Gibbs-Duhem equation at constant T and

The surface concentrations T depend on the thickness of the interfacial region, and we would like to express them through quantities which are independent of it. This can be done for those species which occur both at the interface and in the solution. Usually one of the components of the solution, the solvent, has a much higher concentration then the others. We denote it by the index 0 , and introduce surface excesses with respect to the solvent in the following way In the bulk of the solution the Gibbs-Duhem equation (at constant T and p) is simply E Ni dfri = 0, or ... 

The last expression is based on the definition of the activity coefficient. Substituting this expression in the Gibbs-Duhem equation at constant T and P gives the following ... 

It is not possible to apply the Gibbs-Duhem equation at constant T and p to this system because of the requirement of equilibrium between the solution and pure solid 1. Substitution of Eq. (11-147) for pi and Eq. (11-11) for P2 into Eq. (11-158) results in... 

We now attend to the third term above on the right, beginning with the Gibbs-Duhem relation at constant T and P Equation (1.22.26) reads... 

This set of simultaneous equations, coupled with the Gibbs-Duhem equation at constant T, are all we need to derive the most common forms of the basic theory providing a range of relationships between fluctuating quantities and thermodynamic... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

By combining the two last equations, the Gibbs-Duhem equation for a binary system at constant T and p is obtained ... 

As usual, our goal is to find the minimum of G( ) in order to determine the equilibrium position ( = eq) of the chemical reaction at constant T and P. From (6.10c) [cf. the Gibbs -Duhem equation (6.36a)], the differential dG under these conditions is simply... 

Following our development of the Gibbs-Duhem equation in Chapter 8, we now apply Eq. (45) to a process at constant T and P, where components are added to the surface in an arbitrary ratio. In this process, the chemical potentials will vary ... 

The Gibbs/ Duhem equation provides a relation between the Lewis/Randall rule and Henry s law. Substituting dGt from Eq. (11.28) for dAft in Eq. (11.8) gives, for a binary solution at constant T and P,... 

Frequently it is more convenient or only possible to measure activities or activity coefficients for a component that differs from the one in which the experimentalist is interested. In that case it is expedient to use the Gibbs-Duhem equation for a binary mixture. For a two-component system at constant T and P, we find n dii — —n2dfi2. On account of Eq. (3.6.2) this may be rewritten as... 

For this we employ the Gibbs-Duhem equation (6.39), which at constant T and p is... 

Gibbs-Duhem equations for binary mi.xture at constant T and P... 

Differentiation of the Gibbs-Duhem relation (i.e., equation 25-81) by a>a at constant T and p yields... 

This rearrangement of the Gibbs-Duhem equation in (29-129) is useful to simplify the curvature requirement for gmixture vs. yi at constant T and p ... 

By means of Eqs. (13) and (14) a generalized form of the Gibbs-Duhem equation is obtained (at constant T and P), which is expressed by... 

Equation 3.82 or 3.83 is the Gibbs-Duhem relation. It indicates the number of variables that can be independently varied for a system. At constant T and p, for a homogeneous system containing k components the chemical potentials of (k - 1) components can be chosen at will. 

Also the chemical potentials in the bulk are subject to the Gibbs-Duhem equation which at constant T and P may be written ... 

At constant T and P, the Gibbs-Duhem equation simplifies to Si=i 0. Analogous to chemical potential are partial molar properties such as V,-, Hi, and S. Show that for all partial molar quantities such as V , Hi, and Si, at constant Tand Pone can also write... 

We now use the Gibbs-Duhem equation to investigate the behavior of the solvent in an ideal-dilute solution of one or more nonelectrolyte solutes. The Gibbs-Duhem equation applied to chemical potentials at constant T and p can be written xt dp,i = 0 (Eq. 9.2.43). We use subscript A for the solvent, rewrite the equation as xa d/iA + = 0,... 

Notice that va and are not independent. To see this we simply must realize that we can carry out the steps from Eq. (2.166) to the Gibbs-Duhem equation (2.168) with G replaced by V and /u. replaced by Vi. At constant T and P this means... 

We have already derived the Gibbs-Duhem equation in Chapter 1.4. At constant p and T ... 

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