Equal-areas 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. 

The symmetry is not exact, however, as a careful examination of the figure will show. This choice of variables also satisfies the equal-area condition for coexistent phases here the horizontal tie-line makes the chemical potentials equal and the equal-area construction makes the pressures equal. 

Figure A3.3.5 Thermodynamic force as a function of the order parameter. Three equilibrium isotherms (full curves) are shown according to a mean field description. For T < T, the isotherm has a van der Waals loop, from which the use of the Maxwell equal area construction leads to the horizontal dashed line for the equilibrium isotherm. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T

Figure 8.7 Three interpretations of the integrations (8.2.21) and (8.2.22) that determine the vapor-liquid saturation pressure for a pure substance. In each panel the solid curve is the 250 K isotherm from Figure 8.5. The filled circles locate the saturated volumes at = 21.9 bar. Top The shaded area is that given by the integral in (8.2.21). Middle The shaded region is the rectangular area P v - v ). According to (8.2.21), the shaded regions in the top two panels have the same area. Bottom The shaded area is that provided by the equal-area construction (8.2.22) in this panel, the positive and negative areas cancel one another.

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. 

For the bubble-T calculation in the phi-phi form, a viable alternative to Newton-Raphson is presented in Figure 11.1. This algorithm is composed of three principal parts an initialization, an outer loop that searches for the unknown T, and an inner loop that searches for the vapor-phase mole fractions y. The algorithm can be used for any number of components, but it is restricted to equilibrium between two phases. In the special case of a single component, the algorithm is equivalent to the Maxwell equal-area construction given in (8.2.22). 

In order to calculate the critical field value He and consequently the full equihbrium solution (stable branch), the Maxwell construction (Callen, 1985) is applied, which consists of matching the energy of the two phases, in the so-called equal-area construction (Fig. 2(b)). 

The analytic M(p) for a fixed T[Pg.54]

II. This is the equivalent, in the ii, p-plane, of the equal-areas construction in the p, u-plane shown in Fig. 1.8, and is the once differentiated version of the double-tangent construction in Fig. 3.1. 

A critical point of order p is a thermodynamic state in which p phases become identical p = 2 for ordinary critical points, of the kind we have been discussing p = 3 for trkritical points, the subject of 9.5 etc. The mean-field-theoiy value Sd of the exponent 5 for a critical point of order p may be inferred from Fig. 1.8 or 3.4 and their obvious generalizations. When p = 2, as in Hg. 1.8 or 3.4, every isotherm at Tthree intersections with the tieline in the equal-areas construction at T=T those previously distinct intersections coalesce in a sin e, three-fold root, which makes tte critical isotherm cubic at the critical point fid = 3. For p-phase equilibrium, the analogue of the curve in Hg. 1.8 or 3.4 has 2p—2 loops the horirontal line that determines the equilibrium pressure or chemk potential makes 2p -1 intersections with the curve, in such a way that the areas of the 2p-2 loops are equal in pairs and at T-T the 2p—1 roots coalesce, so that the critical isotherm is of degree 8d = 2p-l ( = 5 for a tricritical point). Then from (9.64), the bcndeiiine dimensionality for a critical point of order p is... 

Comparison of Eq. (5.4-7) with Eqs. (5.4-8) and (5.4-9) shows that when G ie) — Gm(a) = 0, area 1 and area 2 are equal to each other. The adjustment of the locations of points a and e to make these areas equal gives a tie hne between the coexisting liquid and gas states and is known as the Maxwell equal-area construction. 

For type II superconductors, the zero-temperature critical field quoted is obtained from an equal-area construction The low-field magnetization is extrapolated linearly to a field chosen to give an enclosed area equal to the area under the actual magnetization curve. 

Figure 5, Steady states of the cooperative isomerization model, showing sub-critical (t) = 1), critical (t) — 4), and supercritical (t) = 5 curves of mole fraction x as a function of the forward activation energy t. The deterministic transitions for r/ = 5 are indicated by arrows the dashed line denotes an equal areas construction which determines the unique equilibrium transition (22).

In the simulations of Figure 7 it is apparent that most of the points studied on the metastable branches for n = 5 are in fact stable on the time scale of these computations. This should not be surprising, since the "size" of a destabilizing fluctuation is large near the "equal-areas" construction (dashed line. Figures 5 and 7), and decreases to microscopic dimensions only at... 

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