Gibbs fundamental relation

Equation (3.53) is known as Gibbs fundamental relation. Transforming (3.53) into... 

The definitions of enthalpy, H, Helmholtz free energy. A, and Gibbs free energy, G, also give equivalent forms of the fundamental relation (3) which apply to changes between equiUbrium states in any homogeneous fluid system ... 

Gibbs s fundamental relation governing an infinitesimal, reversible transformation can be written... 

The fundamental thermodynamic equation relating activity coefficients and composition is the Gibbs-Duhem relation which may be expressed as ... 

From the Gibbs fundamental equation f(U,S,V,N), we have the three functions of S, V, and N, the respective differential relations, and the Euler equations given by... 

Entropy depends explicitly only on energy, volume, and concentrations because the Gibbs relation is a fundamental relation and is valid even outside thermostatic equilibrium. 

Excess Gibbs energy and activity coefficients are linked. From the equations (6.30) and (6.33) the following fundamental relation is obtained ... 

Equation (17.23) is the fundamental relation between the cell potential and the Gibbs energy change accompanying the cell reaction. 

Thus, for any given pair potential U R), we can calculate the PP (2.5.7), and from there all the thermodynamic quantities. The fundamental relation between A(T,P,N) and the Gibbs... 

Equation 21 can be applied to any part of the system (a subsystem) if only this part contains a large enough number of molecules to provide the validity of the Gibbs fundamental equation. In this case, the other part of the system relates to the thermostat. 

The fundamental relation in the T, P, N ensemble is between the Gibbs free energy and the isothermal-isobaric PF, namely,... 

In Sect. 4 we present several adsorption isotherms which are solutions of the Maxwell relations of the Gibbs fundamental equation of the multicomponent adsorbate [7.15]. These isotherms are thermodynamically consistent generalizations of several of the empirical isotherms presented in Sect. 3 to (energetically) heterogenous sorbent materials with surfaces of fractal dimension. In Sect. 5 some general recommendations for use ofAIs in industrial adsorption processes are given. 

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. 

Equation (4.5-3) is called the Gibbs equation or the fundamental relation of chemical thermodynamics. We could also choose to write... 

Instead of measuring the equilibrium cell voltage Aeoo at standard conditions directly, this can be calculated from the reaction free energy AG for one formula conversion. In this context one of the fundamental equations is the GIBBS-HELMHOLTZ relation [7]. 

In the next section, we discuss hydrodynamic forces between fluid interfaces. The interaction between fluid interfaces is strongly influenced by surfactants and contaminants at the interfaces. Therefore, we first need to introduce a fundamental relation between the amount of substance adsorbed at a fluid interfaces and the surface tension. This is quantitatively expressed in the Gibbs adsorption isotherm. We only introduce the Gibbs adsorption isotherm for a two-component system, that is, a liquid and one dissolved substance. It is... 

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. 

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 ... 

Finally, a Gibbs/Diihem equation is associated with each fundamental property relation ... 

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. 

These equations can be used to derive the four fundamental equations of Gibbs and then the 50,000,000 equations alluded to in Chapter 1 that relate p, T, V, U, S, H, A, and G. We should keep in mind that these equations apply to a reversible process involving pressure-volume work only. This limitation does not restrict their usefulness, however. Since all of the thermodynamic variables are state functions, calculation of AZ (Z is any of these variables) by a reversible path between two states gives the same value as would be obtained for all other paths between those states. When other forms of work are involved, additions can be made to the equations to account for the additional work. The... 

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