Equations Gibbs fundamental equation

In order to better understand the physical nature of the chemical potential jxt of a chemical substance, let us first review the major mathematical features of the Gibbsian thermodynamics formalism. The starting point is the Gibbs fundamental equation for the internal energy function... 

The starting point for thermodynamic description, whether in the calculus-based or the geometry-based formalism, is the Gibbs fundamental equation for a given equilibrium state S. In the energy representation, this is expressed as... 

Consider an adsorbate phase consisting of na moles of a nonvolatile adsorbent (surface) and ns moles of an adsorbate (gas phase). They are assigned internal energy U, entropy S and volume V. The surface A of the adsorbent is assumed to be proportional to the adsorbent volume. The Gibbs fundamental equation for the full system is then... 

Nernst equation — A fundamental equation in -> electrochemistry derived by - Nernst at the end of the nineteenth century assuming an osmotic equilibrium between the metal and solution phases (- Nernst equilibrium). This equation describes the dependence of the equilibrium electrode - potential on the composition of the contacting phases. The Nernst equation can be derived from the - potential of the cell reaction (Ecen = AG/nF) where AG is the - Gibbs energy change of the - cell reaction, n is the charge number of the electrochemical cell reaction, and F is the - Faraday constant. 

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... 

The work done consists of the reversible part and the dissipated work. We want to calculate these contributions individually with the help of an entropy balance. For the sake of simplicity we will only consider pure substances. According to Gibbs fundamental equation, we have... 

Perhaps the most important, concepts of the axiomatic foundation of ther modynamics are the ones referred to as the First and Second Laws dealing with the internal energy U and the entropy S. They are essentially statements dealing with energy conservation and the transformation of one form of energy (e.g., work) into another one (e.g., heat). If combined, the First and Second Laws give rise to the so-called Gibbs fundamental equation... 

Equation (1.37) is the Gibbs fundamental equation specialized to a fluid confined to a slit-pore with chemically homogeneous, (infinitesimally) smooth substrate surfaces. 

From Gibbs fundamental equation [see Eq. (1.22)], it follows that... 

Another difficulty is caused by the application of Gibbs thermodynamic concept to nonequilibrium conditions of an adsorption layer, as it is done in many theoretical models (cf. Chapter 4). From non-equilibrium thermodynamics we learn (cf Eq. (2C.8) derived by Defay et al. (1966), Appendix 2C) that the diffusional transport causes an additional term to the Gibbs equation. However, this term seems to be negligible in many experiments. The experimental data discussed in Chapter 5, obtained from measurements in very different time windows, support the validity of Gibbs fundamental equation (2.33) also under non-equilibrium conditions. 

When the system is in equilibrium the Gibbs fundamental equation (2.33) is obtained. Further symbold used in this appendix are ... 

Substitution of Pj into the total differential of G produces the Gibbs fundamental equation... 

The Gibbs fundamental equation, therefore, takes the following form for the closed system... 

We shall next show how the Gibbs fundamental equation for a surface phase may be obtained from the above approach. Since U, S, and V are additive properties, it is possible to write... 

From the Gibbs fundamental equation it follows that... 

With dq = T-ds, Gibbs fundamental equation follows for reversible processes within the system ... 

The starting point for the description of the system considered is the Gibbs fundamental equation, Eq. (1), shown in Fig. 2.91 in terms of the specific quantities ... 

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state ... 

An alternative formulation of the alxtve expressions can be obtained by treating the entropy in terms of the internal energy, volume, and composition, that is, 5( /,K A )- This leads to the entropy formulation of Gibbs fundamental equation and, while not conunonly used in thermodynamics, provides certain advantages in statistic thermodynamics. The following expressions are then obtained ... 

Let us consider an open thermodynamic system consisting of v components, i.e. containing particles of u kinds. I he first and second principles of thermodynamics written together for a quasi-static process in such a system represent the Gibbs fundamental equation in its energetic expression ... 

In an external field with an intensity A, the Gibbs fundamental equation takes the form... 

From the Gibbs fundamental equation in its entropy form, one can similarly derive the condition of stability to infinitely small perturbations ... 

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. 

Gibbs fundamental equation 1.1.1-1 for a pivot upon tension can be written as... 

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