Phase Equilibria. Clapeyron Equation

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. 

Clapeyron-Clausius equation A thermodynamic equation applying to any two-phase equilibrium for a pure substance. The equation states ... 

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

The Clapeyron equation expresses the dynamic equilibrium existing between the vapor and the condensed phase of a pure substance ... 

This equation is called the Clapeyron equation and can be applied to any two phases in equilibrium, e.g., solid and liquid, liquid and vapor, solid and vapor or two crystalline forms of the same solid. Thus for the equilibrium... 

Equation 3 is analogous to the Clausius-Clapeyron equation for equilibrium of a substance in the vapor and condensed phases (4). 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

The worst deviations from the Clapeyron equation occur when one of the phases is a gas. This occurs because the volume of a gas depends strongly on temperature, whereas the volume of a liquid or solid does not. Accordingly, the value of A Vm is not independent of temperature when the equilibrium involves a gas. 

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. 

So far we have made no special assumptions as to the nature of the Phases A and B in deriving Equation (8.10). Evidently the Clapeyron equation is applicable to equilibrium between any two phases of one component at the same temperature and pressure, and it describes the functional relationship between the equilibrium pressure and the equilibrium temperature. 

The Clapeyron equation can be reduced to a particularly convenient form when the equilibrium between A and B is that of a gas (g) and a condensed (cond) phase [liquid or solid]. In this situation... 

A confirmation of this conclusion also is provided by an examination of the solid-liquid equilibrium in the neighborhood of 0 K. As shown in Equation (8.9), a two-phase equilibrium obeys 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. 

The Clausius-Clapeyron equation describes the univariant equilibrium between crystal and melt in the P-Tfield. Because molar volumes and molar entropies of molten phases are generally greater than their crystalline counterparts, the two terms and AFfusion both positive and we almost invariably observe an... 

Clapeyron equation deals with the change in the equilibrium pressure with a change in the equilibrium temperature. Suppose Tand/ are temperature and pressure at equilibrium. At equilibrium, free energies of the two phases are equal. 

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

The vapor pressure, pv, is the pressure exerted by fluids and solids at equilibrium with their own vapor phase. The vapor pressure is a strong function of T, as expressed in the Clausius-Clapeyron equation [1] ... 

Finally, Section 4.6 concerns the relationship of phase equilibrium to other hydrate properties. The hydrate application of the Clapeyron equation is discussed... 

However, the condensed three-phase P-T locus is not exactly vertical. Ng and Robinson (1977) measured the Lw-H-Lhc equilibrium for a number of structure II hydrate mixtures and suggested that a better estimation of the slope dP/dT might be obtained through the Clapeyron equation ... 

In hydrate equilibrium, it may seem slightly unusual to apply it to binary systems (water and one guest component) of three-phase (Lw-H-V or I-H-V) equilibrium to obtain the heats of dissociation. As van der Waals and Platteeuw (1959b) point out, however, the application of the Clapeyron equation is thermodynamically correct, as long as the system is univariant, as is the case for simple hydrates. 

Roberts et al. (1940), Barrer and Edge (1967), Skovborg and Rasmussen (1994) present similar, detailed derivations to consider the use of the Clapeyron equation for hydrate binary and multicomponent systems. The reader is referred to the work of Barrer and Edge (1967) for the precise meaning of dP/dT and the details of the derivation. Barrer and Stuart (1957) and Barrer (1959) point out that the problem in the use of the Clapeyron equation evolves from the nonstoichio-metric nature of the hydrate phase. Fortunately, that problem is not substantial in the case of hydrate equilibrium, because the nonstoichiometry does not change significantly over small temperature ranges. At the ice point, where the hydrate number is usually calculated, the nonstoichiometry is essentially identical for each three-phase system at an infinitesimal departure on either side of the quadruple point. 

The de Forcrand method has been found to be much more accurate than Villard s Rule. One reason for its accuracy is related to the determination of AHf and AH2 from three-phase (Lw-H-V or I-H-V) equilibrium measurements of pressure and temperature via the Clapeyron equation ... 

This indirect method avoids problems of metastability and occlusion in the direct method because the P-T measurements are at equilibrium and they are not dependent on the amounts of each phase present. The question about the validity of the method centers on the validity of the Clapeyron equation to the three-phase hydrate equilibrium, as discussed in the following section. 

Table 4.10 shows the literature values for hydrate numbers, all obtained using de Forcrand s method of enthalpy differences around the ice point. However, Handa s values for the enthalpy differences were determined calorimetrically, while the other values listed were determined using phase equilibrium data and the Clausius-Clapeyron equation. The agreement appears to be very good for simple hydrates. Note also that hydrate filling is strongly dependent on... 

In a thorough review of calorimetric studies of clathrates and inclusion compounds, Parsonage and Staveley (1984) presented no direct calorimetric methods used for natural gas hydrate measurements. Instead, the heat of dissociation has been indirectly determined via the Clapeyron equation by differentiation of three-phase equilibrium pressure-temperature data. This technique is presented in detail in Section 4.6.1. 

The Clausius-Clapeyron equation" is an integrated version of the Clapeyron equation that applies to equilibrium between an ideal gas vapor phase and a condensed phase, with the conditions that the volume of the... 

The fundamental relationship that allows the determination of the equilibrium vapor pressure, P, of a pure condensed phase as a function of temperature is the Clausius-Clapeyron equation... 

This equation is the well-known Clapeyron equation, and expresses the pressure of the two-phase equilibrium system as a function of the temperature. Alternatively, we could obtain dp/dT or dp/dP. When three phases are present, solution of the three equations gives the result that dT, dP, and dp are all zero. Thus, we find that the temperature, pressure, and chemical potential are all fixed at a triple point of a one-component system. 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

Gas and condensed phase equilibrium the Clausius-Clapeyron equation... 

The Clapeyron equation can be simplified to some extent for the case in which a condensed phase (liquid or solid) is in equilibrium with a gas phase. At temperatures removed from the critical temperature, the molar volume of the gas phase is very much larger than the molar volume of the condensed phase. In such cases the molar volume of the condensed phase may be neglected. An equation of state is then used to express the molar volume of the gas as a function of the temperature and pressure. When the virial equation of state (accurate to the second virial coefficient) is used,... 

If the gas phase activity of the host is controlled by the presence of a pure condensed phase, solid or liquid, the equilibrium between host and guest in a stoichiometric clathrate can be described in terms of the gas phase pressure of the guest. This is, in effect, a vapor pressure for the guest. At higher pressures the guest will condense to form clathrate, and at lower pressures the clathrate will decompose. Temperature variation of this pressure will follow the Clapeyron equation which, with the usual assumptions (ideal gas behavior of the vapor and negligible volume of the condensed phase), reduces to the Clausius-Clapeyron equation ... 

This is the Clapeyron equation, which apphes to any type of phase equilibrium. 

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