Classical Clapeyron equation

The classical Clapeyron equation adequately predicts the features of first-order phase transitions, and this has been established for a number of examples of first-order transitions effected by the deliberate variation of temperature or pressure. Second- or higher-order transitions are not readily explained by classical thermodynamics. Unlike the case of first-order transitions, where the free-energy surfaces of the two phases... 

Correlation Methods Vapor pressure is correlated as a function of temperature by numerous methods mainly derived from the Clapeyron equation discussed in the section on enthalpy of vaporization. The classic simple equation used for correlation of low to moderate vapor pressures is the Antoine S equation (2-27). 

There is another important law that follows from the classical theory of capillarity. This law was formulated by J. Thomson [16], and was based on a Clausius-Clapeyron equation and Gibbs theory, formulating the dependence of the melting point of solids on their size. The first known analytical equation by Rie [17], and Batchelor and Foster [18] (cited according to Refs. [19,20]) is... 

Thermodynamics and kinetics can surely be counted—along with transport phenomena, chemistry, unit operations, and advanced mathematics—as subjects that form the foundation of Chemical Engineering education and practice. Thermodynamics is of course a very old subject. For example, it was the same Rudolf Clausius, who in 1865 coined two immortal sentences (1) "The energy of the universe is constant" and (2) "The entropy of the universe tends to a maximum," that developed the famous Clausius-Clapeyron equation, one of the most basic physico-chemical relationships. Classical thermodynamics was largely complete in the 19th century, before even the basic structure of the atom was understood. 

This expression reduces to the classical Clausius-Clapeyron equation when the difference in compressibility, thermal expansion and heat capacity vanish as is observed for most phase transitions in lipids [80]. 

If we leave the foundations of statistical theory alone, we see that the applications depend on taking the spectroscopic data and plugging them into equation 25. Thus in actual practice, statistical thermodynamics accepts the classical theory without question and provides methods for the calculation of thermodynamic properties. Many classical thermodynamic relations cannot be derived from statistical theory. For instance, the phase rule simply falls out of classical theory. Derivation of the phase rule from statistical theory, without a large set of assumptions, does not seem to be possible. Popular equations such as the Clapeyron and Clausius-Clapeyron equations cannot be derived from statistical theory without the aid of classical thermodynamics. 

Another classical method for determining the heat of adsorption is based on the application of the Clausius-Clapeyron equation ... 

Note too, that while the unfrozen water content as a function of temperature is dependent on the soil (Fig. 1.1), the suction or stress states are independent of the type of soil. Indeed, the stress states p and are essentially described by equation 1.4, which is often known as the modified Clausius-Clapeyron equation It can indeed be derived from that purely general equation as Edlefsen and Anderson (1943) in their classic work showed (although interestingly, they concluded, wrongly, that a different variant of the equation would describe soil freezing) ... 

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