Gibbs energy fundamental property relations

Equation 54 implies that U is a function of S and P, a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Hehnholt2 energy,, and Gibbs energy, G ... 

Equations 54 and 58 through 60 are equivalent forms of the fundamental property relation apphcable to changes between equihbtium states in any homogeneous fluid system, either open or closed. Equation 58 shows that ff is a function of 5" and P. Similarly, Pi is a function of T and C, and G is a function of T and P The choice of which equation to use in a particular apphcation is dictated by convenience. Elowever, the Gibbs energy, G, is of particular importance because of its unique functional relation to T, P, and the the variables of primary interest in chemical technology. Thus, by equation 60,... 

For convenience, the three other fundamental property relations, Eos. (4-16), (4-80), and (4-82), expressing the Gibbs energy and refated properties as functions of T, P, and the are collected nere ... 

Equation (10.2), the fundamental property relation for single-phase systems, provides an expression for the total differential of the Gibbs energy ... 

Fundamental Property Relations Based on the Gibbs Energy. 4-21... 

FUNDAMENTAL PROPERTY RELATIONS BASED ON THE GIBBS ENERGY... 

If a mixture of monomers is not in equilibrium, any polymerization that occurs at constant temperature, T, and constant pressure, P, must lead to a decrease in the total Gibbs free energy of the system. A reaction variable may be defined. The fundamental property relation for a single phase system for the total differential of the Gibbs free energy can be written as a function of temperature, pressure, and chemical potential as follows ... 

Apply the fundamental property relation for Gibbs energy and other tools of the thermodynamic web to predict how the pressure of a pure species in phase equilibrium changes with temperature and how other properties change in relation to one another. Write the Clapeyron equation and use it to relate Tand Pfor a pure species in phase equilibrium. Derive the Clausius-Clapeyron equation for vapor-liquid mixtures, and state the assumptions used. Relate the Clausius-Clapeyron equation to the Antoine equation. 

We can also get thermodynamic property data to solve for the left-hand side of Equation (7.7) through an equation of state. At constant temperature (as mandated by the definition of fugacity), we can write the fundamental property relation of the Gibbs energy of pure species i as ... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

This fundamental equation shows that S = - BG /dT, Ar G = dG /9f, and f 71n(10)no(H) = BG VdpH. These equations are not directly useful because there is no experimental way to determine G but Maxwell relations for this and other fundamental equations do provide equations for experimental determinable properties. For the system being discussed the standard transformed Gibbs energy of reaction is given by... 

The chemical potential provides the fundamental criteria for determining phase equilibria. Like many thermodynamic functions, there is no absolute value for chemical potential. The Gibbs free energy function is related to both the enthalpy and entropy for which there is no absolute value. Moreover, there are some other undesirable properties of the chemical potential that make it less than suitable for practical calculations of phase equilibria. Thus, G.N. Lewis introduced the concept of fugacity, which can be related to the chemical potential and has a relationship closer to real world intensive properties. With Lewis s definition, there still remains the problem of absolute value for the function. Thus,... 

The fugacity, an auxiliary property, is related to the Gibbs free energy, the fundamental thermodynamic property at equilibrium. To establish this relationship, first we write the Gibbs energy in the form,... 

In this chapter we developed the theoretical tools for the study of saturated phases, namely, phases that coexist in equilibrium with each other. The fundamental property is the Gibbs free energy, which has the same value in both phases. This fundamental equality is the basis of all the results obtained in this chapter. Fugacity and the fugadty coefficient are auxiliary variables introduce for convenience. They are both related to the Gibbs free energy but are more convenient to work with because they do not require a reference state. In addition, they reduce the very simple expressions in the ideal-gas state. 

The temperature dependence of the Gibbs energy of formation allows one to determine the entropy of formation according to the Gibbs-Helmholtz relation AS = -(dAG/dT), i.e., the temperature dependence of the cell voltage relates to the entropy of the cell reaction, AS = nq(dE/dT). The enthalpies of formation and cell reaction follow according to AH = AG + TAS. Knowledge of these fundamental thermodynamic quantities allows the comprehensive determination of the thermodynamic properties of the system. 

Since atoms or molecules in the vicinity of the surface of a condensed phase have different bonding from those in the hulk they have different thermodynamic properties. In this chapter the concept of the Gibbs dividing surface and the two fundamental quantities for describing the thermodynamic properties of surfaces and interfaces, surface energy (y) and surface stress (a ), are defined. The relations between y and the other thermodynamic variables for surfaces are established. Finally, methods for obtaining y and cr are described and representative values of both are presented. 

Consider a system with N identical particles contained in volume V with a total energy E. Assume that N,V, and E are kept constant. We call this an NVE system (Fig. 1.1). These parameters uniquely define the macroscopic state of the system, that is all the rest of the thermodynamic properties of the system are defined as functions of N, V, and E. For example, we can write the entropy of the system as a function S = S(N,V, E), or the pressure of the system as a function p = P N, V, E). Indeed, if we know the values of N, V, and E for a single-component, single-phase system, we can in principle find the values of the enthalpy H, the Gibbs free energy G, the Helmholtz free energy A, the chemical potential jx, the entropy S, the pressure P, and the temperature T. In Appendix B, we sununarize important elements of thermodynamics, including the fundamental relations between these properties. 

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