Clapeyron diagram

Fig. 2. Clapeyron diagram showing saturated refrigerant and isosteres...

Fig. 3 shows an idealised solar collector (generator) containing adsorbent which is connected to a condenser that rejects heat to the environment and an insulated box containing a liquid receiver and a flooded evaporator. Fig. 4 shows the p-T-x (pressure - temperature - concentration or Clapeyron diagram) for the adsorbent-adsorbate pair with typical temperatures. 

It appears from (2.11) that, when Al is negative, amorphization becomes possible at high pressures as shown experimentally for ice [2.129] and silicon [2.130]. For positive AVm, which describes generally the behavior of metallic systems, the principal features of a polymorphous Clapeyron diagram is presented in Fig. 2.25 in T-P-C space [2.20]. In the T-C plane, an increase in... 

Fig. 2.25. Polymorphous Clapeyron diagram in T-P-c space for a binary solid solution displaying three isothermal sections to the surface T(P, c)...

Barrer s discussion4 of his analog of Eq. 28 merits some comment. Equation 28 expresses the equilibrium condition between ice and hydrate. As such it is valid for all equilibria in which the two phases coexist and not only for univariant equilibria corresponding with a P—7" line in the phase diagram. (It holds, for instance, in the entire ice-hydratell-gas region of the ternary system water-methane-propane considered in Section III.C.(2).) In addition to Eq. 28 one has Clapeyron s equation... 

While the Gibbs phase rule provides for a qualitative explanation, we can apply the Clapeyron equation, derived earlier [equation (5.71)], in conjunction with studying the temperature and pressure dependences of the chemical potential, to explain quantitatively some of the features of the one-component phase diagram. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

We are permitted to assume that dp is directly proportional to dT because AH and AV are regarded as constants, although even a casual inspection of a phase diagram shows how curved the solid-gas and liquid-gas phase boundaries are. Such curvature clearly indicates that the Clapeyron equation fails to work except over extremely limited ranges of p and T. Why ... 

The Clapeyron equation, Equation (5.1), yields a quantitative description of a phase boundary on a phase diagram. Equation (5.1) works quite well for the liquid-solid phase boundary, but if the equilibrium is boiling or sublimation - both of which involve a gaseous phase - then the Clapeyron equation is a poor predictor. 

A sublimation process is controlled primarily by the conditions under which phase equilibria occur in a single-component system, and the phase diagram of a simple one-component system is shown in Figure 15.30 where the sublimation curve is dependent on the vapour pressure of the solid, the vaporisation curve on the vapour pressure of the liquid, and the fusion curve on the effect of pressure on the melting point. The slopes of these three curves can be expressed quantitatively by the Clapeyron equation ... 

If two phases of one component are present, only one degree of freedom remains, either temperature or pressure. Two phases in equilibrium are represented by a curve on a T — P diagram, with one independent variable and the other a function of the first. When either temperature or pressure is specified, the other is determined by the Clapeyron Equation (8.9). If three phases of one component are present, no degrees of freedom remain, and the system is invariant. Three phases in equUibiium are represented on a T — P diagram by a point called the triple point. Variation of either temperature or pressure will cause the disappearance of a phase. 

Phase Diagrams Construct phase diagram from Tfu, and Tvap measurements Phase transitions Identification of phase boundaries Comparison with Clausius-Clapeyron equation... 

Chemical potential, chemical equilibrium (Kp, Kc, Kx), Phase equilibrium (1 component), Phase diagrams Vapor pressure equation from Clapeyron eqn,... 

Despite the complexities introduced by metastable solid and liquid phases, topological features of the phase diagram can be thermodynamically interpreted in a standard manner from Clapeyron-type equations. Thus, from the forward slopes of a-fi, /3-liquid (stable), and a-liquid (metastable) phase boundaries, we can infer from (7.32) that... 

Neutron diffraction studies under pressure [84] on the 70/30 composition have revealed that transitions in this copolymer are displaced towards higher temperature with increasing pressure, as can be seen in the phase diagram of Fig. 11. In addition, it is worth noting the non-linear increase of the Curie temperature with pressure. By considering the Clausius-Clapeyron relation dTc/dP = TCAVC/Ahc, this effect can be related to a decrease in the volume... 

The key to the study of phase equilibria is the phase diagram, interpreted by the phase rule and the Clapeyron equation. Such phase diagrams will serve as the basis for our discussion. 

This is a further form of the Clapeyron Equation which gives the slope of the P versus T line (BC) for the liquid-vapour equilibrium in the phase diagram. 

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