Gibbs analogy

We will illustrate the Gibbs analogy for a very simple mechanical system, namely, the one-dimensional macroscopic particle system illustrated in Figure 3.1 and discussed in Section ILA.l. 

For this system the Gibbs analogy may be stated as follows. The thermodynamic equilibrium values of the macroscopic parameters Ap . A are analogous to the static equilibrium values of the particle coordinate x. The removal of the constraints on Ap+1,..,. A is analogous to the release of the constraint force F on x. The thermodynamic process T Tp is analogous to the particle relaxation process xf = 0. Finally, the entropy... 

Working in the spirit of the Gibbs analogy, the solution of this problem apparently requires an extension of equilibrium thermodynamics that parallels the extension of particle statics needed to predict trajectories x t), namely, its extension to particle dynamics. 

Onsager has partially overcome the dual problem of formulating a thermodynamic equation of motion and of choosing the thermodynamic forces, thus enabling him to develop a nonequilibrium extension of classical thermodynamics in the spirit of the Gibbs analogy. 

The relevance of Eqs. (A.7) asd (A.8) to Onsager s theory arises since these results, in the spirit of the Gibbs analogy, are the thermodynamic equivalents of the particle mechanics eqs. (3.13) and (3.14). For example, the most probable value of the particle coordinate a found by maximizing Eq. (3.13) coincides with its static equilibrium value xf=q S) determined from Eq. (3.11). Similarly, the most probable values of the unconstrained A s found by maximizing Eq. (A.7) coincide with their thermodynamic equilibrium values A +, i A determined from Eq. (A.l). (Note, however, that this analogy between particle mechanics and thermodynamics is not perfect. This is because while for microscopic particles, typical fluctuations x x of the particle s coordinate a can be comparable in magnitude to its initial displacement from flnal equilibrium xf S) - a the macroscopic validity of thermodynamics requires that typical fluctuations A j-... 

Analogous to the defining equation for the residual Gibbs energy is the definition of a partial molar residual Gibbs energy (eq. 161) ... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

Adsorbed-Solution Theoiy The common thennodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is... 

The Gibbs-Helmholtz equation can be used to calculate the standard free energy of formation of a compound. This quantity, AGf, is analogous to the enthalpy of formation, AH . It is defined as the free energy change per mole when a compound is formed from the elements in their stable states at 1 atm. 

Heavy water is deuterium oxide, D20. The standard reaction Gibbs free energy for the autoprotolysis of pure deuterium oxide is +84.8 kj-mol 1 at 298 K. (a) If pD is defined analogously to pH, what is the pD of pure D20 at 298 K ... 

This equation for the surface excesses is the analog of the Gibbs-Duhem equation for bulk phases. 

We have now derived the phase boundary between the two liquids. By analogy with our earlier examples, the two phases may exist as metastable states in a certain part of the p,T potential space. However, at some specific conditions the phases become mechanically unstable. These conditions correspond to the spinodal lines for the system. An analytical expression for the spinodals of the regular solution-type two-state model can be obtained by using the fact that the second derivative of the Gibbs energy with regards to xsi)B is zero at spinodal points. Hence,... 

A binary ionic solution must contain at least three kinds of species. One example is a solution of AC and BC. Here we have two cation species A+ and B+ and one common anion species C . The sum of the charge of the cations and the anions must be equal to satisfy electro-neutrality. Hence NA+ + NB+ = N(. = N where NA+, AB+ and Nc are the total number of each of the ions and N is the total number of sites in each sub-lattice. The total number of distinguishable arrangements of A+ and B+ cations on the cation sub-lattice is M/N A, JVg+ . The expression for the molar Gibbs energy of mixing of the ideal ionic solution AC-BC is thus analogous to that derived in Section 9.1 and can be expressed as... 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

Equation 17.23 has the form of an adsorption isotherm since it relates the amount adsorbed to the corresponding pressure. This is known as the Gibbs Adsorption Isotherm. For it to be useful, an expression is required for T. Assuming an analogy between adsorbed and liquid films, Harkins and Jura(15) have proposed that ... 

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