The critical point

The critical point always corresponds to the minimum value of x on the spinodal curve. From eq. (IV. 12) this is obtained when d = d c, where d c satisfies 

In the symmetrical case 4 c = 1/2. On the other hand, when Nb becomes much smaller than Nj, the critical point shifts toward low concentrations of A. (Ultimately, when Nb — 1, we are led back to a polymer + solvent problem (discussed in the next section). From eq. (IV.12) the critical value of X is  

When Nb = Na, Xc is very small and compatibility is exceptional. However, when the situation is unsymmetrical (Nb N ), then Xe (— 1/2 Nb) becomes somewhat larger, and compatibility is more frequent. We do not discuss the full coexistence curves near the critical point for the dissymetric case, but their qualitative aspect is shown in Fig. IV.6. 

Returning to a phase diagram such as that shown in Fig. IV.S we now focus on the area of the one-phase region that is near the critical point. In this domain, the local concentration d has large fluctuations which can be detected by light scattering experiments. At present this kind of data is not very abundant for polymer-polymer systems (because of the long times required to reach equilibrium) but we hope that the situation will improve. 

The complete calculation of correladcms can then be performed by a random phase method described in Chapter X. Here we quote the results. They can be expressed simply in terms of the Debye function go(N, q) for the scattering by an ideal chain of N monomers (defined in Section 1.1). Explicitly, one finds the simple formula 

The Critical Point is Where Coexisting Phases Merge 

The critical point is the intersection of the binodal and spinodal curves. At the critical point, there is no longer a distinction between the A-rich and the F-rich 

EXAMPLE 2 5.4 Lattice model critical point. To find the critical point (Xc, Xc, Tc) for the lattice mixture model, determine the point where both the second and third derivatives of the free energy (given by Equation (15.14)) equal zero  

The boiling of water would seem to be a very different type of phase equilibrium from the mixing of oil and water. But in both cases, phase boundaries are concave downw ard with a two-phase region inside and a critical point at the top. Remarkably, both can be described, to a first approximation, by the same physical model. 

To create the phase diagram for boiling, put N particles in a container of volume V at temperature T. Follow- the system pressure p T,V,N) as you decrease the volume, starting with low densities p, that is large volumes per molecule, v = VIN = p  

The critical isotherm goes through an inflection point at the critical point. Mathematically, this condition can be written as  

The partial derivatives in Equations (1.16) and (1.17) specify that we need to keep the temperature constant at its value at the critical point. 

The isotherm above the critical point is representative of a supercritical fluid. This isotherm continuously decreases in pressure as the volume increases. A supercritical fluid has partly hquidlike characteristics (e.g., high density) and partly vaporlike characteristics (compressibflity, high-diffusivity). Not surprisingly, there are many interesting engineering appUcations for substances in this state. There can be confusion between the terms gas and vapor. We refer to a gas as any form of matter that fills the container it can be either subcritical or supercritical. When we speak of vapor, it is gas that if iso-thermaUy compressed will condense into a liquid and is, therefore, always subcritical. 

Determination of Location of a Two-Phase System on a Phase Diagram 

Consider a two-phase system at a specified T that contains 20% vapor, by mass, and 80% liquid. Identify the state on a Tv phase diagram. Explain why graphical determination of the state is termed the lever rule. 

Every substance has a certain temperature, the critical temperature, above which only one fluid phase can exist at any volume and pressure (Sec. 2.2.3). The critical point is the point on a phase diagram corresponding to liquid-gas coexistence at the critical temperature, and the critical pressure is the pressure at this point. 

At temperatures above the critical temperature and pressures above the critical pressure, the one existing fluid phase is called a supercritical fluid. Thus, a supercritical fluid of a pure substance is a fluid that does not undergo a phase transition to a different fluid phase when we change the pressure at constant temperature or change the temperature at constant pressure.  

A fluid in the supercritical region can have a density comparable to that of tbe liquid, and can be more compressible than the liquid. Under supercritical conditions, a substance is often an excellent solvent for solids and liquids. By varying the pressure or temperature, the solvating power can be changed by reducing the pressure isothermally, the substance can be easily removed as a gas from dissolved solutes. These properties make supercritical fluids useful for chromatography and solvent extraction. 

The critical temperature of a substance can be measured quite accurately by observing the appearance or disappearance of a liquid-gas meniscus, and the critical pressure can be measured at this temperature with a high-pressure manometer. To evaluate the density at the critical point, it is best to extrapolate tbe mean density of the coexisting liquid and gas phases, (p + p )/2, to the critical temperature as illustrated in Fig. 8.8 on page 207. The observation that the mean density closely approximates a linear function of temperature, as shown in the figure, is known as the law of rectilinear diameters, or the law of Cailletet and Matthias. This law is an approximation, as can be seen by the small deviation of tbe mean density of SFe from a linear relation very close to the critical point in Fig. 8.8(b). This failure of the law of rectilinear diameters is predicted by recent theoretical treatments.  

Thermodynamics and Chemistry, second edition, version 3 2011 by Howard DeVoe. Latest version www. chem.vund.edu/thermobook 

Rowlinson, Liquids and Liquid Mixtures , Butterworths, London, 2nd edn., 

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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 experiment could be repeated at a number of different temperatures and initial pressures to determine the shape of the two-phase envelope defined by the bubble point line and the dew point line. These two lines meet at the critical point, where it is no longer possible to distinguish between a compressed gas and a liquid. 

Moving back to the overall picture, it can be seen that as the fraction of ethane in the mixture changes, so the position of the two-phase region and the critical point change, moving to the left as the fraction of the lighter component (ethane) increases. 

The initial condition for the dry gas is outside the two-phase envelope, and is to the right of the critical point, confirming that the fluid initially exists as a single phase gas. As the reservoir is produced, the pressure drops under isothermal conditions, as indicated by the vertical line. Since the initial temperature is higher than the maximum temperature of the two-phase envelope (the cricondotherm - typically less than 0°C for a dry gas) the reservoir conditions of temperature and pressure never fall inside the two phase region, indicating that the composition and phase of the fluid in the reservoir remains constant. 

For both volatile oil and blaok oil the initial reservoir temperature is below the critical point, and the fluid is therefore a liquid in the reservoir. As the pressure drops the bubble point is eventually reached, and the first bubble of gas is released from the liquid. The composition of this gas will be made up of the more volatile components of the mixture. Both volatile oils and black oils will liberate gas in the separators, whose conditions of pressure and temperature are well inside the two-phase envelope. 

The expansion is done around the principal axes so only tliree tenns occur in the simnnation. The nature of the critical pomt is detennined by the signs of the a. If > 0 for all n, then the critical point corresponds to a local minimum. If < 0 for all n, then the critical point corresponds to a local maximum. Otherwise, the critical points correspond to saddle points. 

The types of critical points can be labelled by the number of less than zero. Specifically, the critical points are labelled by M. where is the number of which are negative i.e. a local minimum critical point would be labelled by Mq, a local maximum by and the saddle points by (M, M2). Each critical point has a characteristic line shape. For example, the critical point has a joint density of state which behaves as = constant x — ttiiifor co > coq and zero otherwise, where coq corresponds to thcAfQ critical point energy. At... 

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. 

Figure A2.3.2 (a) P-V-T surface for a one-component system that contracts on freezing, (b) P-Visothenns in the region of the critical point.

Q.lb), T = 2,al(21kb). This follows from the property that at the critical point on die P-V plane there is a... 

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. 

At the critical point p(5 Pldp)j. = 0, and the integral of the direct correlation fiuictioii remains finite, unlike the integral of h r). 

It is accurate for simple low valence electrolytes in aqueous solution at 25 °C and for molten salts away from the critical point. The solutions are obtained numerically. A related approximation is the following. 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

The assumption that the free energy is analytic at the critical point leads to classical exponents. Deviations from this require tiiat this assumption be abandoned. In mean-field theory. 

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

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... 

Domb C 1996 The Critical Point. A Historical Introduction to the Modern Theory of Critical Phenomena (London Taylor and Francis)... 

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. 

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

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. 

Figure A2.5.12. Typical temperatxire T versus mole fraction v diagrams for the constant-pressure paths shown in figure A2.5.11. Note the critical points (x) and the horizontal tliree-phase lines.

Figure A2.5.17. The coefficient Aias a fimction of temperature T. The line IRT (shown as dashed line) defines the critical point and separates the two-phase region from the one-phase region, (a) A constant K as assumed in the simplest example (b) a slowly decreasing K, found frequently in experimental systems, and (c) a sharply curved K T) that produces two critical-solution temperatures with a two-phase region in between.

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