Gibbs function INDEX

In Frame 13, section 13.2 we established the Gibbs function, G, as the index of spontaneous change. Equation (13.10) Frame 13 gave the relationship between G and the enthalpy, H and entropy, S namely ... 

State functions derivable therefrom (such as ASd or AHd) are the fundamental quantities of interest, the arbitrariness of K or Kq causes no difficulty other than being a nuisance. It should be remembered that, once a choice of units and of standard state has been made, a value of /C or 1 implies that AG is a large negative quantity, and hence, that AGd is also likely to be large and negative. Thus, equilibrium will be established after the pertinent reaction has proceeded nearly to completion in the direction as written. Conversely, for values of K, or Kq equilibrium sets in when the reaction is close to completion in the opposite direction. Thus, the equilibrium constant serves as an index of how far and in what direction a reaction will proceed, and this prediction does not depend on the arbitrariness discussed earlier. It should be clear that the equilibrium constants do not in themselves possess the same fundamental importance as the differential Gibbs free energies. However, the full utility of equilibrium constants will not become clear until some illustrative examples are provided below. 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

Here T is the temperature, p is the pressure, a is the surface tension, A is the surface area, V is the volume, 5 is the entropy density, Pi are the particle densities, and pi the chemical potentials of the different components, R is the radius of the critical cluster referred to the surface of tension, the index a specfies the parameters of the cluster while p refers to the ambient phase. The equilibirum conditions coincide with Gibbs expressions for phase coexistence at planar interfaces (R oo) or when, as required in Gibbs classical approach, the surface tension is considered as a function of only one of the sets of intensive variables of the coexisting phases, either of those of the ambient or of those of the cluster phase. In such limiting cases, Gibbs equilibrium conditions... 

E = constant in scaling equation G = Gibbs free energy h = scaling function defined by Equation 8 M = molecular weight n = index of refraction P = pressure... 

The 1978 publication is a 456 page monograph containing selected values for the entropy, molar volume, and for the enthalpy and Gibbs energy of formation for the elements, 133 oxides, and 212 other minerals and related substances at 298.15 K. Thermal functions are also given for those substances for which heat-capacity or heat-content data are available. The thermal functions are tabulated at 100 K intervals for temperatures up to 1800 K. The monograph includes detailed references to the source literature and a compound index. 

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