Equation of state near the critical point

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... 

Widom B 1965 Equation of state near the critical point J, Chem. Phys. 43 3898... 

Figure 3. The minimum value of F along the critical Figure 4. Schematic representation of the p — T plan isotherm calculated from various equations of state, near the critical point in terms of the parametric...

The above derivation leads to an equation for y in terms of (Sp / dz). Van der Waals paper also contains a variety of other items, such as stability considerations, the pressure in the interfacial layer, spherical interfaces, the value of y near the critical point, a discussion on the thickness of the transition layer, the effect of higher terms in the series expansion of the profile and corresponding state features. Van der Waals also showed that his theory agreed with Gibbs adsorption law, an issue that was later discussed in more detail by Widom J. In the present context we shall not discuss these features further, except to mention that for the temperature dependence close to the critical point T van der Waals predicted... 

The above observations have been made primarily in the study of the new measurements for parahydrogen. Because of the wide use of this equation in other data compilations, it is of interest to note the results of the comparison of the preliminary calculations for parahydrogen using this equation of state with more rigorous calculations that have been made subsequently. These comparisons are given by Roder et al. [ ] and indicate an average absolute error of 0.16% with maximum errors of 4% near the critical point. With the exception of the area near the critical point the density errors do not exceed 1 %. 

It should be noted that the validity of the critical power laws (6.10)-(6.11) is restricted to a very small range of temperatures near the critical point (Levelt Sengers Sengers 1981). Equations of state that incorporate these critical power laws but which remain valid in an appreciable range of densities and temperatures in the critical region have been developed (Sengers 1994). 

The results of Amagat s and Raveau s work may be summed up in the statement that, whereas the theorem of corresponding states holds good very approximately, the equation of van der Waals gives results quite inconsistent with the experimental values, especially near the critical point. 

In both equations, k and k are proportionality constants and 0 is a constant known as the critical exponent. Experimental measurements have shown that 0 has the same value for both equations and for all gases. Analytic8 equations of state, such as the Van der Waals equation, predict that 0 should have a value of i. Careful experimental measurement, however, gives a value of 0 = 0.32 0.01.h Thus, near the critical point, p or Vm varies more nearly as the cube root of temperature than as the square root predicted from classical equations of state. 

The Peng-Robinson equation is related to the Redlich-Kwong-Soave equation of state and was developed to overcome the instability in the Redlich-Kwong-Soave equation near the critical point Peng and Robinson (1970). 

The EOSs are mainly used at higher pressures or when some of the components are near or above their critical point. The most commonly used cubic equations of state are the Peng-Robinson [3] and the Soave-Redlich-Kwong [4] equations. 

This relation is analogous to the equation of state for spin systems in the famous Landau theory of phase transitions. It reads, H = A0(T — Tc)t]/ + u0ij/3, where H is a magnetic field and i// is an order parameter, A0 and u0 being constants. In our problem, the left hand side of Eq. (2.41), which is a function of T and corresponds to H in spin systems. On the right hand side, the coefficient Vic — Vi which is determined by the degree of ionization, corresponds to the temperature parameter A0(T — Tc) in spin systems. Therefore, at yt = yu, we find T — Tc cc — c a with 8 = 3 near the critical point of ionic gels, where T— Tc plays the role of H in spin systems. 

First-principle calculations of the thermodynamic properties are more or less hopeless enterprise. One of the most famous phenomenological approaches was suggested by van der Waals [6, 8, 9]. Using the dimensionless pressure it = p/pc, the density v = n/nc and the temperature r = T/Tc, the equation of state for the ideal gas reads it = 8zzr/(3 -u) — 3zA Its r.h.s. as a function of the parameter v has no singularities near u = 1 v = it = t = is the critical point) and could be expanded into a series in the small parameter 77 = [n — nc)/nc with temperature-dependent coefficients. Solving this... 

Before preceding, it is useful to consider the form of the force-force correlation function, which is given in Equation (21), with Equations (22), (24), (25), (26) and (27). The form of the force-force correlation function, derived using density functional formalism, is employed because it permits the use of very accurate equations of state for solvents like ethane and CO2 to describe the density dependence and temperature dependence of the solvent properties. These equations of state hold near the critical point as well as away from it. Using the formalism presented above, we are able to build the known density and temperature-dependent properties of the... 

For a pure supercritical fluid, the relationships between pressure, temperature and density are easily estimated (except very near the critical point) with reasonable precision from equations of state and conform quite closely to that given in Figure 1. The phase behavior of binary fluid systems is highly varied and much more complex than in single-component systems and has been well-described for selected binary systems (see, for example, reference 13 and references therein). A detailed discussion of the different types of binary fluid mixtures and the phase behavior of these systems can be found elsewhere (X2). Cubic ecjuations of state have been used successfully to describe the properties and phase behavior of multicomponent systems, particularly fot hydrocarbon mixtures (14.) The use of conventional ecjuations of state to describe properties of surfactant-supercritical fluid mixtures is not appropriate since they do not account for the formation of aggregates (the micellar pseudophase) or their solubilization in a supercritical fluid phase. A complete thermodynamic description of micelle and microemulsion formation in liquids remains a challenging problem, and no attempts have been made to extend these models to supercritical fluid phases. 

Apparent hydration numbers have been derived from experimental measurements assuming the formation of a hydration complex studied as a chemical reaction. xhe change of volume for the reaction is calculated from an equation of state which includes variation of the dielectric constant based on the solvent isothermal compressibility, while the bare ion and the complex are assumed spherical with crystallographic and Stokes-Einstein radii respectively. The latter radius is obtained from conductance measurements. Due to these assumptions, the apparent hydration numbers increase when temperature increases and diverge near the critical point due to the divergence of the solvent compressibility. Furthermore, negative values are obtained when the Stokes-Einstein radius for the complex is smaller than the crystallographic radius. 

High-pressure systems in the vicinity of critical points, such as synthesis gas and air separation systems, remain a challenge. Our flash algorithm has difficulty in identifying the correct phase state, or converging to the correct vapor-Uquid solutions. This problem may be exacerbated by the difficulty in obtaining the equation of state volume root in the vicinity of the critical points. Further work to improve the algorithm and the equation of state volume root determination is required. It is believed that the homotopy continuation methods are probably better suited for calculations near the critical points. 

The uncertainties in the equation of state are 0.1% in density (except near the critical point), 0.25% in vapor-pressure, 1% in heat capacities, 0.2% in the vapor-phase speed of sound, and 3% in the liquid speed of sound. The liquid speed of sound uncertainty is an estimate and cannot be verified without experimental information. The uncertainties above 290 K in vapor pressure may be as high as 0.5%. 

The equation of state is valid from the triple point to 500 K with pressures to 100 MPa. At higher pressures, the deviations from the equation increase rapidly, and it is not recommended to use the equation above 100 MPa. The uncertainties in the equation are 0.3% in density (approaching 1% near the critical point), 0.2% in vapor pressure, and 2% in heat capacities. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4-6%. 

The equation of state determined by Z (A, V, T ) is not known in the sense that it carmot be written down as a simple expression. However, the critical parameters depend on s and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 b) illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenheim [19], the curvature near the critical point is consistent with a critical exponent P closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the form associated with van der Waals equation. Figure A2.3 5 (b) shows that P/pkT is approximately the same function of the reduced variables and R... 

In 1976, Hall and Eubank (12,13) published two papers which have direct bearing upon the present equation of state. In the first paper, they noted the rectilinear behavior for the mean of the vapor and liquid isochoric slopes issuing from the same point on the vapor pressure curve near the critical point and the power law behavior for the difference in these slopes. The second paper presented an empirical description of the critical region which generally agreed with the scaling model but differed in one significant way—the curvature of the vapor pressure curve. 

Iteration for Coexisting Densities. Orthobaric densities near the critical point generally cannot be obtained accurately from isochoric PpT data by extrapolation to the vapor-pressure curve because the isochore curvatures become extremely large near the critical point. The present, nonanalytic equation of state, however, can be used to estimate these densities by a simple, iterative procedure. Assume that nonlinear parameters in the equation of state have been estimated in preliminary work. For data along a given experimental isochore (density), it is necessary merely to find the coexistence temperature, Ta(p), by trial (iteration) for a best, least-squares fit of these data. 

This result is the analog of the Clapeyron equation (8.2.27) extended from pure substances to binary mixtures. It gives the slope of a saturation line of constant composition plotted on a PT diagram. The differences in partial molar enthalpies and volumes in (9.3.19) are usually positive, so the slope given by (9.3.19) is usually positive (see Figure 9.8). However, negative values of those slopes are observed for some mixtures at states near mixture critical points these are usually caused by negative partial molar volumes of the heavier component. 

P4.21 (a) The Dieterici equation of state is purported to have good accuracy near the critical point. It does fail... 

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