Gibbs Duhem equation at constant temperature and pressure

From the Gibbs-Duhem equation at constant temperature and pressure [Equation (11.34)], we can write... 

The chemical potential can be written in terms of the activity coefficient (see Eq (6.4)). Subsituting this expression into the Gibbs-Duhem equation, at constant temperature and pressure, we find... 

For the special case where G is the free energy, Eq. (2-13) is referred to as the Gibbs-Duhem equation. At constant temperature and pressure, Eq. (2-13) reduces to... 

The pressure at which standard-state fugacities are most conveniently evaluated is suggested by considerations based on the Gibbs-Duhem equation which says that at constant temperature and pressure... 

In equation (5.27), we used the Gibbs-Duhem equation to relate changes in the chemical potentials of the two components in a binary system as the composition is changed at constant temperature and pressure. The relationship is... 

The Gibbs-Duhem equation is one of the most extensively used relations in thermodynamics. It is written in the following equivalent forms for a binary solution at constant temperature and pressure ... 

GIBBS-DUHEM EQUATION, In a system of two or more components at constant temperature and pressure, the sum of the changes for the various components, of any partial molar quunlily. each multiplied by the number of moles of the component present, is zero. The special case of two components is ihe basis of the Gibbs-Duheni equation of the form ... 

The similarity to the Gibbs-Duhem equation is quite apparent, and indeed this equation is the Gibbs-Duhem equation if X refers to the Gibbs energy. We should note that the differential dXt, the differential that appears in Equations (6.13) and (6.14), depends upon the differential quantities of the temperature, the pressure, and the mole fractions as expressed in Equation (6.7). At constant temperature and pressure Equation (6.12) becomes a special case of Equation (6.14). 

Using the forces and flows identified in Eq. (7.1), and the Gibbs-Duhem equation for an n-component system at constant temperature and pressure, we obtain... 

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 ... 

The Gibbs-Duhem equation provides a relation between the chemical potentials of each of the chemical species in a given phase. At constant temperature and pressure, this relation simplifies to... 

In order to calculate the distribution coefficient by Equation 1.29, the activity coefficient Y must be evaluated. The activity coefficients are generally determined from the experimental data and correlated on the basis of thermodynamic phase equilibrium principles. The relationship most often used for this purpose is the Gibbs-Duhem equation (Equation 1.7). At constant temperature and pressure, this equation becomes... 

Since in much of the remainder of this book we are concerned with equilibrium at constant temperature and pressure, the Gibbs energy will be of central interest. The Gibbs-Duhem equations for the Gibbs energy, obtained by setting 0 = G in Eqs. 8.2-8, are... 

We first prove that, if a solution is ideal with respect to r - 1 of its components, it is ideal with respect to component r. At constant temperature and pressure, the Gibbs-Duhem equation [Eq. (6-59)] becomes... 

Equation 5.72 is called the Gibbs-Duhem equation and expresses how the chemical potentials for the different components depend on each other at constant temperature and pressure. If the amount of one component, n, is changed, we obtain from Equation 5.72... 

Now, derived from the second law of tliermodynamics, the Gibbs-Duhem equation relates the activities of the components in a phase (McQuarrie and Simon 1997 Berry, Rice, and Ross 2000 Levine 2008 Atkins and Paula 2009). At constant temperature and pressure, for a two-component system, it reads... 

The above relationship can be derived by taking the derivative of Eq. (4.39) with respect to X2 and using the Gibbs-Duhem equation ZLi = 0 at constant temperature and pressure. Finally, by... 

At constant temperature and pressure, from eq 2.100 the Gibbs-Duhem equation can be obtained ... 

The Gibbs-Duhem equation for any extensive function of state Z at constant temperature and pressure can be expressed as... 

Boiling point elevation and freezing point depression can be tied to the osmotic coefficient, (p, and are practical means for its measurement. We start with the Gibbs-Duhem equation at constant pressure and temperature ... 

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