Equilibria Clapeyron equation

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 ... 

As pointed out earlier, the equilibrium constant of a system changes with temperature. The form of the equation relating K to T is a familiar one, similar to the Clausius-Clapeyron equation (Chapter 9) and the Arrhenius equation (Chapter 11). This one is called the van t Hoff equation, honoring Jacobus van t Hoff (1852-1911), who was the first to use the equilibrium constant, K. Coincidentally, van t Hoff was a good friend of Arrhenius. The equation is... 

Clausius-Clapeyron equation An equation expressing the temperature dependence of vapor pressure ln(P2/Pi) = AHvapCl/Tj - 1/T2)/R, 230,303-305 Claussen, Walter, 66 Cobalt, 410-411 Cobalt (II) chloride, 66 Coefficient A number preceding a formula in a chemical equation, 61 Coefficient rule Rule which states that when the coefficients of a chemical equation are multiplied by a number n, the equilibrium constant is raised to the nth power, 327... 

Line db in Figure 8.1 represents the equilibrium melting line for C02. Note that the equilibrium pressure is very nearly a linear function of T in the (p, T) range shown in this portion of the graph, and that the slope of the line, (d/ /d7 )s ], is positive and very steep, with a magnitude of approximately 5 MPa-K-1. These observations can be explained using the Clapeyron equation. For the process... 

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. 

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,... 

A rate of reaction usually depends more strongly on temperature than on concentration. Thus, in a first-order (n = 1) reaction, the rate doubles if the concentration is doubled. However, a rate may double if the temperature is raised by only 10 K, in the range, say, from 290 to 300 K. This essentially exponential behavior is analogous to the temperature-dependence of the vapor pressure of a liquid, p, or the equilibrium constant of a reaction, K. In the former case, this is represented approximately by the Clausius-Clapeyron equation,... 

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... 

This is known as the Clausius-Clapeyron equation. It is a state relationship that allows the determination of the saturation condition p = p(T) at which the vapor and liquid are in equilibrium at a pressure corresponding to a given temperature. 

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. 

As we have already stated, the Gibbs free energy modification connected with a polymorphic transition is zero. The univariant equilibrium along the transition curve is described by the Clapeyron equation ... 

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... 

Equation 6.56 is known as the equation of lowering of freezing point and is valid for solid mixtures crystallizing from multicomponent melts. Like the Clausius-Clapeyron equation, it tells us how the system behaves, with changing T, to maintain equilibrium on the univariant curve. However, whereas in the Clausius-Clapeyron equation equilibrium is maintained with concomitant changes in 7) here it is maintained by appropriately varying the activity of the component of interest in the melt and in the solid mixture. 

Because H fusion is difficult to measure as a result of the high value of 7), it may be derived indirectly through calculations involving the vitreous state (see Berman and Brown, 1987) or through the Clausius-Clapeyron equation for the crystal-melt equilibrium (cf equation 6.48 and section 6.3). 

Using the Gibbs-Helmholtz equation obtained Clapeyron equation for the equilibrium solid liquid. 

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. 

In chemisorption where severe surface perturbations can occur, the Clausius-Clapeyron equation cannot be applied, since equilibrium pressures are low and often unobtainable. Nonetheless, a differential heat analogous to the isosteric heat can be obtained from heats of immersion without recourse to pressure data where the amounts adsorbed prior to immersion can be measured gravimetrically (Sec. VII,A). 

The approach we follow is essentially that used to derive the Clapeyron equation (Atkins 1994). Suppose we consider an infinitesimal temperature change for a system in which adsorbed gas and unadsorbed gas are in equilibrium. The criterion for equilibrium is that the free energy of both the adsorbed (subscript s) and unadsorbed (subscript g) gas change in the same way ... 

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. 

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