Critical point equation

The critical point Equation is derived from the spinodal Equation using the critical condition given earlier. The differential of the spinodal Equation with respect to is zero at the critical point. This calculation is perfectly feasible but so far no one has made any attempt to use these Equations to predict the critical point. In the past it has usually been approximated using the simple Flory-Hi jns expression for the critical point. 

The spinodal, binodal and critical point Equations derived on the basis of this theory will be discussed later. When the theory has been tested it has been found to describe the properties of polymer blends much better than the classical lattice theories 17 1B). It is more successful in interpreting the excess properties of mixtures with dispersion or weak attraction forces. In the case of mixtures with a strong specific interaction it suffers from the results of the random mixing assumption. The excess volumes observed by Shih and Flory, 9, for C6H6-PDMS mixtures are considerably different from those predicted by the theory and this cannot be resolved by reasonable alterations of any adjustable parameter. Hamada et al.20), however, have shown that the theory of Flory and his co-workers can be largely improved by using the number of external degrees of freedom for the mixture as ... 

Equation (4-26) was developed by Hougen and Watson. More recently, Mehra, Brown, and Thodos utUized it to determine /C-values for binary hydrocarbon systems up to and including the true mixtiu-e critical point. Equation (4-27) has received considerable attention. Applications of importance are given by Benedict, Webb, and Rubin Starling and Han " and Soave. Two unsymmetrical formulations are ... 

The right-hand side of Eq. (10-60) is the entropy of a unit area of surface minus the entropy of the surface material content of liquid. (3y/5r),(.) is negative for liquid-vapor systems. The surface tension vanishes at the critical point. Equation (10-60) may be rewritten by use of the relation... 

If Gm(T, p, x), the appropriate thermodynamic function for a binary mixture, is analytic, the deductions about the behaviour of the various thermodynamic properties are entirely analogous to those for the one-component fluid. The same conclusions arise from any general Taylor series expansion in which all the coefficients are non-zero except those two required to define the critical point [equations (6a, b)]. In particular, the coexistence curve (T vs. x at constant p) should be parabolic, the critical isotherm vs. x at constant T and p) should be cubic, and the molar heat capacity C, ,m should be everywhere finite. 

I he spinodal (Equation 3.2 -2) and critical point (Equations 3.2-2,-3) conditions lead, in this case, to... 

Near the critical point, Equation 6 holds true, and... 

Because the compressibility of pure water has a singularity at the critical point. Equation (2.11) predicts that the standard partial molar volume should diverge in the critical point. The sign of the critical divergence, determined by the spatial integral of the direct correlation function, C 2, depends on the nature of the solute-solvent interaction. 

Theoretically Based Equations.— Although semi-theoretical derivations can be produced for some vapour-pressure equations, for example equation (12), these are valid at fairly low pressures only, and there are no theoretically based equations which relate vapour pressure to temperature over the whole range from the triple point to the critical point. Equations can be... 

No restriction has been imposed on the parameter X. We take advantage of this freedom by selecting X = (—or = (—r) with f< 0, and then take the limit P —> Pc, i.e.,p —> 0. Such a choice is consistent with the requirement that correlations become very large near the critical point. Equation... 

The average accuracy of the Lee and Kesler model is much better than that of all cubic equations for pressures higher than 40 bar, as well as those around the critical point. 

At low temperatures, using the original function/(T ) could lead to greater error. In Tables 4.11 and 4.12, the results obtained by the Soave method are compared with fitted curves published by the DIPPR for hexane and hexadecane. Note that the differences are less than 5% between the normal boiling point and the critical point but that they are greater at low temperature. The original form of the Soave equation should be used with caution when the vapor pressure of the components is less than 0.1 bar. In these conditions, it leads to underestimating the values for equilibrium coefficients for these components. 

The importance of the van der Waals equation is that, unlike the ideal gas equation, it predicts a gas-liquid transition and a critical point for a pure substance. Even though this simple equation has been superseded, its... 

A2.3.2.2 EQUATIONS OF STATE, THE VIRIAL SERIES AND THE LIQUID-VAPOUR CRITICAL POINT... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

Ebeling W and Grigoro M 1980 Analytical calculation of the equation of state and the critical point in a dense classical fluid of charged hard spheres Phys. (Leipzig) 37 21... 

Wdom B 1965 Equation of state near the critical point J. Chem. Phys. 43 3898 Neece G A and Wdom B 1969 Theories of liquids Ann. Rev. Phys. Chem. 20 167... 

The van der Waals p., p. isothenns, calculated using equation (A2.5.3), are shown in figure A2.5.8. It is innnediately obvious that these are much more nearly antisynnnettic around the critical point than are the conespondingp, F isothenns in figure A2.5.6 (of course, this is mainly due to the finite range of p from 0 to 3). The synnnetry is not exact, however, as a carefiil examination of the figure will show. This choice of variables also satisfies the equal-area condition for coexistent phases here the horizontal tie-line makes the chemical potentials equal and the equal-area constniction makes the pressures equal. 

If the small temis in p- and higher are ignored, equation (A2.5.4) is the Taw of the rectilinear diameter as evidenced by the straight line that extends to the critical point in figure A2.5.10 this prediction is in good qualitative agreement with most experiments. However, equation (A2.5.5). which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

Finally, we consider the isothennal compressibility = hi V/dp)y = d hi p/5p) j, along tlie coexistence curve. A consideration of Figure A2.5.6 shows that the compressibility is finite and positive at every point in the one-phase region except at tlie critical point. Differentiation of equation (A2.5.2) yields the compressibility along the critical isochore ... 

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

It is curious that he never conuuented on the failure to fit the analytic theory even though that treatment—with the quadratic fonn of the coexistence curve—was presented in great detail in it Statistical Thermodynamics (Fowler and Guggenlieim, 1939). The paper does not discuss any of the other critical exponents, except to fit the vanishing of the surface tension a at the critical point to an equation... 

However, the discovery in 1962 by Voronel and coworkers [H] that the constant-volume heat capacity of argon showed a weak divergence at the critical point, had a major impact on uniting fluid criticality widi that of other systems. They thought the divergence was logaritlnnic, but it is not quite that weak, satisfying equation (A2.5.21) with an exponent a now known to be about 0.11. The equation applies both above and... 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

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