applied with the spirit of the Bethe-Peierls-Prigo­gine “lattice-gas” model.[35, 55-57]  In equation (5), Zm represents the product of the number of “near-neighbor” molecular interactions and the statistical Jacobian.  As such, Zm is a statistical form-factor[38, 48] which depends only weakly upon the details of the van der Waals the inter-molecular potential, in principle, can be calculated independently of its details.  In this work, Zm is set equal to 18 in accordance with the analysis by van Pelt, Peters, and de Swaan Arons.[58]

          The canonical partition function generated by the SPHCT gives analytic expressions for the pressure, the chemical potentials, and the entropy of a fluid.  The volume derivative of the logarithm of the SPHCT canonical partition function SPHCT formalism gives the pressure:



          Similarly, its derivative with respect to molar abundance develops the chemical potential:



In equation (7), the “thermal de Broglie wavelength,” l = h/Ö(2pmkBT),  where h is Planck's constant and kB Boltzmann's constant;  T is the absolute temperature and m the molecular mass;  t is the close-packing parameter Ö2p/6 » 0.7405, c the Prigogine c-factor,[35] h the “free-volume” parameter h = tnv*/V, and Y the van der Waals fac­­tor Y = exp(T*/2T) - 1, and T* =  /kB where  represents a statistically-weighted mean value of the long-range, attractive, van der Waals intermolecular potential integrated over its effective range.

          It is emphasized that the three parameters in the SPHCT canonical partition function, c, V*, and T*, are not adjustable parameters.  The values of c, V*, and T* can be determined independently by measurement of such as the neutron scattering cross-section,[59] the molecular transport mean-free-path,[60] gas viscosity,[61] the x-ray scattering form-factor,[62] Raman line shapes,[63] or the temperature dependence of molecular spectroscopic absorption lines or of the isochoric specific heat.[64]  Because there are three SPHCT parameters to be determined, they could in principle be obtained from the three equations for the pressure and its first and second volume derivatives at the critical point and the three critical state variables, pc, Tc, Vc.  In practice, the SPHCT parameters are often taken from vapor-pressure measurements combined with critical point data.

          Thus it is seen that the SPHCT partition function bears a formidable provenance of some of the most significant advances in statistical mechanics developed in this century.  Its partition function includes as its reference system the results of one of the few exactly-solvable problems in quantum statistical mechanics, the hard-sphere gas;  and its minor factor contains the solution of the mean-field problem of molecules which interact through a long-range but relatively weak interaction.  The equation of state generated by the SPHCT does not at all invoke adjustable parameters (as do, for examples, the Redlich-Kwong or Soave equations of state), and therefore is valid for use in general regimes of temperature and density.

          For calculating the density and chemical potential of n-octane, the values of its critical temperature, pressure, volume and reference chemical potential have been taken from standard reference tables of the United States Bureau of Standards,[65, 66] and the values of its SPHCT constants from the calculations by van Pelt et al.[58], which are give in Table 1.


Critical parameters

SPHCT parameters


569.4 K


311.449 K


486 cm3/mole


66.001 cm3/mole


24.6 atm








4.29 kcal/mole



Table 1.  Critical and SPHCT parameters for n-octane. [58, 65, 66]


          The contributions to the chemical potential given by equation (7) represent the thermodynamic components of those variables.  The contributions to the chemical potential attributable to the internal rotations and vibrations and to the electronic binding energy must be measured or calculated separately.



The electronic contribution to the chemical potential obtains from the energies of the internal molecular bonds which form the respective molecules and may be accurately assumed to be independent of temperature or density.  The “internal” vibrations and rotations are, by definition, volume preserving and therefore independent of density.  Furthermore, those excitations are typically of relatively high energies corresponding to infrared wavelengths and to characteristic temperatures on the order of thousands of degrees.  Therefore, although contribution to the chemical potential from the “internal” vibrations and rotations has a formal temperature dependence, such can be neglected for considerations of the thermodynamic evolution of systems in the terrestrial regimes of temperature.  With such considerations,




Accordingly, the total chemical potential can be calculated from its measured value at standard temperature and pressure:



The reference values of the chemical potential can either be taken from the tabulated measured values or calculated from that of another molecule for which such value has been measured, using the methods of, for examples, Franklin,[67] or van Krevelen,[68] or Souders, Matthews and Hurd.[69]  Thus the chemical potential, and thereby both the Gibbs free enthalpy and the entropy may be determined for arbitrary density and pressure by combining its reference value with that calculated using the SPHCT.


3.       The pressure, density, and chemical potential of supercritical n-octane in its normal state and as a function of clustering.

Fig. 1 Indication of the different effects upon the molecular covolume diameter, b, of com­­bination by clustering and by chemical bonding.

          In order to study the response of a fluid to configurational changes involving clustering, a “quasi-chemical” approach has been used, in which the system is treated similarly as a chemically reactive system for which the individual molecules combined to form larger, but fewer, entities, as



Unlike the degree of polymerization by ordinary chemical bonding, which is usually limited by the saturation nature of the chemical bond, the degree of clustering has no such restriction, and the combination process for clustering is described more generally as,



where Nc is the clustering parameter.  Furthermore, in the case of combination by clustering for which the bonding is not a chemical one, and there is no activation energy.  However, each clustered  “quasi-polymer” which results from such combination will possess a chemical potential which will in general be quite different from a simple multiple of that of a molecule of the fluid in its original unclustered configuration.

Fig. 2 Examples of the difference in molecular covolume caused by efficient and inefficient clustering.

          In the case of chemical combination, the specific molar covolume (co­vo­lume per mole) of the product is almost al­ways smaller than that of the reagents;  such that for the example given by equation (12), v*(X) > 1/2v*(X2).  Such property is an inevitable consequence of the overlapping of the electronic bonding orbitals of the polymer and the extent to which the nuclei of their respective constituent atoms are drawn together.  However, in the case of combination by clustering, there is no diminution of the internuclear separation, which remains no smal­ler then the distance of closest approach for the free atomic, gas system.  This property is indicated pictorially in Fig. 3 for a simple 2-cluster of spherical molecules.  For a clustered system, the “free-volume” para­me­ter, h = tnv*/V, is given by the new parametric variable, in which the molar specific covolume, vc*, of the clustered Nc-mer is taken as a linear com­bi­ nation of its constituents, hc(nc) = tnc vc*/V, where nc, is given simply by the ratio of the original molecular abundance, n0 , to the cluster index, Nc as required to satisfy the constraints of stoichiometry,  nc n0/Nc , and the molar specific covolume for the cluster characterized by Nc is given by vc* = ncv0*.  This assumption that the covolume of the clustered system is a linear combination of that of the pure fluid overestimates its magnitude for low values of the clustering parameter, even for clusters characterized by “efficient clustering.”  The effect upon the molecular covolume effected by efficient and inefficient packing is indicated pictorially in Fig. 4.  Thus the system is described explicitly in terms of molecular clustering which subsumes overall inefficient packing, short-range order, and increased density, which qualities are pecu­liar in their combination uniquely to the liquid state.


          The Gibbs free enthalpy of n-octane has been calculated at 800oK throughout the range of pressure 1-100,000 atm and as a function of degree of molecular clustering.   Those results are given in Table 1 and also shown graph­ically in Fig. 3.  In Fig. 3, the Gibbs free enthalpy of n-octane is shown both as a pure mono-molecular fluid, designated Nc = 1, and also as a number of pure fluids of fractional molar abundance equal to 1/Nc moles of fluid composed each of “Nc‑clus­tered molecules,” - or Nc-mers.  Thus Fig. 3 is a stability diagram showing the relative thermodynamic stability of the different configurations of n-octane at 800oK as functions of pressure and degree of molecular clustering.  The trace designated Nc = 1 indicates no clustering and is that of the Gibbs free enthalpy of pure n-octane.  The other traces represent the free enthalpy of each partial molar specie of Nc-mers.  The effect of a variation by clustering of the molar abundance at constant temperature upon the free enthalpy of a fluid specie is seen to have the property that, at low pressures in the kinetic regime where the dilute, or ideal gas, charac­ter­is­tics of a fluid dominate its thermodynamic behavior, a decrease in molar abundance causes an increase in the free enthalpy.  However, at elevated pressures typically greater than 10,000 atm, the entropic hard-core characteristics dominate the free enthalpy,

Fig. 5. Gibbs free enthalpy of n-octane at 800 K as functions of pressure and degree of molecular clustering, Nc.

and a decrease in molar abundance causes a commensurate decrease in that thermodynamic potential.  Thus at high pressures, the effects of diminished molar abundance manifest oppositely their effects at low pressures, by decreased free enthalpy with decreased abundance.


          The exclusionary effects of the “free-volume” parameter are manifested strongly at high pressures and densities, and the effect of an increase in that parameter, acting through the third-order singularity in the pressure, chemical potential, and Gibbs free enthalpy, is to increase rapidly the values of those respective thermodynamic variables.  There­fore, the effect of overestimating the magnitude of the specific covolume would be to overestimate also the magnitudes of the chemical potentials of the components to which such approximate modeling of the liquid state by clustering might be applied.  Consequentially, if the analysis were to show that such clustering, as calculated overestimating the magnitude of the specific covolume, were to produce no qualitative change in the evolution of the system, - and particularly no phase transition, - no compelling conclusion could be drawn from such results.  However, as clearly shown to the contrary in Table 2 and Fig. 3, at high pressures and densities, the fluid system strongly favors clustered states.

Fig.  6 Gibbs free enthalpy of n-octane at 800 K as functions of pressure and degree of clustering.

          The Gibbs free enthalpy of several clustered states of n-octane are shown graphically in Fig. 4 for the pressure range of 10,000-50,000 atm on a linear scale.  In Fig. 4, may be noted explicitly that, at high densities and with increasing pressure, phases characterized by increasingly smaller degrees of clustering become preferentially stable contrasted to the normal mono-molecular phase.  Such property demonstrates clearly the powerful predisposition of the fluid system to reconfigure itself at high densities into the largest cluster possible consistent with the liquid state and limited by the onset of the Alder-Wainwright solid state.  In Fig. 4,  the trace for the Gibbs free enthalpy of the clustered phase consisting of 1/24 m of Nc = 24 -mers extends only slightly beyond the pressure at which it intersects that of the pure gas phase of mono-molecular n-octane.   As shown in Fig. 3, the trace for the Gibbs free enthalpy of the very highly clustered state corresponding to clusters of 240 individual molecules does not extend to the trace for that of pure n-octane.  The reason for the termination of that trace is that the fluid consisting of an abundance of 1/240 mole of “240-mers” passes the Alder-Wainwright transition to a solid at a lower pressure, approximately 1,000 atm, but at a higher free enthalpy than that of the n-octane gas at that same pressure.  Having once passed the Alder-Wainwright transition into the equilibrium state characterized by the solid phase, a system possesses no memory of its previous gas phase.  N-octane does not achieve that solid phase below pressures of approximately 112,000 atm at 800 K.  Therefore the system corresponding to 240-mer clusters, which undergoes the Alder-Wainwright transition at a much lower pressure, must be considered as only a “virtual,” or metastable state.  Because the magnitude of the Gibbs free enthalpy at the transition pressure does not depend sensitively upon the degree of clustering for values of Nc > 10, the optimal degree of clustering may be reasonably assumed to coincide with the number of nearest neighbors at close packing.  Thus the optimal value for Nc should be 12 if the constituents of the cluster comprise one nearest neighbor and approximately one next-nearest neighbor.  This model of a liquid is supported by the experimental measurements of the two-particle correlation function of the liquid state in such diverse materials as argon[70] and the liquid metals sodium, potassium, in­dium,[62] and rubidium.[71] Those investigations all show three peaks in the x-ray scattering form-factor, which cor­res­pond to one coordination shell, which strongly suggests that the extent of clustering involves a single coordination shell of nearest neighbors at very nearly the close-packing distance.  The choice of Nc = 12 as the optimal value of the clustering parameter is supported also by the experimental work of Bernal and Mason on clusters of metal spheres at high densities.[72]


4.       The change in volume; the first-order, gas-liquid phase transition.

          A first-order gas-liquid phase transition should manifest a discontinuous volume change at the transition temperature or pressure.  Therefore, the volumes of n-octane have been calculated in both the gas and liquid phases as functions of pressure as a pure mono-molecular fluid, and also as a series of pure clustered fluids composed each of Nc “clustered molecules,” - or Nc-­mers, - and are given in Table 3 and shown in Fig. 5 on double-log scales.  The respective volumes are de­sig­nated

Fig.  7.  The volume of n-octane at 800 K as a function of pressure and degree of clustering , Nc.

by their molar abun­dance, Nc.  In Fig. 5, the volume of one mole of pure n-octane is again represented by the trace designated, Nc = 1, which represents the mono-molecular gas phase with no clustering.  Also shown in Fig. 5 are the representations of the ideal-gas volume for one mole at that temperature, VI.G, and the close-packed volume of n-octane, Vcp = 48.87 cm3, which appear as straight lines.

          As may be observed in Fig. 5, at low pressures the volume of each respective “clus­ter-polymer” gas is determined by its molar abun­dance, and accordingly each volume diminishes inversely with the degree of clustering, nc.  As also to be observed in Fig. 5, at high pressures each volume becomes increasingly less sensitive to pressure and approaches the close-pack volume limit in a “quasi-asymptotic” manner.  The ideal-gas volume has, of course, no lower limit.  At sufficiently high pressure and density, the fluid will undergo the fluid-solid phase transition to a close-packed system first described by Alder and Wainwright.  Inevitably, the volumes associated with each increasing degree of clustering are monotonically smaller because volume is an extensive variable, as the molar abundance depends inversely upon the degree of clustering, Nc.

Fig. 8.  Clausius diagram for n-octane at 800 K in its normal gas phase and clustered liquid phase for which Nc=12.


          As the preceding analysis has shown, there exist a series of pressures at which the Gibbs free enthalpy of the gas state is equal to that of the clustered liquid state, and those “crossing” pressures decrease with increasing extent of clustering, with a negative second derivative of that decrease indicating the approach to an optimal clustering parameter.  For the reasons given in sec tion 3, the configuration for which Nc = 12 is taken as a reasonable approximation to the optimal value of the clustering parameter.   At the supercritical temperature of 800 K, a mole of n-octane will be initially in the gas phase and will manifest monotonically diminishing volume with increasing pressure.  At the approximate pressure p = 20,283 atm, the Gibbs potential for both the unclustered gas and the clustered, liquid state of n-octane, for which Nc = 12, will be 53.45 kcal.  The volume of the unclustered, “normal” fluid is, at that pressure, approximately 100.2 cm3, with a density of 1.14 g/cm3, and those variables for the clustered, liquid are, respectively, approximately 68.9 cm3, and 1.66 g/cm3.  Therefore, as the volume of the sample is compressed further at 800 K and 20,282 atm from its initial value of 100.2 cm3, increasing amounts of gas, at the density of 1.14 g/cm3, will pass into the liquid phase at the density of 1.14 g/cm3, until the volume of 68.9 cm3 is reached;  at which volume the n-octane will be entirely in the clustered, liquid phase.

          This behavior of n-octane at 800 K is shown graphically in the Clausius diagram given in Fig. 6.  Such is the ordinary description of a gas-liquid phase transition.  However, during the course of compression and passing from the gas to the liquid phase, while the temperature, pressure, and chemical potentials of the two phases will have been equal, the mole of n-octane will have absorbed approximately 15.4 kcal of heat, as will be shown in the following section.  This latter property of possessing an inverted latent heat of transformation, is very unlike the ordinary, low-pressure, gas-liquid phase transition and is characteristic of adiabatic demagnetization.


5.       A test for phase separation:  The change of sign of the isothermal compressibility of the metastable, two-component, single-phase system.

          The foregoing analysis has established that, at high pressures, the Gibbs free enthalpy of the normal mono-molecular gas exceeds that of an assemblage of clustered molecules, and that the system becomes then abruptly unstable against clustering.  The clustered system has been shown to occupy a smaller volume at all pressures above the bifurcation pressure.  The rapidity of divergence of the respective Gibbs free enthalpies of the normal and clustered states, as shown in Figs. 3 and 4, together with the break in their respective densities shown in Fig. 5, argue strongly for phase separation.  Gas-gas separation into distinct phases is a well-known phenomenon at high pressures, particularly for mixtures of gases composed of com­ponents which differ much in the magnitudes of their respective molar covolumes.   An additional argument for phase separation obtains from a change of sign of the isothermal compressibility of the (hypothetical) metastable, two-component, single-phase system.  The value of density at which the isothermal compressibility becomes negative determines the point at which the pressure-density curve of the single-phase system intersects the spinodal, and at which the single-phase system becomes absolutely unstable against phase separation.  Of course, the compressibility of the real system is always positive;  the negative property of that thermodynamic function for the imaginary system in the unstable state in the absence of phase separation is a mathematical artifact.  However, the change of sign of the isothermal compressibility is a useful test to examine the absolute instability of the metastable single-phase state.

          There are two different volumes to be considered in connection with the isothermal compressibility of the two-component, single phase system:


where in second equation (13), the partial volumes are given by:


and similarly for the partial volume v2.  For the binary, single-phase system, the variance of the volume with respect to pressure is given by:



such that the isothermal compressibility is given by:



In a simple, non-reactive system, or one which does not undergo clustering, the terms in the first bracket on the right of equation (16) are zero;  for, in such systems, the abundance of each component is fixed.  However, in a chemically reactive or clustering system, the relative abundance of each component is determined by their ratio which minimizes the Gibbs free enthalpy, subject to the constraint of stoichiometry:


In equation (17), n0 is the total number of individual molecules in the system which remains fixed, such that



The partial derivatives in equation (16) of component abundances with respect to pressure are determined by applying the Gibbs relationship,



Therefore, by introducing equations (18) and (19) into the equation for the isothermal compressibility, the sum in the first square bracket of equation (16) becomes,



The volume derivative of the chemical potential has a fourth-order singularity in its free-volume parameter, and its inverse, , is a well-behaved positive function, monotonic at high density.  Therefore, a change in sign of the isothermal compressibility at high pressures for a mixture of a clustered and “normal” molecules must result from the factor (-Ncv1+v2) admitting a real root for some value of the cluster abundance n2, or the abundance ratio n1/n2.  That the factor (-Ncv1+v2) involves two terms of different signs suggests that it admits a real root, and that indeed it does is shown here.

          In order to analyze fully the metastable, two-component, single-phase system, the SPHCT formalism for mixed systems must be used, including particularly for the reference system, the equations from scaled particle theory for mixed systems.  Such will be done in a following article.  For the moment and because those equations are much more complicated, here follows an abbreviated argument which is only approximate but which includes the essential features responsible for the change of sign of the isothermal compressibility, and thereby for also the absolute instability of the two-component single-phase system against phase separation.

          For a multicomponent system, the fundamental equation of state given by the SPHCT which is the generalization of that for a single-component system, equation (6), becomes:



in which the factor, Z({ni},V,T) is given by:



In equation (22),




and the mean molecular radius and the SPHCT molar covolume are related as Ö2vi* = Ri3.  Because the contribution to the pressure attributable to the van der Waals component of the intermolecular potential is, at high pressures, at least an order of magnitude smaller than such attributable to the hard-core component, the van der Waals terms will be neglected.

          The SPHCT gives analytic expressions for the partial volumes and chemical potentials.  When the van der Waals contributions in equation (22) are neglected, the partial volume of the i-th component obtains as:



(In equation (24) and the following, the partial volumes, vi, which are thermodynamic variables, must not be confused with the SPHCT molar covolumes, vi*, which are fixed molecular parameters.)  For the factor (Ncv1-v2) in equation , the circumstance that Ncv1* = v2* causes cancellation of the second, fourth, and last terms.  The first terms in equation (24) give to equation (20) the difference (V/n)(Nc-1), which is always positive.  However, at high pressures, the differences of those first terms will be much smaller than those of the fourth term, the magnitude of which is dominated by its third-order singularity.  The difference in the fourth terms is:



Similarly, the difference of the third terms is,




Thus the factor in equation (20) becomes,



At low pressures, the leading term on its right side, (Nc-1)V/n, will dominate equation (27).  That term is positive for all values of Nc > 1, and the isothermal compressibility will therefore be then positive.  At high pressures, the right side of equation (27) is dominated by the terms in the square bracket which contain factors strongly singular in the free-volume parameter, x.  This term is factored by the quantity (n0 - 2Ncn2) which admits the real root (n2)s = n0/2Nc;  and for all values of n2 > (n2)s, the isothermal compressibility of the metastable single-phase system becomes negative.  The value of the root (n2)s determines the point of intersection of the pressure-volume curve for the single-phase system with that of the spinodal which delineates its absolute instability against phase separation.  Thus the intersection with the spinodal occurs when a minority of the molecules equal approximately to half the clustering fraction have undergone clustering;  for a system characterized by 12-clusters, such would be at roughly n0/24, a quite reasonable result.

          The preceding argument is a reasonable, but not a precise, one.  The effects of the weak van der Waals component to the intermolecular potential have been neglected, as have also such of the differing magnitudes of the Prigogine c-factors and of cluster packing efficiency.   Such effects must influence to some small extent the value of the spinodal intersection, but they cannot eliminate its existence, which is assured by the fourth-order singularity in equation (27).


6.       The change in enthalpy;  the negative latent heat of transformation.

          The latent heat of transition which accompanies the abrupt volume and density change at the gas-liquid phase transition is calculated directly from the change in enthalpy between the two phases:



From the analytic expression given by the SPHCT for the canonical partition function, the internal energy of the system is given as a sum of four parts:



Because the contribution to the internal energy from the electronic bonding energy, U(V,T,n), is independent of density and temperature, (DUIint.)gas-liquid = 0.  Also, because the effective ideal gas contribution to the internal energy is a function of temperature only and independent also of density, (DUI.G.)gas-liquid = 0.  The terms in the SPHCT partition function attributable to the hard-core component of the intermolecular potential are purely entropic and do not contribute to the internal energy;  Uh-c = 0.  Therefore the only energetic contribution to the internal energy obtains from the weak, long-range van der Waals potential, UvdW:



The absolute value of the van der Waals component of the total pressure is, at pressures greater than approximately 2,000 atm, more than an order of magnitude smaller than the hard-core pressure component.  (The van der Waals component of the pressure is always negative.)  From equation (30), the difference in the van der Waals component to the internal energy between the gas and liquid states goes as



Therefore, the change in enthalpy attributable to the high-pressure, supercritical, gas-liquid phase transition may be calculated from equation (28) as



          The change in the enthalpy of n-octane in passing from the gas to the liquid state at the temperature 800 K, calculated from equation (32) is +15.4 kcal/mole.  Two properties of the high-pressure, gas-liquid phase transition are particularly significant:  the magnitude of the change in enthalpy;  and its sign.  The magnitude of the latent heat of transformation for the gas-liquid phase transformation is remarkably large;  by comparison, the latent heat of trans­for­mation for the associated water molecule in passing from the gas to the liquid state is only 0.54 kcal/gm[73];  and the latent heat of the normal gas-liquid phase transition of hydrocarbons, which are non-associated molecules, at modest pressures is approximately 0.07-0.15 kcal/gm[73].  The sign of the change of enthalpy in passing from the gas to the liquid phase establishes that the transition is endothermic, - i.e., with an inverted latent heat of transformation such that it releases heat upon passing from the liquid to the more disordered, gas phase.  In such respect, the high-pressure, supercritical, gas-liquid phase transition bears similarity to adiabatic demagnetization.


7.       Discussion.

          As set forth at the beginning of section 3, the stability and evolution of n-octane has been examined as a function of molecular clustering by applying the “quasi-chemical” perspective which for which the effective process is described as:


Fig.  9 Components of the chemical potential of n-octane at 800 K as functions of pressure.

in which the clustering parameter, Nc, specifies both the (different) molecular specie as well as the molar abundance of the product.  It is seen that the nonlinear response of the Gibbs free enthalpy to variations in the clustering parameter results in that thermodynamic potential becoming smaller at high pressures and densities for the clustered states than for the normal, monomolecular gas state.  Such result is a consequence of both the dominance of the hard-core contribution to the chemical potential at high pressures and densities, and the sensitivity of that function to the magnitude of the covolume.

          The manner in which the hard-core component of the chemical potential dominates that function at pressures grater than approximately 5,000 atm is shown graphically in Fig. 7 where are represented the three terms from equation (7) of the components of the chemical potential throughout the range of pressures 1-100,000 atm.  From Fig. 7, it is clear that the hard-core component determines the chemical potential of the fluid at high pressures;  at pressures above approximately 5,000 atm, that component is at least an order of magnitude greater than either the ideal gas or the van der Waals component.

Fig.  10 Chemical potential of n-octane at 800 K as a function of pressure and magnitude of its covolume.

          The sensitivity of the chemical potential to the magnitude of the covolume is shown graphically in Fig. 8 where is plotted the behavior of the chemical potential of n-octane as a function both of pressure and of the magnitude of its covolume, v0*.  In Fig. 8, it is observed that small changes in the covolume have little effect upon the value of the chemical potential at pressures below approximately 1,000 atm but work large effect at high pressures.  Relatively large changes in the covolume, factors of 8-12, begin to effect the chemical potential at pressures as low as 200 atm.


          Understanding of the high-pressure gas-liquid phase transition obtains from recognizing the interplay between the different mechanisms for, respectively, the behavior shown in these last two figures.  The Gibbs free enthalpy is an extensive function and is seen clearly in Fig. 7, to be positive at high pressures.  Therefore at high pressures, if the system could configure itself so as to have a smaller molar abundance, with no other effects, the magnitude of its Gibbs free enthalpy would be lessened.  (To be emphasized is that fact that this property does not depend upon any arbitrary scale of energy which might be used to set the value of the chemical potential at STP, for the thermodynamic stability of the fluid depends always upon the difference between the free enthalpies of the respective configurations.)  However, from Fig. 8, it is seen equally clearly that, at high pressures if the system could configure itself to have a smaller covolume, with no other effects, the magnitude of its Gibbs free enthalpy would also be lessened.  Thus a fluid system at high pressures would like to have, simultaneously, both the smallest possible abundance and covolume.  However, the requirement of stoichiometry prohibits a reduction of molar abundance except by processes of chemical combination or chemical polymerization or clustering;   and the clustering process is restricted such that the distance of closest approach of the particles in a cluster is never smaller than the distance of closest approach in the gas state, which constraint inevitably dictates that a decrease in molar abundance must be accompanied by an increase in specific molar covolume.   The thermodynamic stability of a real fluid at high pressures is therefore the consequence of competition between these two opposing tendencies:  the tendency to reduce its Gibbs free enthalpy of the fluid by reducing its abundance by clustering, - while (unavoidably) increasing its molar covolume;  and the tendency to reduce its Gibbs free enthalpy by reducing the magnitude of its covolume by un-clustering, - and while thereby increasing its molar abundance.  The behavior of the chemical potential as a function of the magnitude of the system’s specific molar covolume at high pressures, represented in Fig. 8, taken together with the investigations by Gibbons of the behavior of that potential as a function of molecular shape, provide strong argument for the choice of optimal clustering parameter.[74]  Gibbons has shown that the chemical potential is substantially higher for all systems which are not spherical and that, for any system of given covolume, the Gibbs potential will be lowest for a system of spherical molecules.    These two properties taken together argue strongly that the clustering will occur optimally for a cluster which assumes the close-packed configuration in the most nearly spherical form, which in turn indicates a clustering parameter Nc = 12, which has been used.


          The high-pressure gas-liquid phase transition may be understood directly by applying to quasi-chemical perspective the second law of thermodynamics.  The direction of the evolution of a system is determined by De Donder’s inequality:


in which the x is the variable of extent and the thermodynamic Affinity, A, is defined in terms of the stoichiometric coefficients and chemical potentials of the components and phases as



the sum being taken over all components and phases.  In the present analysis for the quasi-chemical, clustering reaction (33), for which the clustering parameter is taken as Nc = 12, the stoichiometric coefficient of the normal, mono-molecular reagent -1 and that of the clustered product is 1/12.   Because the present analysis has been carried out considering one mole of the reagent, normal, mono-molecular n-octane, the thermodynamic Affinity may be calculated directly from the free enthalpies of the two phases.  Below the transition pressure, the thermodynamic Affinity for the “quasi-chemical” reaction (33) is always negative, and the system is, in that regime of pressure,  robustly stable against clustering.  Above the transition pressure, that thermodynamic Affinity is always positive, and the second law of thermodynamics, as expressed by De Donder’s inequality, dictates that the system must then transform into the clustered state.

          The choice of n-octane as the molecule of investigation has been intentionally arbitrary and was taken primarily because of the availability data on that compound.  For reason of generality, a material was chosen which is neither an ideal gas nor approximately spherical.  However, the particular choice of n-octane was perhaps not best for reason of its thermal and compositional instability.  At the temperature applied for this analysis, 800 K, not only is n-octane thermodynamically unstable, at all pressures, but also the kinetic rate coefficients are sufficiently high that it will react and reorganize itself fairly quickly:  At low pressures, n-octane at 800 K will decompose into methane and amorphous carbon;  and at high pressures, it will resolve itself into a hydrocarbon system with the characteristic Planck-like distribution of molecules by carbon number.  Therefore, to a certain extent these calculations on n-octane might be considered as a model analysis.  With such perspective, considerations such as possible variation of the SPHCT parameters with density may be set aside, at least initially.  Although neither the SPHCT parameter of molar covolume, v*, nor Prigogine c-factor, c, should be expected to vary much at high densities, for each is typically determined for a component in its liquid or dense phase, a change of either would not alter the qualitative aspects of the high-pressure, supercritical, gas-liquid phase transition.

          Caution:  Calculation of the relevant Chapman-Jouguet parameters has established that, at 800 K and high pressures, n-octane (and the hydrocarbon system into which it resolves) is highly brisant, with detonation qualities comparable to TNT.


          The present analysis may be considered to represent the third generation of the modern studies of fluids at high densities, and builds upon the advances made previously.  The first generation involved the studies by Alder and his colleagues of the hard-sphere fluid.  Those studies were initially computational investigations, which employed both the monte carlo and molecular dynamics techniques, and which determined the famous Alder-Wainwright high-density, gas-solid phase transition.  The second generation involved the analytical work by Percus, Yevick, Thiele, Wertheim, Reiss, Lebowitz, Frisch, Carnahan, Starling, and others which resulted in the exact, analytical solution of the statistical mechanical problem of the hard-sphere gas.  The present analysis has now taken that analytical solution and applied it to a computational investigation of the real fluid, n-octane, at high pressure and for a series of dense configurations, in the spirit of the original Alder-Wainwright investigations but enhanced by the availability of the analytical solution.  The present analysis has effectively used a more general partition function which includes clustered states with the characteristics of liquids, and has determined that certain of those states describe a phase which possesses, at sufficiently high pressures, a Gibbs free enthalpy which is lower than the normal, monomolecular gas state, a density which is greater, and a large, negative, latent heat of transformation between the phases.


          The high-density, supercritical, gas-liquid phase transition is an example of the class of phenomena designated entropically-driven phase transitions.[75]  Probably the best-known example of an entropically-driven phase transition is the Alder-Wainwright gas-solid phase transition of a hard sphere gas, discussed in the first section.[5, 6, 76, 77]  Other examples of entropically-driven phase transitions include: gas-gas demixing, first predicted by Krichev­skij,[78], and subsequently analyzed by Tsiklis,[79] and other workers,[80-82];  retrograde exsolvation,[83-85];  and the evolution of superlattices in hard-sphere mixtures.[86-88]  The high-density, gas-liquid phase transition has recently been observed experimentally in a physical analogue of polystyrene spheres suspended in a liquid of equal specific gravity as reported by Dinsmore et al.[75, 89, 90].



          The author expresses specific gratitude to his colleagues of the Russian Academy of Sciences at the Joint Institute of the Physics of the Earth for their encouragement, their critical review, and their unstinting generosity in offering guidance and detailed knowledge of the many facets of the physics of multicomponent systems in regimes of high density.  Particular gratitude goes also to Professor V. A. Krayushkin of the Institute of Geological Sciences of the Academy of Sciences of Ukraine in Kiev for his knowledge of the significance of the results of this analysis for problems of Earth sciences and petroleum.  The author is particularly grateful also to Professor Kerson Huang of the Massachusetts Institute of Technology and Professor Howard Reiss of the University of California at Los Angeles for their critical reviews and discussions of this work.  Special gratitude is owed to Professor Reiss for having brought to our attention the recent work by Dinsmore, Kaplan, Pine, Yodh, et al. who observed experimentally the high-density gas-liquid phase transition in a physical analogue of a high-pressure, supercritical fluid.

          This work was supported by the Russian Academy of Sciences and, in part, by Research Grant 6-187-1 from Gas Resources Corporation, Houston.


Nc = 

List of Symbols:

A                 thermodynamic Affinity

c                  Prigogine c-factor

h                 Planck’s constant

kB                Boltzmann’s constant

m                molecular mass

NA               Avagadro’s constant

N                 number of molecules

Nc                clustering parameter

n                 number of moles

p                 pressure

Q                canonical partition function

q                 independent factor in canonical partition function

R                 universal gas constant

Ri                scaled particle theory effective radius of i-th component, = (Ö2vi*)1/3

T                 absolute temperature

T*               SPHCT characteristic temperature

V                 volume

V*               SPHCT characteristic volume

X                 scaled particle theory function, = (p/6)(SniRi2)/V

Xi                representative molecular reagent

vi                 partial volume, = (V/ni)

v*                SPHCT specific molar covolume

Y                 SPHCT van der Waals factor

Zm               quantal Jacobian joining collective and independent-particle coordinates » number of interacting neighbors

Greek letters

b                 Boltzmann factor, = (kBT)-1

                  statistically-weighted, integrated mean value of long-range, attractive van der Waals potential

l                 thermal de Broglie wavelength

h                 SPHCT free-volume parameter

m                 chemical potential

t                  close-packing parameter, = Ö2p/6


cp               close-pack

I.G.             ideal gas

hc                hard core

j                 phase

SPHCT       Simplified Perturbed Hard Chain Theory

vdW            van der Waals

0                 pure component


c                 critical

electronic     electronic

i                  component

internal         internal

Mean-Field  mean-field

Ref              reference



[1]        J.D. van der Waals, , Leiden, Leiden, 1873.

[2]        L. Boltzmann, Vorlesungen uber Gastheorie, Leipzig, 1896.

[3]        B.J. Alder and T.E. Wainwright, J. Chem. Phys., 27 (1957) 1208.

[4]        B.J. Alder and T.E. Wainwright, J. Chem. Phys., 31 (1959) 459.

[5]        B.J. Alder and T.E. Wainwright, J. Chem. Phys., 33 (1960) 1439.

[6]        B.J. Alder and C.E. Hecht, J. Chem. Phys., 50 (1969) 2032.

[7]        J.G. Kirkwood and E. Monroe, J. Chem. Phys., 9 (1941) 514.

[8]        J.I. Goldman and J.A. White, J. Chem. Phys., 89 (1988) 6403.

[9]        E.A. Guggenheim,  (1964) 43.

[10]      E.A. Guggenheim, Mol. Phys., 9 (1965) 199.

[11]      H. Reiss et al., J. Chem. Phys., 31 (1959) 369.

[12]      H. Reiss et al., J. Chem. Phys., 32 (1960) 119.

[13]      M. Ross, J. Chem. Phys., 71 (1979) 1567.

[14]      J.D. Weeks et al., J. Chem. Phys., 54 (1971) 5237.

[15]      J.D. Weeks et al., J. Chem. Phys., 55 (1971) 5422.

[16]      R. Widom, J. Chem. Phys., 39 (1963) 2808.

[17]      B. Widom, Science, 157 (1967) 375.

[18]      J. Lebowitz, Phys. Rev., 133 (1964) A895.

[19]      J.L. Lebowitz and J.S. Rowlinson, J. Chem. Phys., 41 (1964) 133.

[20]      M.S. Wertheim, J. Math. Phys., 5 (1964) 643.

[21]      E. Thiele, J. Chem. Phys., 39 (1963) 474.

[22]      J.K. Percus and G.J. Yevick, Phys. Rev., 110 (1958) 1.

[23]      H. Reiss, in I. Prigogine (Ed.), Adv. in Chem. Phys., Interscience, New York, 1965, pp. 1.

[24]      H. Reiss, J. Chem. Phys., 47 (1967) 186.

[25]      H. Reiss and D.M. Tully-Smith, J. Chem. Phys., 55 (1971) 1674.

[26]      H. Reiss, in U. Landman (Ed.), Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, London, 1977.

[27]      R.J. Speedy and H. Reiss, Mol. Phys., 72 (1991) 999.

[28]      N.F. Carnahan and K.E. Starling, J. Chem. Phys., 51 (1969) 635.

[29]      T.F. Anderson and J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev., 19 (1980) 1.

[30]      S. Beret and J.M. Prausnitz, AIChE. J., 21 (1975) 1123.

[31]      C.C. Chapelow and J.M. Prausnitz, AIChE. J., 20 (1974) 1097.

[32]      M.D. Donohue and J.M. Prausnitz, , Gas Processors Report, Gas Processors Association, 1977.

[33]      M.D. Donohue and J.M. Prausnitz, AIChE. J., 24 (1978) 849.

[34]      J.H. Vera and J.M. Prausnitz, Chem. Eng. J., 3 (1972) 113.

[35]      I. Prigogine, Molecular Theory of Solutions, North-Holland, Amsterdam, 1957.

[36]      N.N. Bogolyubov, Problems of a Dynamical Theory in Statistical Mechanics, GITTL, Moscow, 1946.

[37]      R.P. Feynman and A.R. Hibbs, Path Integrals and Quantum Mechanics, McGraw-Hill, New York, 1965.

[38]      I.M. Idzyk et al., Theoret. Math. Phys., 73 (1987).

[39]      I.M. Idzyk, Inst. Theoret. Phys. Acad. Sci. Ukraine, 87-162P (1988).

[40]      I.R. Yukhnovskii, Phase Transitions of the Second Order:  the Method of Collective Variables, World Scientific Press, Singapore, 1987.

[41]      I.R. Yukhnovskii, Inst. Theoret. Phys. Acad. Sci. Ukraine, Preprint No. 88-30P (1988).

[42]      I.R. Yukhnovskii and O.V. Patsahan, J. Stat. Phys., 81 (1995) 647.

[43]      I.R. Yukhnovskii, Proc. Steklov Inst. Math. (1989) 223.

[44]      D.N. Zubarev and V.G. Morozov, Proc. Steklov Inst. of Math. (1992) 155.

[45]      D.S. Wong et al., Ind. Eng. Chem. Res., 31 (1992) 2033.

[46]      S.I. Sandler, Fluid Phase Equilibria, 19 (1985) 233.

[47]      K.-H. Lee and S.I. Sandler, Fluid Phase Equilibria, 21 (1985) 177.

[48]      C.-H. Kim et al., AIChE. J., 32 (1986) 1726.

[49]      K.-H. Lee et al., Fluid Phase Equilibria, 25 (1986) 31.

[50]      K.-H. Lee and S.I. Sandler, Fluid Phase Equilibria, 34 (1987) 113.

[51]      K.-H. Lee et al., Fluid Phase Equilibria, 50 (1989) 53.

[52]      E. Fermi, Ricerca. Sci., 7 (1936) 13.

[53]      J.M. Blatt and V. Weisskopf, Theoretical Nuclear Physics, J. Wiley, New York, 1952.

[54]      K. Huang and C.N. Yang, Phys. Rev., 105 (1957) 767.

[55]      H.A. Bethe, Proc. Roy. Soc. A., 150 (1935) 552.

[56]      W.L. Bragg and E.J. Williams, Proc. Roy. Soc. A., 145 (1934) 699.

[57]      R.H. Fowler and E.A. Guggenheim, Proc. Roy. Soc., A174 (1939) 189.

[58]      A. van Pelt et al., Fluid Phase Equilibria, 74 (1992) 67.

[59]      J.O. Hirschfelder et al., Molecular Theory  of Liquids and Gases, John Wiley, New York, 1954.

[60]      C.L. De Ligny and N.G. van der Veen, Chem. Eng. Sci., 27 (1972) 391.

[61]      F.H. Mourits and F.H.A. Rummens, Can. J. Phys., 55 (1977) 3007.

[62]      N.W. Ashcroft and J. Lekner, Phys. Rev., 145 (1966) 83.

[63]      K. Tanabe and J. Hiraishi, J. Mol. Phys., 39 (1980) 1507.

[64]      D. Ben-Amotz and D.R. Herschbach, J. Phys. Chem., 94 (1990) 1038.

[65]      U. S. Bureau of Standards, Washington, 1946-1952.

[66]      U. S. Bureau of Standards, Washington, 1947-1952.

[67]      J.L. Franklin, Industrial and Engineering Chemistry, 41 (1949) 1070.

[68]      D.W. van Krevelen and H.A.G. Chermin, Chem. Eng. Sci., 1 (1951) 66.

[69]      M. Souders Jr et al., Ind. Eng. Chem., 41 (1949) 1037.

[70]      A. Eisenstein and N.S. Gingrich, Phys. Rev, 62 (1942) 261.

[71]      L.J. Sundstrom, Phil. Mag., 11 (1965) 657.

[72]      J.D. Bernal and J. Mason, Nature, 188 (1960) 910.

[73]      A.L. Anderson, A Physicist's Desk Reference, American Institute of Physics, New York, 1981.

[74]      R.M. Gibbons, Mol. Phys., 17 (1969) 81.

[75]      P.D. Kaplan et al., Phys. Rev. Let., 72 (1994) 562.

[76]      B.J. Alder, J. Chem. Phys., 40 (1964) 2724.

[77]      B.J. Alder et al., J. Chem. Phys., 49 (1968) 3688.

[78]      I.R. Krichevskij and P. Bolshakov, Acta Physicochim. URSS, 14 (1941) 353.

[79]      D.S. Tsiklis and L.A. Rott, Rus. Chem. Rev., 36 (1967) 351.

[80]      J.A. Schouten et al., Physica, 73 (1974) 556.

[81]      J.A. Schouten, J. Phys: Condens. Matter, 7 (1995) 469.

[82]      R.L. Scott, , Conference of Thermodynamics, Baden, Austria, 1973, pp. 220.

[83]      I.R. Krichevskij, Phase Equilibria in Solutions under High Pressure, Goskhimizdat, Moscow, 1952.

[84]      G.C. Kennedy, Am. J. Sci., 260 (1962) 501.

[85]      K. Todheide and E.I. Franck, Zeit. f. Physic. Chem., Neue Folge. Bol. (1963) 388.

[86]      M.D. Eldridge et al., Nature, 365 (1993) 35.

[87]      M.D. Eldridge et al., Mol. Phys., 79 (1993) 105.

[88]      M.D. Eldridge et al., Mol. Phys., 80 (1993) 987.

[89]      A.D. Dinsmore et al., Phys. Rev. E., 52 (1995) 4045.

[90]      A.D. Dinsmore et al., Nature, 383 (1996) 239.