Maxwell s construction

Fig. 1. The dependence of y on the volume fraction of the dry polymer in the swollen state q>2 for free swelling in water for c = 0 and s = 0.5. Numbers at curves denote the charge fraction on the chain, i. For i = 0.008, the critical value xt — 0-57 was determined using Maxwell s construction. From Ilavsky [34]...

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]

Fig. 4. An example of Maxwell s construction in the dependence of / on

Due to this shift, the experimentally determined extent of the collapse Ae is smaller than the one determined by Maxwell s construction. Also the value Xc found by employing Maxwell s construction is somewhat higher than the respective y of the acetone-water mixture at which the collapse takes place and which have been determined for the uncharged PAAm network. 

Fig. 6. Dependence of the extent of the collapse A and of the critical value of the interaction parameter yc on the content of sodium methacrylate xMNj (—) course determined by Maxwell s construction of data from Fig. 5 (O), ( ) experimental data. Taken from Ilavsky [11]...

By assuming that the effective degree of ionization i — xMNa [Pg.190]

It is seen that the phase transition (condensation) in the monolayer is accompanied by a significant decrease of the slope of the surface pressure isotherm. In contrast to the model based on Frumkin s equation and Maxwell s construction (see Fig. 2.19), the dependence of n on 0 in the phase coexistence region does not generally have a horizontal line, except for very small 6 values. 

O BoHIng point on an isothorm O Dew point on an isotnerm 9-0-0 Bolling point curve (connects all boiling points) e-o-0 Dew point curve (connects all dew points) maxwell s construction State points on theT] Isotherm Liquid phase Two-phase region... 

T< Tc, isotherms show unphysical oscillations analogous to the oscillations that result from the van der Waals equation of state for real gases, which predict an LGPT [37]. The range of densities where dP/dV)r > 0 are thermodynamically unstable and indicate that the system must phase separate into LDL and HDL. The equilibrium isotherm can be obtained from the isotherms obtained in simulations by applying Maxwell s construction (see Fig. 3). At volumes V > Vldl and V < HDL. the equilibrium states are (homogeneous) LDL and HDL, respectively. At volumes Vhdl < F < Vldl> regions of HDL and LDL coexist. The fraction of the system in each phase is determined by the lever rule [37]. 

FIGURE 7.2. Free energy F e) as a function of the thickness e of a liquid film deposited on a non wet-table substrate. Maxwell s construction describes the coexistence between the dry solid (e = 0) and a puddle (e = Cc), and shows that a film of thickness e < 6c lowers its energy by breaking up into two phases (dry solid and puddle), as indicated by the arrow. 

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...

Traditionally, transitions with phase coexistence are analyzed by means of Maxwell s construction. It was introduced to cure and interpret the unphysical backbending branch that occurs for real gases described by the van der Waals equation in the pressure (p) versus... 

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