Constructing thermodynamic functions limits

But what the pressure p(o), diemical potential M(p), etc., in the constrained system are, depend on the distance L that defines the constraint. If L is very large, the flrrduations within can almost amount to phase separation foe van der Waak loops in p(v) and M(p) would then enclose only small areas, and foe analytic functions p(o), ip(p), etc., would be dose to foe non-analytic functions obtained from them by the equal-areas, double-tangent, or convex-envelope constructions. Tire effect of the constraint with such large L is minimal and in the limit in which L is macroscopic foe thermodynamic properties become those of foe unconstrained fluid. But when L is small, the deviation of p(t>) from the equilibrium pressure in foe unconstrained system at that temperature is considerable, and similarly for foe other thermodynamic functions. 

From the above said, it may be concluded that a detailed kinetic model of coal combustion process that combines all three basic processes can not virtually be constructed, as it is impossible to do for each process separately. Therefore, the empirical models based on separation and experimental study of the limiting stages are extensively used. Such models separately do not reveal general regularities and do not allow the generalized conclusions to be drawn. The thermodynamic model makes it possible to study the whole attainability region and hence to consider states of the considered system as a whole and to keep track of the variation in the amounts of any component as a function of some or other kinetic constraints. The latter are written, as was shown above, easily enough even for such complex processes as coal combustion. 

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. 

There undoubtedly exists a best (and as yet undiscovered) cubic equation of state, but best only in a coarse statistical sense. Such an equation, constructed so as to avoid manifestly unreasonable predictions of all possible thermodynamic properties of interest, probably would not produce really adequate estimates of any single property (except perhaps over very limited ranges of the variables of state). The search for this equation, if successful, would yield an expression of limited usefulness as a cubic equation. [It could, of course, provide the basis for more precise, noncubic expressions, obtained e.g. by coupling the cubic equation with deviation functions (see, e.g., Gray et al. (17) and Redlich (18)).]... 

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