Phase equilibrium Clausius-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). 

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 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... 

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] ... 

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... 

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... 

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

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 ... 

Note that any of the equations in (2.3.5) or (2.3.4) specifies P as a function of T or vice versa. It is not always recognized that the equilibrium constraints manifested in the Clausius—Clapeyron equation are commonly employed to fix the temperature of a helium bath in the range 0.3 to 4.2 K, by adjusting the vapor pressure of the helium gas above the liquid phase to correspond to the desired temperature. 

Gaseous anaesthesia in mice is an equilibrium process between the anaesthetic gas and the phase in which the gas exerts its effect (the biophase). As such, it should be amenable to treatment by the Clausius-Clapeyron equation. Modification of this equation is required when the distribution of a series of gases is to... 

In this case it is easy to show that (29.84) or its equivalent (29.86) can be put in the form of a Clausius-Clapeyron equation. For in the indifferent state considered, the system can undergo an equilibrium transformation without any effect on the composition of the phases. We can show, just as in 2 of chapter XXVIII that this change takes place at constant T andp. The intensive variables v, are thus constant during the transformation, so that on differentiating... 

Solids Below the triple point, the pressure at which the solid and vapor phases of a pure component are in equilibrium at any given temperature is the vapor pressure of the solid. It is a monotonic function of temperature with a maximum at the triple point. Solid vapor pressures can be correlated with the same equations used for liquids. Estimation of solid vapor pressure can be made from the integrated form of the Clausius-Clapeyron equation... 

Clausius-Clapeyron equation The differential equation relating pressure of a substance to temperature in a system in which two phases of the substance are in equilibrium. Also referred to as Clapeyron equation, or Clapeyron-Clausius equation. 

If you plot the temperature and vapor pressure data given in Table 1, you reconstruct the liquid-vapor equilibrium line in the phase diagram of that liquid (Fig. 174). The equation of this line, and you might remember this from your freshman chemistry course, is the Clausius-Clapeyron equation ... 

In the previous section it was observed that the Langmuir postulates of sites of equal activity and no interaction between occupied and bare sites were responsible for nonagreement with experimental data. It might be surmised that these assumptions correspond to a constant heat of -ad-sm-pt-ion—Indeed.-it-is-p.QssibIe to derive the Langmuir isotherm by assuming that is independent of d. The heat of adsorption can be evaluated from adsorption-equilibrium data. First the Clausius-Clapeyron equation is applied to the two-phase system of gas and adsorbed component on the surface ... 

Although the heat of adsorption or enthalpy change accompanying adsorption is directly obtained by calorimetry, it can conveniently be evaluated from the adsorption isostere. According to thermodynamics, the relationship between temperature T and pressure P under a state of -(J> phase equilibrium can generally be expressed with the Clausius-Clapeyron equation ... 

Clausius-Clapeyron equation - An approximation to the Clapeyron equation applicable to liquid-gas and solid-gas equilibrium, in which one assumes an ideal gas with volume much greater than the condensed phase volume. For the liquid-gas case, it takes the form d(lnp)/dT = A HIRV- where R is the molar gas constant and A H is the molar enthalpy of vaporization. For the solid-gas case, A H is replaced by the molar enthalpy of sublimation, A H. 

Thermodynamic analysis of the equilibrium between a condensed phase (solid or liquid) and the vapor is summarized by the Clausius-Clapeyron equation ... 

The ideal gas law characterizes the relationship between pressure, temperature, and volume for gases. Both the Clausius-Clapeyron and Antoine equations characterize the vapor-liquid equilibrium of pure components and mixtures. At atmospheric pressure and ambient temperature, water is a liquid but an equilibrium exists with its vapor phase concentration—its vapor pressure. The vapor pressure is a function of temperature. The formula for the Clausius-Clapeyron equation is ... 

The criteria for equilibria involving solid phases are exactly those given in 7.3.5 for any phase-equilibrium situation phases in equilibrium have the same temperatures, pressures, and fugacities. Moreover, pure-component solid-fluid equilibria obey the Clapeyron equation (8.2.27). This means the latent heat of melting is proportional to the slope of the melting curve on a PT diagram and the latent heat of sublimation is proportional to the slope of the sublimation curve. In the case of solid-gas equilibria, the Clausius-Clapeyron equation (8.2.30) often provides a reliable relation between temperature and sublimation pressures, analogous to that for vapor-liquid equilibria. 

At thermodynamic equilibrium the chemical potential of each component i in both liquid and solid phases has to be equal. For simple systems and certain simplifications, like pure crystalline solid phase of component b (see Walas 1985), thermodynamic considerations lead to the well-known Clausius-Clapeyron equation... 

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