Critical behavior and gas-liquid coexistence

It is well known that the van der Waals equation of state is qualitatively correct in the sense that it is capable of predicting ga.s-liquid phase equilibria as well as critical phenomena. Mathematically speaking this is because the van der Waals equation may be [jerc eived as a third-order polynomial in p regardless of whether we consider the bulk or a confined fluid. As for the bulk, the location of the critical point of the confined fluid is determined by the conditions 

Expanding P in a Taylor series around the critical point (i.e., around T = p = 1) and retaining terms up to third-order derivatives, the resulting (reduced) equation of state can be east as 

Consider now coexisting gas and liquid phases at densities p and pj, respectively. If we take these densities to he sufficiently close to the critical density, wc may write 

However, if gas and liquid are in eoexisteiiee previous expression can be rearranged to give 

T(j illustrate the range of temperatures over which the power law in Eq. (4.37) is valid, we plot the order parameter as a function 

Consider now coexisting gas and liquid phases at densities and respec- 

T(j illiLstrati tlu range of tcmiJcraturcs over which the powi r law in Eq, (4.37) is valid, we plot the order parameter - p as a function of r/Tcb in Fig. 4.4. The comparison shows that the critical behavior as predicted by Eq. (4,37) prevails for temperatures that are about 10% lower than the (bulk) critical temperature. However, as one would have guessed, the scaling law docs not hold for much lower temperatures as one can also SCO from Fig. 4.4. 

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