Gibbs-Duhem equation fugacity coefficient

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

Two limitations are involved in the derivation of the above equation (1) the compositions of mixed solvents (points c and d) should be close enough to each other for the trapezoidal mle used to integrate the Gibbs-Duhem equation to be valid, (2) the solubility of the solid should be low enough for the activity coefficients of the solvent and cosolvent to be taken equal to those in a solute-free binary solvent mixture. In addition, the fugacity of the solid phase in Eq. (4) should remain the same for all mixed solvent compositions considered. 

The activity coefficients of the various components in a solution are related by the Gibbs-Duhem equation. Converting p to in Equation (4.128) gives the Gibbs-Duhem equation for fugacity at constant [T,p],... 

Activity coefficient is a function of the state of a mixture. An activity-coefficient equation is required to calculate the fugacities of real solutions. The interrelationship of the activity coefficients through the Gibbs-Duhem equation implies that the activity-coefficient equations of aU components are derivatives of a common thermodynamic function. Since the activity coefficient is an expression of the nonideal behavior of a component, a thermodynamic function is needed to express the nonideality of the total solution and then to obtain from it the activity-coefficient equation. 

Thus, the activity coefficient of component i in solution is determined from knowledge derived from measurements of the vapor fugacity of component i in equilibrium with the liquid solution. If the vapor fugacities of components other than i are known, the vapor fugacity of / can be determined by application of the Gibbs-Duhem equation and, again, the activity coefficient of component i can be found. 

Note that the temperature derivative of lntemperature derivative of ln/( cf. (4.3.14) with (4.3.24). Further, the fugacity coefficients must obey a Gibbs-Duhem equation. Letting IRT be the generic function /in the Gibbs-Duhem equation (3.4.10), we find... 

Table 6.3 The chemical potential, fugacity, fugacity coefficient, activity, and activity coefficient are all constrained by a form of the Gibbs-Duhem equation.

Equations (1.3-47)-(1.3-49), with K = 380, produce the versus >a curves shown in Fig. 1.3-S, for t — 40 C and P 0.02S atm. As expected, a differs significantly unity. Perhaps surprisingly, so does The behavior of i, however, is conditioned by that of a> because the two fugacity coefficients mu satisfy the following form of the Gibbs-Duhem equation ... 

Apply the Gibbs-Duhem equation to relate activity coefficients of different species in a mixture. Evaluate whether a set of activity coefficient data is thermodynamically consistent. Given values of the pure species fugacity and the Henry s law constant, convert between activity coefficients based the Lewis/Randall rule, yj, and Henry s Law, y, " . 

Since the activity coefficient derives from the relative amount of unlike vs. like interactions, its value changes with changing composition. The value also depends on the specific choice of reference state. Similarly the activity of species i, Oi compares the fugacity of species i in the liquid to the fugacity of the pure species at its reference state. The Gibbs—Duhem equation allows us to relate the activity coefficients of different species in a mixture and, thereby, to test for thermodynamic consistency. 

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