Van der Waals Model of Condensation and Critical Behavior

Despite its quantitative flaws, the Van der Waals equation (2.13) provides an intriguing description of condensation and critical behavior that mimics certain aspects of real fluid behavior. We therefore wish to examine some detailed aspects of the Van der Waals description of condensation and critical phenomena as a starting conceptual model. 

It was shown by J. C. Maxwell that a horizontal line can be drawn through the Van der Waals loop region in such a way that the area enclosed above the line in the upward loop exactly matches that enclosed below the line in the downward loop ( Maxwell s equal-area construction ). As shown in Fig. 2.10b, this horizontal line (say, at pressure P0) can be taken as the Van der Waals approximation to the actual condensation plateau, bounded on the left by the steeply sloping liquid branch, and on the right by the more gently sloping gaseous branch of the isotherm. The three points where this horizontal line P = P0 crosses the Van der Waals isotherm may be obtained as the roots of the cubic polynomial P = P(V) for P = P0, i.e., as solutions of the equation 

According to the general theory of cubic equations, this equation will have three distinct real roots V1 V2, V3 whenever the discriminant D, 

The critical point (Tc, Pc) can therefore be uniquely identified with the inflection point that occurs at the top of the coexistence dome, i.e., the point at which both first and second derivatives of the isotherm vanish, 

This analytic characterization allows us to find Vc, Tc (and therefore Pc) as solutions of the two differential equations in (2.42). (A noncalculus method for finding these same solutions is sketched in Sidebar 2.11 below.) 

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