Maxwell equal-area construction

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

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). 

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. 

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. 

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)). 

Phase equilibrium requires that A2 = Al and hence that the integral vanish. All conditions are satisfied if the points 1 and 2 are located such that the areas A = B. This geometry defines the Maxwell construction. It shows that stable liquid and vapour states correspond to minima in free energy and that AL = Ay when the external pressure line cuts off equal areas in the loops of the Van der Waals isotherm. At this pressure that corresponds to the saturated vapour pressure, a first-order phase transition occurs. 

Fig. 51. Plol of — t)(iiZ)/i)Z z=u versus /j(0) for the cases of a second-order wetting transition (a) and a first-order wetting transition (b). The solution consistent with the boundary condition always is found by intersection of the curve /x2(0) - 1 with the straight line [h + gq.(0)]/y. In case (a) this solution is unique for all choices of k /y (keeping the order parameter g/y fixed). Critical wetting occurs for the case where the solution (denoted by a dot) occurs for y (0) = +1, where then ) i(Z)/dZ z u = I) and hence the interface is an infinite distance away from the surface. For ft] > ftlc the surface is non-wet while for fti > ft C the surface is wet. In case (b) the solution is unique for ft] < ft (only a non-wet state of the surface occurs) and for ft > ft (only a wet state of the surface occurs). For ft > ft] > ft three intersections (denoted by A, B, C in the figure) occur, B being always unstable, while A is stable and C metastable for ft]c > ft > ft and A is metastable and C stable for ft " > ft > ftic. At ftic where the exchange of stability between A and C occurs (i.e., the first-order wetting transition) the shaded areas in fig. 51b are equal. This construction is the surface counterpart of the Maxwell-type construction of the first-order transition in the bulk lsing model (cf. fig. 37). From Schmidt and Binder (1987).

We return once more to the van der Waals isotherms (Fig. 11.3) The van der Waals curves describe fairly exactly compression of gases up to the dew point. If the volume is decreased beyond this point, pressure does not increase, but condensation sets in. This occurs at constant pressure, meaning that the corresponding piece of the curve must be horizontal until the gas phase completely disappears at the boiling point. These lines have been constructed so that the areas enclosed by the van der Waals curves above and below the straight line are equal (Maxwell construction or equal area rule) (compare Fig. 11.3 where the two areas in the case of the T2 isotherm are dark gray). Both gas and liquid exist simultaneously along these lines. The subsequent steep rise in pressure with further decrease of volume is characteristic of the low compressibility of liquids. 

Pathways A and B are identical in the beginning and terminal regions. Pathway B, however, stems from the Maxwell construction applied to A the horizontal line divides the loop region into equal areas i and ii. The result is that the gas pressure does not alter over the region bounded approximately by 1 x 10- and 4 x 10" meter. Constant pressure manifests in spite of the volume tuning along B. 

Therefore, the two shaded parts in Figure 2.12 have an equal area. This situation is the same as the constant-pressure line for the vapor-liquid coexistence in the isothermal process of a single-component system (Maxwell construction). We can find (f>i and < 2 from this equahty of the areas. It is, however, easier to find 4>i and < 2 from A/tp(2) and Afisi4>i) = although solving these two... 

The left side of this equation is the sum of the two shaded areas, one positive and one negative, in the upper graph in Fig. 4.1. The equation states that the two areas between the van der Waals pressure p(t, v), and the constant pressure, p t), which replaces it between Vg and vi, are equal. Therefore Vg and v/ may also be found graphically via this equal area or Maxwell construction. We remark that between Vg and vi the van der Waals pressure isotherms are said to exhibit a van der Waals loop. 

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... 

The non-physical van der Waals isotherm may be improved using the so-called Maxwell construction. It involves drawing the horizontal section AD, for which (8p/8V)T = 0, joining the two branches of the isotherm, EA and DF, corresponding to the liquid and gaseous phase of a system, respectively. It follows from the condition of equality of chemical potentials at a critical point that the section AD should be thus selected that the areas and S2 be equal. Between the points A and D the system is nonhomo-geneous, i.e. separated into two phases coexisting in equilibrium. The... 

Three different roots of Eq. (2.37) exist for a >2. The region enclosed by the dashed lines (shown in Fig. 2.19 for a = 2.5) corresponds to unstable states of the system. The surface pressure that corresponds to the coexistence of a condensed and gaseous state can be determined by using the so-called Maxwell construction [1, 104, 105]. That is, the areas enclosed by the dashed lines corresponding to this coexistence pressure and the two portions of the loop should be equal to each other on the fl vs A dependence. For a = 2.5 this pressure is marked in Fig. 2.19 by the solid line. 

Pv( > ) is called the a-phase (liquid phase) and the interval where again p (v,0) <0, is called the 0-phase (vapor phase). The interval (a,lS), where p (v, ) >0, is called the spinodal region and assiomed to be unstable. Also, in the isothermal case it should be noted that the horizontal line p = pm for which areas A and B are equal is called the Maxwell line and in the steady state case two phases coexist when the pressure is at p j. This construction of steady state solution with two coexistent phase is called the Maxwell line construction and the derivation is discussed in Fermi [12], Also, in the steady state case the states (b, n,) and ()8ni>°°) stable, [a ,a] and metastable... 

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...

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