Maxwell construction

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. 

The Maxwell construction produces a Van der Waals phase diagram that resembles the experimental results of argon, shown in figure 1, in all respects. 

In the region of a first-order transition ip has equal minima at volumes V and V2, in line with the Maxwell construction. The mixed phase is the preferred state in the volume range between V and V2. It follows that the transition from vapour to liquid does not occur by an unlikely fluctuation in which the system contracts from vapour to liquid at uniform density, as would be required by the maximum in the Van der Waals function. Maxwell construction allows the nudeation of a liquid droplet by local fluctuation within the vapour, and subsequent growth of the liquid phase. 

Figure 5.30 Chemical potential as a function of the hydrogen concentration x and the temperature. The coexistence curve (TJ and the Maxwell construction between the two points where/...

Figure 12 Surface free energy along the axis a of Fig. 11. (a) At low temperatures the free energy is not stable for the whole range of orientations 0

The Maxwell construction would determine the condition of two phase coexistence or the points on the curves where the first-order phase change occurs [6,7]. It is the condition that the two phases have the same value of g or j d II = 0 from Eq. (2.6) at zero osmotic pressure, v2 and vx being the values of v in the two phases. However, this criterion is questionable in the case Kcritical point). This is because the shear deformation energy has not been taken into account in the above theory. See Sect. 8 for further comments on this aspect. 

Fig. 1. 2 X versus v at II = 0 from Eq. (2.33). The ordinate is (i — xMPc and the abscissa is vpys. The dotted line represents a discontinuous change determined by the Maxwell construction, which is not reliable for K % n, however, (see the text)...

Fig 1 shows that the frequency of the main component of the sound falls continuously as the length of the tube above the burning surface increases. Maxwell constructed a constant-frequency whistle by applying the coachman s... 

The fluctuations of the pearl number smear out the force oscillations of the pearl opening for p>(b/lBf2)2/3z as found previously. The fluctuations of the pearl sizes are reflected by the prefactors p ll2(lnf2lby2pl3. As in the previous simple argument where the fluctuations have been ignored, the force shows oscillations with unstable decreasing branches that should be replaced by plateaus or pseudo-plateaus using the Maxwell construction. These results have been confirmed by detailed variational calculations [70-73]. 

Since gels which lack hydrophobic interactions are thought to show a thermo-swelling type of the transition, we applied our model to such systems to confirm its applicability. Calculated results are plotted against the reduced temperature (7 ) at various ft values in Fig. IS. The transition points were determined from the Maxwell construction. The calculated swelling curves reveal the thermoswelling behavior. An increase in fi enhances the ma tude of the volume change and lowers the transition temperature. 

A schematic plot of the variation of the pressure with volume, as predicted by the van der Waals equation of state, at various temperatures is given in Fig. 10.2. At temperatures above the critical temperature, the pressure-volume variation is monotonic and qualitatively similar to that of an ideal gas (see dotted-line). At temperatures below tire critical temperature, the pressure-volume curve begins to oscillate, exhibiting a van der Waals" loop (see dashed-line). This behavior is unphysical, but represents tire vapor-liquid transition, and should be replaced by the solid line. The precise location of the solid line is given by the Maxwell construction. 

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... 

Figure 38 illustrates the family of isotherms of equation (3.14) plotted using the Maxwell construction. 

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. 

The dependencies of IT on B for various 0 values, calculated from Eqs. (2.118) and (2.119), are shown in Fig. 2.21. It is seen that the phase transition in the monolayer is characterised by a decreased slope of the n (B)-curves, in contrast to the two-dimensional condensation model based on the Gibbs equation (2.114) [119-128] or that derived in [1, 104, 105] employing the Frumkin equation and Maxwell construction. With increasing B, all isotherms exhibiting a... 

The equihbrium chemical potential is shifted from the Maxwell construction, p = p,Q in the proximity of the solid surface. In the sharp interface theory, this shift, called disjoining potential [25], is defined as... 

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. 

With the chemical potential and pressure obtained in the form of the closed expressions (4.A.9) and (4.A.11) in Chapter 4, the phase coexistence envelope can be localized directly by solving the mechanical and chemical equilibrium conditions (1.134) and (1.135) for the vapor and liquid phase densities, Pvap and puq, whether or not the solution exists for all intermediate densities. Provided the isotherm is continuous across all the region of vapor-liquid phase coexistence, Eqs.(1.134) and (1.135) are exactly equivalent to the Maxwell construction on either pressure or chemical potential isotherm. This stems from the fact that the RISM/KH theory yields an exact differential for the free energy function (4.A. 10) in Chapter 4, which thus does not depend on a path of thermodynamic integration. 

Figure 1.18. Pressure isotherms of SPC water (solid lines) obtained by the RISM/KH theory supplemented with the Maxwell construction. Metastable and unstable states (thin solid and short-dash portions, respectively). Binodal and spinodal (bold solid and dashed lines, respectively).

Marcus theory 37 Maxwell construction 46, 50, 51 mean spherical approximation (MSA) 8,49-51,171, 172,178-180 -closure 9,49,50,171,180 mechanical instability 51 melittin 122... 

The above results agree with the classical results. Namely, (i) agrees with the Maxwell construction of steady state solutions with coexistent phases and (ii) agrees with the fact that for any transformation occurring in an isolated system the entropy of the final state can never be less than that of the initial state. Therefore, in order to support the feasibility of the assumption of existence of one-parameter family of solution (Fig. 3.1 and 3.2), it is desirable to have nontrivial solutions which minimize locally the entropy rate among the solutions in Fig. 3.1 and 3.2. For this purpose, considering that the entropy rate also depends on Ug, u, and etc., we express as follows... 

Ferri, J.K. and Stebe, K.J., Soluble surfactants undergoing surface phase transitions a Maxwell construction and the dynamic surface tension, J. Colloid Interface Sci., 209, 1, 1999. 

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)). 

Fig. 2. a) The multiple solution branches from the roots of Eq. 22, for a first-order transition from the Bean-Rodbell model, and b) the Maxwell construction for determining the critical field He and the full equilibrium solution, for a first-order magnetic phase transition system. 

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