Thermodynamics Clausius-Clapeyron equation

Fundamental Property Relation. The fundamental property relation, which embodies the first and second laws of thermodynamics, can be expressed as a semiempifical equation containing physical parameters and one or more constants of integration. AH of these may be adjusted to fit experimental data. The Clausius-Clapeyron equation is an example of this type of relation (1—3). 

Enthalpy of Vaporization The enthalpy (heat) of vaporization AHv is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liqmd of a pure component i.e., at a temperature (below the critical temperature) anci corresponding vapor pressure. AHy is related to vapor pressure by the thermodynamically exact Clausius-Clapeyron equation ... 

Although thermodynamically it is relatively simple to determine the amount of water vapor that enters the atmosphere using the Clausius-Clapeyron equation (see, e.g.. Chapter 6, Equation (1)), its resultant atmospheric residence time and effect on clouds are both highly uncertain. Therefore this seemingly easily describable feedback is very difficult to quantify. 

The slope of the line allows for the determination of the enthalpy of vaporization of water, A//Vap, and the y intercept yields the entropy of vaporization, A. S vap As both the enthalpy and the entropy of water increase as the phase change liquid — vapor occurs, the slope and y intercept of the Clausius-Clapeyron equation are negative and positive, respectively. At 373 K these thermodynamic quantities have values of AHvap = 40.657 kJ mol-1 and ASvap = 109.0 J K-1 mol-1. The leavening action due to water vapor or steam arises from the increased amount of water vapor that forms as pastry temperatures initially rise in the oven and then from the increased volume of the water vapor as temperatures continue... 

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 thermodynamics at a first-order transition ... 

The Clausius-Clapeyron Equation expresses the relationship between vapor pressure and temperature. It is the equation for the vapor-pressure line. We will develop this equation with the Clapeyron Equation, which was developed using thermodynamic theory. 

To check the phase transformation isotropic -> nematic, the validity of the Clausius Clapeyron equation is examined. It has been shown 38), that within the experimental error the results fulfill Eq. 1 in analogy to the low molar mass l.c. The phase transformation isotropic to l.c. is therefore of first order with two coexisting phases at the transformation point. Optical measurements on the polymers confirm these thermodynamical measurements (refer to 2.3.1.3). 

The Clausius-Clapeyron equation is an exact thermodynamic relationship between the slope of the vapor pressure curve and the molal heat of vaporization ... 

For wider temperature ranges, Hv (T) can be expressed as a polynomial or some other function of T. Integration of the Clausius-Clapeyron equation then leads to expressions given in the Handbook of Vapor Pressure (Yaws 1994) or in the Physical and Thermodynamic Properties of Pure Chemicals (Daubert et al. 1994). 

Problem 6 Give the thermodynamic derivation of Clapeyron equation and Clausius-Clapeyron equation. Discuss their applications also. 

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. 

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. 

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... 

The requirement of thermodynamic reversibility also applies to the chromatographic method, but in this case it is necessary to work at very low surface coverage (at zero coverage ) in the Henry s law region. Values of the specific retention volume, Vs, determined at different temperatures are inserted in the Clausius-Clapeyron equation in place of the equilibrium pressures to obtain A h. Provided that a number of conditions are observed, the method is capable of providing a fairly easy and rapid assessment of the adsorbent—adsorbate interaction energy. 

Although we shall not be concerned experimentally with measuring heats of adsorption, it is appropriate to comment that Mi for the physical adsorption of a gas is always negative, since the process of adsorption results in a decrease in entropy. The isosteric heat of adsorption (the heat of adsorption at constant coverage 6) can be obtained by application of the Clausius-Clapeyron equation if isotherms are determined at several different temperatures the thermodynamics of adsorption have been fully discussed by Hill. ... 

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. 

The Clausius-Clapeyron Equation. This equation expresses the relationship between vapor pressure and temperature. In 1834 Clapey-ron, using thermodynamic iJieory, developed the following equation which will be accepted without derivaticm... 

This expression, known as the Clausius-Clapeyron Equation, is of great historic significance, being a very early derivation that links seemingly unrelated variables. This was considered to be a noteworthy example of the power of thermodynamic theory and may be considered a precursor to later theoretical developments. 

Summarizing, an attempt has been made to provide a systematic account of the thermodynamic properties of the adsorbed phase. The Gibbs adsorption equation, as an extension of the Clausius-Clapeyron equation, has played a key role in linking experimental isotherm data to the determination of molar or differential entropies and enthalpies. Similarly, calorimetric measurements can be systematically applied to obtain the same type of information. 

Preliminary to such a search we examine several thermodynamic properties of fluids at or close to criticality, that clearly show why and how fluctuations dominate under such conditions, (i) Consider first the isothermal compressibility, kj = —(dV/dP)T/V. At the critical point the isotherm dP/dV)r has zero slope thus, Ki grows indefinitely as T —> Tc. (ii) Using Eq. (1.3.13) and the definition for K one finds that (dV/dT)p = -(dV/dP)TidPldT)v = KiV dP/dT)y, wherein (dP/dT)v does not vanish. Therefore, the coefficient of thermal expansion, = i /V) dV/BT)p also grows without limit as the critical point is approached, (iii) According to the Clausius-Clapeyron equation in the form AH = T(Vg — Vi)(dP/dT), the heat of vaporization of the fluid near the critical point becomes very small, since Vg — Vi 0, whereas dP/dT remains finite. 

Phase trcmsitions in monolayers may be treated thermodynamically analogously to those in three-dimensional systems. As will be derived in sec. 3.4, the Clausius-Clapeyron equation, relating the variation of pressure with temperature for a two-dimensional situation, reads... 

A new thermodynamic derivation of eq. (4.8) has been proposed making use of a modified Clausius-Clapeyron equation. The derivation of this equation is based on the assumption that plastic deformation involves a partial melting of the polymer crystals (Hirami et al, 1999). 

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. 

For the adsorption of a mole of any gaseous molecule onto an inert surface (one that is not changed by the adsorption itself), it can be shown from thermodynamic principles (see Chapter 1) that this vapor-adsorbate phase change is described by the Clausius-Clapeyron equation ... 

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