Gibbs energies internal energy equation

Differentiation of eq. (6.77), when combined with eq. (6.14), gives the Gibbs-Duhem equation in internal energy for a system where the surface energy is not negligible ... 

Gibbs postulated the fundamental equation for the surface excess of internal energy as [10,11]... 

However, Gibbs demonstrated the equal importance of a second fundamental equation that reveals a beautiful duality of the thermodynamic formalism the deep symmetry between entropy (5.28) and internal energy U (5.29) ... 

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... 

Gibbs (1873) showed how to include the contributions of added matter to the fundamental equation by introducing the concept of the chemical potential of species i and writing the fundamental equation for the internal energy of a system involving PV work and changes in the amounts n, of species as... 

This is referred to as a Gibbs-Helmholtz equation, and it provides a convenient way to calculate H if G can be determined as a function of 71 P, and ,. There is a corresponding relation between the internal energy U and the Helmholtz energy, which is defined by equation 2.5-2 ... 

Derivation of the expression for the minimum production of S in the systems with constant T and V (volume) differs from the one above only by replacement of enthalpy by internal energy (U) and the Gibbs energy by the Helmholtz energy in the equations. When we set S and P or S and V dissipation turns out to be zero according to the problem statement. In the case of constant U and V or H and P, the interaction with the environment does not hinder the relaxation of the open subsystem toward the state max Sos. 

Thermodynamic Functions for Solids.—In the preceding section we have seen how to express the equation of state and specific heat of a solid as functions of pressure, or volume, and temperature. Now we shall investigate the other thermodynamic functions, the internal energy, entropy, Helmholtz free energy, and Gibbs free energy. For the internal... 

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... 

The above equations are all based on the internal energy. Similar equations can be written with the enthalpy since the surface excess enthalpy and energy are identical in the Gibbs representation when 1 =0 (Harkins and Boyd, 1942). Therefore the various energies of immersion defined by Equations (5.6)—(5.8) are all virtually equal to the corresponding enthalpies of immersion, i.e. (A inmH°, AimmHr and Ah 1), thus ... 

For the entropy and internal energy, the canonical variables consist of extensive parameters. For a simple system, the extensive properties are S, U, and V. and the fundamental equations define a fundamental surface of entropy S = S(U,V) in the Gibbs space of S, U, and V. 

For an isothermal fluid flow described by the Redlich-Kwong equation of state, develop expressions in terms of the initial temperature and the initial and final volumes for the changes in internal energy, enthalpy, entropy, and the Gibbs free energy. 

In extended nonequilibrium thermodynamics of polymer solutions, the generalized extended Gibbs equation for a fluid characterized by internal energy U and viscous pressure Pv is... 

Tliis equation defines the partial molar property of species i in solution, where the generic symbol Mt may standfor the partial molar internal energy t/, the partial molar enthalpy //, the partial molar entropy 5,, the partial molar Gibbs energy G,, etc. It is a response function, representing the change of total prope ity n M due to additionat constant T and f of a differential amount of species i to a finite amount of solution. 

The successive Legendre transformations of E yield a state function, G, for which the natural variables p and T, are both intensive properties (independent of the size of the system). Furthermore, for dp = 0 and dT = 0 (isobaric, isothermal system), the state of the system is characterized by dG. This is clearly convenient for chemical applications under atmospheric pressure, constant-temperature conditions (or at any other isobaric, isothermal conditions). Then, in place of equation (21) for internal energy variation, we state the conditions for irreversible or reversible processes in terms of the Gibbs energy as... 

To describe the state of a reaction in a phase, we need to know the stoichiometric coefficients, j, and the chemical potential, pi, for each species in the reaction. For reaction equilibrium, the quantity AG = E Vi pi = 0 (as is T diS). For a possible, or spontaneous, reaction, AG < 0. For multireaction systems, complete equilibrium corresponds to dG = 0 for the system, that is, the Gibbs energy of the phase is a minimum. The total internal entropy production must vanish for the entire system. Similar consideration apply to multiphase systems. An expression analogous to equation 39 for dE, but for fixed T and p conditions, is ... 

The internal energy may be obtained using the Gibbs-Helmholtz equation... 

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... 

The most accurate route to the thermodynamic properties from the SSOZ equation seems to be the energy equation." The integral from which the internal energy is obtained (see Eq. (2.3.1)) seems to be relatively insensitive to errors in the predicted site-site correlation functions. It might on this basis be reasonably assumed that calculations of the Helmholtz free energy via integration of the Gibbs Helmholtz equation... 

Similar to the derivation of the Gibbs-Duhem equation, it is also possible to show the dependence of surface tension on the chemical potentials of the components in the interfacial region. If we integrate Equation (201) between zero and a finite value at constant A, T and nb to allow the internal energy, entropy and mole number to almost from zero to some finite value, this gives... 

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