Points critical -

The critical point can be obtained from the van der Waals equation by resolving into p and setting 

Note that we form the differentials at constant temperature. It emerges for the critical point, 

If we use the adiabatic van der Waals equation, Eq. (4.38), we can likely form the first and second derivative of the pressure with respect to the volume at constant entropy and set them to zero. However, then the constants a and b will emerge in a different way. For example, b = Vd2 — k)/3. 

The reduced adiabatic van der Waals equation looks like 

The internal pressure and the co-volume change their signs, when turning from k = 5/3 to K = 7/3. Thus, for /c 2 the p —V plot with S as parameter shows an unusual behavior. The situation becomes different, when we try to use the values for a and b obtained from the isothermal van der Waals equation. A drawback is that the reduced form looks somewhat complicated. 

---

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. 

When the two components are mixed together (say in a mixture of 10% ethane, 90% n-heptane) the bubble point curve and the dew point curve no longer coincide, and a two-phase envelope appears. Within this two-phase region, a mixture of liquid and gas exist, with both components being present in each phase in proportions dictated by the exact temperature and pressure, i.e. the composition of the liquid and gas phases within the two-phase envelope are not constant. The mixture has its own critical point C g. 

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. 

There has been extensive activity in the study of lipid monolayers as discussed above in Section IV-4E. Coexisting fluid phases have been observed via fluorescence microscopy of mixtures of phospholipid and cholesterol where a critical point occurs near 30 mol% cholesterol [257]. 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

The band energy difference or co (k) can be expanded around a critical point k p as... 

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... 

CO = coq, has a discontinuity in tire first derivative. In figure A1.3.18 the characteristic structure of the joint density of states is presented for each type of critical point. 

Figure Al.3.18. Typical critical point structure in the joint density of states.

For a given pair of valence and conduction bands, there must be at least one and one critical points and at least tluee and tluee critical points. However, it is possible for the saddle critical points to be degenerate. In the simplest possible configuration of critical points, the joint density of states appears as m figure Al.3.19. 

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

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... 

An exception to this conclusion is found in the innnediate vicinity of critical points, where fluctuations become much more significant, although—with present experimental precision—still not of the order of N.)... 

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. 

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. 

---