Gibbs Duhem equation enthalpy

The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

To calculate the relative partial enthalpy of the salts in sea water it was necessary to integrate the Gibbs-Duhem equation graphically. 

The entropy, enthalpy, and volume quantities appearing in Equations (10.1) and (10.2) are molar quantities, but care must be taken that such quantities refer to the same mass of material. This requirement is readily proved. Consider one phase, the primed phase, to contain ( ) moles of component based on a specified chemical formula, and assume that no chemical reaction takes place in this phase. Assume that the double-primed phase contains (h,)" moles of component but that a reaction, possibly polymerization or decomposition, takes place in this phase. The chemical reaction may be expressed as = i vfii = 0, where the sum is taken over all reacting species. The two Gibbs-Duhem equations may then be written as... 

At constant temperature and pressure, the concentration-dependent activity coefficient can be determined from the free excess enthalpy by differentiation through the mole fraction. These equations are the basis for the methods of Wilson and Prausnitz to calculate the activity coefficient [19, 20], The Gibbs-Duhem equation is again a convenient method for checking the obtained equilibrium data ... 

Show that the partial molar volumes computed from Eqs. 8.6-4 and the partial molar enthalpies computed from Eqs. 8.6-9 must satisfy the Gibbs-Duhem equation. 

We treat the CjdHj term by deriving the Gibbs-Duhem equation, for enthalpy as follows. We have H = because Hj is a partial... 

AH and AV are molar enthalpy and molar volume of mixing, respectively. The heat of mixing is negligible in most cases. In an isobaric system the second term vanishes, and the Gibbs-Duhem equation takes the form ... 

According to the first Gibbs-Duhem equation, free enthalpy of the entire solution is equal... 

Melting Temperature and Enthalpy Depression with Crystal Radius Brun and coworkers [252], starting from the Laplace and Gibbs-Duhem equations, found the following relation ... 

The entropy flow rate from the upstream side is given as 5u = msu- The R-field represents the change in the intensive variables between the upstream and the downstream sides. The temperatures, pressures and the chemical potentials of the gas mixture in the upstream and the downstream sides are imposed by the storage elements on the corresponding sides. Due to this, there is an enthalpy difference between the upstream and the downstream sides, which can be represented as the relation between the changes in the intensive variables by using the Gibbs-Duhem equation [5] as... 

As was shown in Chapter 2 (equations (2.4.10) and (2.4.11)), the molar enthalpy of a pure compound Hm T) can be obtained from the tabulated values of Cp T), the heat capacity at constant pressure. For ideal mixtures the Hjok are the same as that of a pure compound. For nonideal mixtures a detailed knowledge of the molar heat capacities of the mixture is needed to obtain the //mi-For a pure compound, knowing p(po> T) at pressurepo and temperature T, the value of p(p, T) at any other pressure p can be obtained using the expression dp = —SjndT+ Vradp which Allows from the Gibbs-Duhem equation (5.2.4), where the molar quantities 5m = S/N and Vm = V/N- Since Tis fixed, dT = 0 and we may integrate this expression with respect to p to obtain... 

Here, is the excess enthalpy (heat of mixing) and V is the excess volume (volume change of mixing). Both and V are subject to direct experimental determination. Taking the total differential of Eq. (7), and comparing the result with Eq. (8), we obtain the Gibbs-Duhem equation ... 

In fluid mechanics it might be natural to employ mass based thermodynamic properties whereas the classical thermodynamics convention is to use mole based variables. It follows that the extensive thermodynamic functions (e.g., internal energy, Gibbs free energy, Helmholtz energy, enthalpy, entropy, and specific volume) can be expressed in both ways, either in terms of mass or mole. The two forms of the Gibbs-Duhem equation are ... 

Verify that the expressions developed in Example 6.12 for the partial molar enthalpies of sulfuric acid and water in a binary mixture at 21°C satisfy the Gibbs-Duhem equation. 

SOLUTION The Gibbs-Duhem equation can be written for the partial molar enthalpies in this... 

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