Maxwell’s equal area construction

Figure 2.10 Representative Van der Waals PV isotherms for CO2 near the critical point (x), showing (a) contrasting monotonic behavior above Tc (at T = 31 OK) compared with oscillatory loops below Tc (at T = 280 and 290K) (b) Maxwell s equal-area construction for finding pressure P0 (horizontal dashed line) that cuts off equal areas in the upper loop (between VM and VG) and the lower loop (between VL and VM) P0 = 52.2 atm for T = 280K, 60.4 atm for 290K.

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... 

If we were to utilize Maxwell s equal-area construction, since the H-h plot in Figure 1.15(d) resembles a van der Waals isotherm in liquid-vapor equilibria [93], then Figures 1.15(c) and 1.15(d) would be qualitatively similar. Related issues can be further investigated via a stability analysis as described in Section II.D. 

This form is called Maxwell s equal area construction and is illustrated in the bottom panel of Figure 8.7. The form (8.2.22) states that the van der Waals loop and the tie line bound two areas whose magnitudes cancel when combined algebraically. 

One feature of the two-phase region can be determined by cubic equations. Maxwell s equal-area rale (which will be verified in Chapter 6) provides a graphical means to determine for a given T. It states that the saturation pressure is the pressure at which a horizontal line equally divides the area between the real isobar and the solution given by the cubic equation. Such a construct is illustrated in Figure 4.12, where the equal areas above and below the isobar fix the value for P. This procedure can be achieved by trial and error. If a higher saturation pressure were predicted, the upper area would be too small. Conversely, too low a value for P would make the upper area too large. 

In principle, all the molecular parameters in Eq. (6) can be determined independently, so that the theory can be quantitatively compared with experimental data. An example of Maxwell s construction in the dependence of x °n critical value of interaction parameter %c of charged PAAm network with the degree of ionization equals to the molar fraction of the sodium methacrylate in the chain i = xMNa = 0.012 are given in Fig. 4 (data of series D from Fig. 5). The compositions of the phases

critical value of Xc were determined by the condition that areas St and S2 defined in Fig. 4 are equal The experimental (p2e is higher and 2 determined by Maxwell s construction (Eq. 13). Thus, the experimental values of (p2e and metastable region the limits of which (p2s and (p2s are determined by the spinodal condition (two values

[Pg.182]

A thermodynamic example may be illustrative. Consider Maxwell s model of the Gibbs USV surface for water (Fig. 1.1), as depicted schematically in Fig. 9.1. In this model, the physical (77, S, V) coordinates are associated with mutually perpendicular axes, and three chosen points on this surface form a triangle whose edges, angles, and area are as shown in Fig. 9.1a. However, the model might have been constructed (with equal thermodynamic justification) in a skewed /io/ orthogonal axis system (Fig. 9.1b) in which the... 

Figure 7.10 Maxwell s construction specifies the physically realized flat part LP with respect to the theoretical isotherm given by an equation of state such as the van der Waals equation. At equilibrium, the chemical potentials at the points L and P must be equal. As shown in the text, this implies at the area I must be equal to area II, specifying the line LP...

---