Coexistence critical slope

Povodyrev et aJ [30] have applied crossover theory to the Flory equation ( section A2.5.4.1) for polymer solutions for various values of N, the number of monomer units in the polymer chain, obtaining the coexistence curve and values of the coefficient p jj-from the slope of that curve. Figure A2.5.27 shows their comparison between classical and crossover values of p j-j for A = 1, which is of course just the simple mixture. As seen in this figure, the crossover to classical behaviour is not complete until far below the critical temperature. 

Figure 5.23 Pressure composition isotherms for critical temperature 7. The construction of the hydrogen absorption in atypical metal (left). The van t Hoff plot is shown on the right. The slope of solid solution (a-phase), the hydride phase the line is equal to the enthalpy of formation (p-phase) and the region ofthe coexistence ofthe divided by the gas constant and the intercept with two phases. The coexistence region is the axis is equal to the entropy of formation...

Figure 2.9 Phase diagram for C02, showing solid-gas (S + G, sublimation ), solid-liquid (S + L, fusion ), and liquid-gas (L + G, vaporization ) coexistence lines as PT boundaries of stable solid, liquid, or gaseous phases. The triple point (triangle), critical point (x), and selected 280K isotherm of Fig. 2.8 (circle) are marked for identification. Note that the fusion curve tilts slightly forward (with slope 75 atm K-1) and that the sublimation and vaporization curves meet with slightly discontinuous slopes (angle < 180°) at the triple point. The dotted and dashed half-circle shows two possible paths between a liquid (cross-hair square) and a gas (cross-hair circle) state, one discontinuous (dashed) crossing the coexistence line, the other continuous (dotted) encircling the critical point (see text).

An isochoric equation of state, applicable to pure components, is proposed based upon values of pressure and temperature taken at the vapor-liquid coexistence curve. Its validity, especially in the critical region, depends upon correlation of the two leading terms the isochoric slope and the isochoric curvature. The proposed equation of state utilizes power law behavior for the difference between vapor and liquid isochoric slopes issuing from the same point on the coexistence cruve, and rectilinear behavior for the mean values. The curvature is a skewed sinusoidal curve as a function of density which approaches zero at zero density and twice the critical density and becomes zero slightly below the critical density. Values of properties for ethylene and water calculated from this equation of state compare favorably with data. 

Figure 4. Difference between the reduced isochoric slopes at the coexistence curve for ethylene as a function of t on a logarithmic scale. The slope is the critical exponent A.

We now discuss the behavior of phase boundaries. When t critical line w = LOc(u,t) which is the phase boundary between the DS and AS phases, is almost linear with positive slope. Beyond the multicritical point, slope of the line separating the AS and the DC phases rises rather sharply. In a region specified hy Uc < u < Uci (the value of Mq depends on t) we have the coexistence between the adsorbed SAW and the collapsed globule phase. This region is shown in Fig 18 by a dotted line. For u > Uq (t) the line w = w u, t) becomes almost flat. The value of Uq (t) decreases as t is increased and becomes equal to that of Uc at t = t = 0.34115. Att = t the multicritical point becomes a symmetric desorbed and collapsed pentacritical point having four eigenvalues greater than one. 

Equation 8.4.10 is the Clausius-Clapeyron equation. It gives an approximate expression for the slope of a liquid-gas or solid-gas coexistence curve. The expression is not valid for coexisting solid and liquid phases, or for coexisting liquid and gas phases close to the critical point. 

Figure 4.5 shows the phase diagram for water over a range of temperatures and pressures. Note the three curves, AB, AC, and AD. Curve AB indicates the temperature and pressure conditions at which ice and water vapor can coexist at equilibrium. Curve AC indicates the temperature and pressure conditions at which liquid water and water vapor coexist at equilibrium. Similarly, curve AD indicates the temperature and pressure conditions at which ice and liquid water coexist at equilibrium. Because ice is less dense than liquid water, an increase in pressure lowers the melting point. (Most substances have a positive slope for this curve.) Point A is the triple point of water. The triple point of a substance indicates the temperature and pressure conditions at which the solid, liquid, and vapor of the substance can coexist at equilibrium. Point C is the critical point of water. The critical point of a substance indicates the critical temperature and critical pressure. The critical temperature (t is the temperature above which the substance cannot exist in the liquid state. 

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