Phase rule Gibbs-Duhem equation

Turning now to adsorption equilibrium, let us apply algebraic methods to a two component 1,2 phase system. From the phase rule there will be two degrees of freedom, but we shall reduce this to one by maintaining the temperature constant. Then for the total system there exists a Gibbs-Duhem equation... 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

Applications of the Gibbs-Duhem equation and the Gibbs phase rule... 

The Gibbs-Duhem equation is applicable to each phase in any heterogenous system. Thus, if the system has P phases, the P equations of Gibbs-Duhem form a set of simultaneous, independent equations in terms of the temperature, the pressure, and the chemical potentials. The number of degrees of freedom available for the particular systems, no matter how complicated, can be determined by the same methods used to derive the phase rule. However, in addition, a large amount of information can be obtained by the solution of the set of simultaneous equations. 

The Gibbs-Duhem equation is the basis for the phase rule of Gibbs. According to the phase rule, the number of degrees of freedom F (independent intensive variables) for a system involving only PV work, but no chemical reactions, is given by... 

Gibbs-Duhem Equation and the Phase Rule at Chemical Equilibrium... 

GIBBS-DUHEM EQUATION AND THE PHASE RULE AT CHEMICAL EQUILIBRIUM... 

The Gibbs-Duhem equation (50.6), derived below, proves to be a useful starting point for an alternative derivation of the Clausius Claperyron equation to that offered in Frame 26) and offers an alternative proof of the Phase Rule to that given in Frame 30. 

Derivation of the Phase Rule using the Gibbs-Duhem Equation... 

The phase rule can be derived in an alternative fashion from a consideration of the Gibbs-Duhem equations (6-60) for the phases of the heterogeneous system and the restrictions imposed on the variations of the intensive variables and/ = 1,. . . , r a = l,. . . v,... 

The next step is to make clear the Gibbs adsorption amount, using the above relative adsorption. Recall (8.27), in which the interfacial tension is a function of i -H 2 independent variables. However, the Gibbs phase rule permits only i independent variables for two phases including i components. Therefore, the problem is how to reduce the number of intensive variables by two while keeping thermodynamical consistency. The Gibbs-Duhem equations for two homogeneous phases a and respectively, are... 

This is the Gibbs-Duhem equation for surface adsorption. Now y can be expressed in terms of i independent intensive variables, which is consistent with the phase rule. The important point is that the coefficient of the second term is the relative adsorption of component i with respect to component 1. Then, we have... 

Equation (15) is called the general Gibbs—Duhem relation. Because it tells us that there is a relation between the partial molar quantities of a solution, we will learn how to use it to determine a Xt when all other X/ il have been determined. (In a two-component system, knowing Asolvent determines Asolute.) This type of relationship is required by the phase rule because, at constant T, P, and c components, a single-phase system has only c — 1 degrees of freedom. 

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