Waals Fluids

Ideal gases do not have critical phenomena since there are no interactions between the gas molecules. In the case of a van der Waals fluid, the mean-field type critical phenomenon is expected because of the existence of the excluded volume and attractive interaction between the fluid molecules, i.e. van der Waals interaction [47]. The equation of state for a van der Waals fluid is given by 

Equation (2.1) is written in a virial expansion form by introducing the density p = 1/V and the reduced variables for density, temperature, and pressure 

The Boyle temperature T is defined as the temperature for which the second virial coefficient is zero, i.e. TR = 1. The ratio T /Tc = 27/8 is well known for a van der Waals fluid. In this transition, two opposing contributions to the free energy, i.e. translational entropy of the fluid molecules and the van der Waals attractive interaction, are balanced. 

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Figure A2.5.6. Constant temperature isothenns of redueed pressure versus redueed volume for a van der Waals fluid. Full eiirves (ineluding the horizontal two-phase tie-lines) represent stable situations. The dashed parts of the smooth eurve are metastable extensions. The dotted eurves are unstable regions.

Figure A2.5.9. (Ap), the Helmholtz free energy per unit volume in reduced units, of a van der Waals fluid as a fiinction of the reduced density p for several constant temperaPires above and below the critical temperaPire. As in the previous figures the llill curves (including the tangent two-phase tie-lines) represent stable siPiations, the dashed parts of the smooth curve are metastable extensions, and the dotted curves are unstable regions. See text for details.

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.

Again, as in the case of Cyfor the van der Waals fluid, there is a linear increase up to a finite value at the... 

For analytic theories, y is simply 1, and we have seen that for the van der Waals fluid F / F equals 2. Divergences with exponents of the order of magnitude of unity are called strong . 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 523-35... 

Show that for a Van der Waals- fluid at die critical point... 

Fig. 3. Phase diagram of van der Waals fluids. At temperature T > Tc, the pressure P is a convex function of the volume V, where Tc is the critical temperature. At T < Tc, a phase transition occurs from liquid to gas or gas to liquid. At T = T ( respectively...

Similarly to the van der Waals fluid, polymer gels were found to have a volume-phase transition. In the case of gels, the gas and liquid phases correspond to the swollen and collapsed (shrunken) phases, respectively. The prototype of the free energy expression was given by [9, 10, 18]. 

Fig. 4. Phase diagram of gels, fl and v denote the osmotic pressure and the microscopic volume of gel. (v = a3/< >). This Il-v plane corresponds to P-V plane of van der Waals fluids. At % < Xc. n is a convex function of v. At > Xc. a volume phase transition from collapsed to swollen states or vice versa occurs. Note II can be either positive or negative...

This is a virial expansion form of the osmotic pressure analogous to the van der Waals fluid. Dusek and Patterson examined this equation and predicted the presence of two phases, i.e. collapsed and swollen phases. % is temperature dependent and is given by,... 

The critical pressure P = PoAso/q2 is more difficult to evaluate. In the earlier literature there is a large spread of values [17]. The recent MC simulations of Orkoulas and Panagiotopoulos [52] yield P c = 8 x 10-5 near the lower limit of earlier estimates, along with a critical compressibility factor of Zc — PJ(pcTc) = 0.024 which is one order of magnitude lower than observed for nonionic fluids (e.g., Zc = 3/8 = 0.375 for the van der Waals fluid). 

It should be emphasized that the comparatively large change obtained in more recent work is mainly caused by the application of finite-size scaling. Under these circumstances, one certainly needs to reconsider how far the results of analytical theories, which are basically mean-field theories, should be compared with data that encompass long-range fluctuations. For the van der Waals fluid the mean-field and Ising critical temperatures differ markedly [249]. In fact, an overestimate of Tc is expected for theories that neglect nonclassical critical fluctuations. Because of the asymmetry of the coexistence curve this overestimate may be correlated with a substantial underestimate of the critical density. 

Using these equations and your values of and U, calculate a and b. How well do your values agree with values tabulated in the literature For a van der Waals fluid, the critical... 

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