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In this paper equilibrium models for the calculation of the excess Gibbs free energy of binary liquid mixtures are developed, the component A of which undergoes chain-forming self-association whilst the component B acts as an 'inert' solvent. It is shown that the extension of the well-known chain-association model of Mecke and Kempter, in which the probability of chain prolongation is assumed to be independent of chain length, is unable to establish satisfactory results because it does not exhibit sufficient unsymmetry. Reduction of the probability of chain growth with in-creasing chain length leads to an improved model with the geometric series replaced by the exponential series. This model, in which only two parameters are used, i. e. the equilibrium constants K for mutual solvation of A and B, and ρ for self-association of A, allows fitting of isothermal experimental GE /R T literature data on cycloalkanol-cycloalkane, alkanol-alkane, and NMF -CCl4 systems within the limits of experimental error. Compared with the two-parameter Wilson equation which gives equally small standard deviations, our equilibrium model has the advantage of allowing passage from GE to HE data and of being applicable to liquid-liquid equilibria.
The enthalpies of mixing at 25° of diethyl ether, di-n-propyl ether, di-n-butyl ether, di-isopropyl ether, propylene oxide, tetrahydrofuran, and tetrahydropyran with chloroform are determined by an isothermal titration method. As a result, the functions HM-f(N CHCl3) are obtained with a step width of 0.025 of the mole fraction and a relative accuracy of 1 per cent or better. Evaluation of the heat of mixing data by means of equilibrium models ("ideal associated mixture") shows that the systems of aliphatic ethers with chloroform behave rather precisely as one-step equilibria of the type A + B = AB (A = ether; B = chloroform). In the systems of cyclic ethers with chloroform, a second equilibrium step, AB + B = AB2 , must be considered, the importance of which decreases with increasing ring size of the ether. The equilibrium data calculated for the seven ether-diloroform systems are discussed.
Phasentrennung als Folge der Konkurrenz zwischen "statistischer" und "chemischer" Vermischung
(1977)
The fact that common thermodynamic conditions are valid for all known types of critical phases (liquid-liquid, liquid-gas, and "gas-gas") suggests that a common principle for the interpretation of material phase instability from a molecular point of view must exist. In this paper we show that the principle of competition between "statistical mixing" (i. e. random mixing) and "chemical mixing" (i. e. mixing effected under the influence of chemical interactions) can give this common inter pretation. If the equilibrium states resulting from both types of mixing are sufficiently different, phase separation occurs. We refer to our earlier papers (since 1972) in which we have applied this principle to describe liquid-liquid phase equilibria by "chemical" models, using the equilibrium constants of exchange equilibria between nearest-neighbour complexes as a measure of "chemical" mixing. In this paper we show that the well-known reduced gas-liquid coexistence curve, T/Tc =f(q/qc), can accurately be fitted by a very simple "mixture" model of molecules A with "vacan cies", provided that the contributions of both statistical and chemical mixing are incorporated into the formula for GE. From a discussion of the application to "gas-gas" phase equilibria in the hyper critical region it results that the weight factor r, by which the contribution of statistical mixing enters into GE, must depend on the density of the gas mixture. Phase separation can only occur if, by increasing pressure, the contributions to GE of statistical and chemical mixing have reached the same order of magnitude. From an attempt to apply the same principle to solid-liquid equilibria it is shown under which external conditions a critical point for this type of phase transition can be expected.
A thermodynamic theory of liquid mixtures based on a simple molecular model is developed which describes the equilibrium state as the result of a coupling between a "chemical" and a "statistical" equilibrium. The intermolecular interactions are taken into account by considering "complexes" formed between a given molecule and its z nearest neighbours. The equilibrium mole fractions of these complexes are calculated by application of the ideal law of mass action to an appropriate set of "exchange equilibria". Formulae for the excess functions GE and HE and for the activities of the components are derived for the cases z=1 and z=4. GE depends on an equilibrium constant K describing the deviation from random distribution of the equilibrium mole fractions of the complexes. HE depends on K and on an energy parameter w which is related to differences of pair interactions. K and w are independent parameters, and there is no limitation in respect to amount and sign of the excess functions. The conditions for the existence of a critical solution point are formulated; at this point GE has a value of about 0.56 R T. If a model with two equilibrium constants is used allowing for instance competition between "self-association" and "complex-formation", the existence of closed miscibility gaps becomes possible. Closed miscibility curves are calculated and the conditions for their appearance are discussed. The relations between this theory and Guggenheim's statistical lattice theory of symmetrical mixtures are pointed out.
As we have shown in a recent paper, the principle of competition between "statistical" and "chemical" mixing represents a molecular thermodynamic approach to all known types of phase separation. This principle is effective if the contributions of two independent spontaneous processes enter into the thermodynamic potential by which the resulting equilibrium state of the system is determined. This is equivalent with the statement that two different forms of entropy exist which are not interchangeable, and for which the law of increasing entropy independently must be valid. As "cooperativity" is introduced by this principle, critical phenomena may be described by simple equilibrium models in which only nearest-neighbour interactions are considered.
Starting from the molar Gibbs free energy GM of the most simple binary equilibrium model z = 1 with nearest-neighbour pairs, nonclassical critical-point exponents α = 0.33 of the molar heat capacity, β = 0.33 of the coexistence curve, γ = 1.33 of the isothermal compressibility, and δ = 4.33 of the critical isotherm, are derived, which are consistent with the well-known exponent in equalities. These non-classical critical-point exponents are independent of the chemical nature of the particles because they are obtained by applying thermodynamic arguments on the coupling constant τ, by which the contribution of "statistical mixing" to GM is weighted.
LANGEVIN equations of the type dn× (t)/dtn+...+c × (t)=K (t) constitute the starting point of a phenomenological fluctuation theory of irreversible processes. These equations are not constructed from transport equations (as in the older theory), but via a generalized MASTER equation from phase space mechanics. The MARKOFF processes of first and higher order defined by the various LANGEVIN equations are studied by the prediction theory of stationary stochastic processes. Instead of the variation principle of the ONSAGEB–MACHLUP theory one has the minimization of the prediction error. The mean relaxation path and the entropy of the considered processes are calculated. It is shown that the entropy consists of one part which is given by the relaxation path and another which is determined by the prediction error.
The relations of the theory of real gases which have first been derived by Mayer and his co-workers can be obtained in a simple way by the functional method. In this case the assumption of the pairwise additivity of the intermolecular potential can be dropped. Apart from some new relations for distributions functions the expansion of the direct correlation functions is obtained as a power series in density with coefficients consisting of integrals over Husimi functions.
Es werden allgemeine Gleichungen zur Berechnung der Verteilung der Nullstellen von der großen kanonischen Verteilungsfunktion Ξ (y) eines Gittergases in der komplexen Ebene der Fugazität y in Bethescher Näherung angegeben, die nach einem graphischen Verfahren gelöst werden können. Für das zweidimensionale, quadratische Gitter und das dreidimensionale, kubische Gitter werden die Bestimmungsgleichungen für die Nullstellenverteilung mit Hilfe des graphischen Verfahrens explizit gelöst. Die Nullstellenverteilung von Ξ (y) des eindimensionalen Gittergases wird in geschlossener Form angegeben.
The cooperative problem for a lattice gas on a plane, square lattice and on a simple cubic lattice is solved by a system of two coupled, transcendental equations, derived by a combinatorial method, which describes a homogeneous or periodical particle density on the lattice as a function of the temperature and the chemical potential of the lattice-gas.
For the particle interaction a Hard-Core potential (nearest neighbour exclusion) with a soft long-range tail is assumed. The zero-component of the Fourier-transform of this long-range interaction part can be positive or negative.
The system of transcendental equations is solved by a graphic method. As a result, the complete pressure-density state diagram and the pressure-temperature phase diagram can be drawn.
The lattice-gas exists in three stable phases: gas, liquid and solid. Three phase changes are possible: condensation, crystallization and sublimation.
Critical points of condensation and freezing are examined. The number of possible phases and phase changes at a fixed temperature depends on the geometric structure of the particle interaction.
Die „Selbstenergien“ Gn in 1 wurden durch näherungsweise Auswertung einer größeren Klasse von Selbstenergiediagrammen approximativ berechnet. Das Gleichungssystem (3.3) in 1 für die renormierten Semiinvarianten wurde umgeformt und durch zusätzliche Näherungsannahmen vereinfacht. Durch Näherungsansätze für die Semiinvarianten M2, M3,... konnten einfache Gleichungen für die Magnetisierung M1 hergeleitet werden. Diese Gleichungen wurden numerisch gelöst. Auf der Grundlage der Beziehungen (3.5) und (4.8) in 1 wurden ferner die innere Energie, die freie Energie und die Atomwärme des zweidimensionalen Ising-Ferromagneten sowie die Druck-Dichte-Isothermen des zweidimensionalen Gittergases numerisch ausgerechnet.