Clausius - Clapeyron equation generalized

Equation (2.5) is the Clausius-Clapeyron equation. General integration, assuming to be constant, gives... 

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,... 

Generally, vapour pressure measurements are fitted to a form of the Clausius-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... 

In general, the molar enthalpy of vaporization is obtained from the Clausius-Clapeyron equation, representing the difference per mole of the enthalpy of the vapour and of the liquid at equilibrium with it ... 

Equation (9) is sometimes known as Clausius-Clapeyron equation and is generally spoken to as first latent heat equation. It was first derived by Clausius (1850) on the thermodynamic basis of Clapeyron equation. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

As a useful thermodynamic property, the isosteric heat of adsorption has been generally applied to characterize the adsorbent surface. The isosteric heat of adsorption is evaluated simply by applying the Clausius-Clapeyron equation if one has a good set of adsorption equilibrium ta obtained at several temperatures. 

This rule is very similar to the general law of Wrewsky, as would be expected from the Clausius-Clapeyron equation. The difference between the two rules is that Wrewsky s is based on a knowledge of the latent heats of evaporation of the components from the azeotropic solution, while the present rule is expressed in terms of more readily available quantities, the latent heats of evaporation of the pure liquids. 

There is a more or less generalized agreement that the isosteric adsorption heat is strongly affected by the microstructure of the adsorbent, particularly in the case of porous solids. This magnitude is better suited for structural analysis than other thermodynamic quantities. The use of the Clausius—Clapeyron equation to determine the isosteric adsorption heat has several limitations both theoretical and experimental, that are well known. 

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 ... 

There are several ways to separate an azeotropic mixture into two components of the desired purities, and these are discussed in. other chemical engineering courses. However, one method will be mentioned here, and it is based on the fact that in general the two components will not have the same heat of vaporization, so that by the Clausius-Clapeyron equation the temperature dependence of their vapor pressures will be different. Since the dominant temperature dependence in vapor-liquid equilibrium is that of the pure component vapor pressures, the azeotropic composition will also change with temperature (and pressure). Therefore, what can be done is to use two distillation columns operating at different pressures. 

The Clausius-Clapeyron equation may be considered a special case of the more general van t Hoff equation... 

Equation (6.72) is now very general. If we keep x[ as constant, we get a generalized Clausius - Clapeyron equations. If we keep the temperature constant, we get... 

The enthalpies of phase changes of low-volatility compounds are not generally determined directly, but are derived fi om die measured relation between vapour pressure and temperature with the help of the Clausius-Clapeyron equation (13). 

This is a remarkable result, for it is merely the Clapeyron equation (8.2.27) extended from pure substances to azeotropic mixtures. The derivative on the Ihs represents the slope of the locus of azeotropes on a PT diagram. We can use (9.3.21) as a basis for correlating azeotropic temperatures and pressures, just as we used it in 8.2.6 for correlating pure-component vapor pressures. We obtain the same generalized form of the Clausius-Clapeyron equation. 

Because the calorimetric methods of measurement of enthalpy of vapor formation are very difficult, the indirect mefliods are used, especially for less volatile substances. The application of generalized expression of the first and second laws of thermodynamics to the heterogeneous equilibrium between a condensed phase in isobaric- thermal conditions is given in the Clausius-Clapeyron equation that relates enthalpy of a vapor formation at the vapor pressure, P, and temperature, T. For one component system, the Clausius-Clapeyron equation has the form ... 

The general relationship between the amount of gas (volume, V) adsorbed by a solid at a constant temperature (T) and as a function of the gas pressure (P) is defined as its adsorption isotherm. It is also possible to study adsorption in terms of V and T at constant pressure, termed isobars, and in terms of T and F at constant volume, termed isosteres. The experimentally most accessible quantity is the isotherm, although the isosteres are sometimes used to determine heats of adsorption using the Clausius-Clapeyron equation. In addition to the observations on adsorption phenomena noted above, it was also noted that the shape of the adsorption isotherm changed with temperature. The problem for the physical chemist early in the twentieth century was to correlate experimental facts with molecular models for the processes involved and relate them aU mathematically. 

This is the common form of the Clausius-Clapeyron equation, also known as the Clapeyron equation. The equation is generally valid and describes all forms of two phase equilibria for one component systems. 

The melting pressure curve may be given based on the general form of the Clausius-Clapeyron equation in a similar manner to the vapor pressure and sublimation curves (Eq. (1-71))... 

Equilibrium concentrations and temperatures 7 are additionally influenced by the pressure. According to the Clausius-Clapeyron equation, a change in the molar volume Al " = V i — at the ceiling temperature T, generally gives... 

In general, when a solution freezes there is a separation of solute and solvent. That leads to the creation of a new triple point where the solvent, in solution, is in equilibrium with both its own pure vapor and pure solid. The relationship between the decrease in the freezing temperature and the lowering of the vapor pressure is given by the Clausius-Clapeyron equation for solid-liquid equilibria ... 

Pepekin s study of the thermodynamic properties of difluoramino and nitro compounds [74,75] included many organic difluoramines besides the products of electrophilic difluoramination cited above. Properties reported include heats of combustion, formation, and atomization, Clausius-Clapeyron equation parameters, and the enthalpies and entropies of evaporation and sublimation. This collection of properties allowed estimation of group additivity parameters for general calculations of thermodynamic properties of organic difluoramines, which were compared to those of corresponding nitro groups. 

In the previous chapter we obtained a number of relations, such as the Clausius-Clapeyron equation, which are correct whatever the natme of the phases which are in equilibrium, provided that there is only a single component. The question arises whether any equations of a comparable generality may be obtained for the case of a multi-component system. 

Substance A must not dissociate in the liquid state and be fully immiscible with the other component in the solid state. Derived from the formulation of the fundamental equilibrium condition, for such solid-liquid equilibria a special form of the generally valid Clausius-Clapeyron equation applies ... 

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