FREISE, V (1991) THE THERMODYNAMICS OF CRITICAL PHASES FROM THE PERSPECTIVE OF CALCULUS OF VARIATIONS. ZEITSCHRIFT FUR PHYSIKALISCHE CHEMIE-INTERNATIONAL JOURNAL OF RESEARCH IN PHYSICAL CHEMISTRY & CHEMICAL PHYSICS, 174. pp. 163-177. ISSN 0942-9352,
Full text not available from this repository.Abstract
Earlier considerations [1] on the stability of non-critical phases are extended to simple critical systems. Mathematically, the concept, which involves a maximum of two dependent variables, is based on the following variational problem: J(alpha = 0) = integral-b/a F(...y(i)' + alpha-g(i)'...)dx = min. (*) Besides the variational parameter, alpha, only the derivatives of the extremals, y(i)'(x), and those of the varying functions. g(i)'(x), occur explicitly. Consequently, the problem (*) can be reduced to differential calculus, not only with respect to the necessary conditions, but also with respect to the (so-called) sufficient conditions. Hence, for a minimum of J(alpha) at alpha = 0, it is necessarv and sufficient that the first non-vanishing derivative, partial derivative(n) J/partial derivative-alpha(n), be of even order and positive at the minimal point. First, the Euler equations result from partial derivative J/partial derivative-alpha = 0. Furthermore, if partial derivative 2J(alpha = 0)/partial derivative-alpha-2 > 0, an ordinary minimum is present geometrically with respect to the parameter-alpha. If, however, both partial derivative 2J/partial derivative alpha-2 and partial derivative 3J/partial derivative-alpha-3 vanish at alpha = 0, a flat minimum is present. From these higher-order differential quotients, the sufficient conditions can then be obtained without further mathematical effort by elimination of alpha and g(i)'. For the transition to thermodynamics, the function F in (*) is replaced by molar thermodynamic variables. With regard to the parameter-alpha, which has been notably upgraded in this conjunction, the stability of a normal phase is described by an ordinary minimum, and that of a critical phase is described by a flat minimum. On the one hand, the conditions for the critical (step) point then result. On the other hand, the two "Gibbs equations for critical phases", described, for instance, in [3], are also obtained. Within the given mathematical limits, extensions for miscible phases up to ternary systems then follow.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | ; THERMODYNAMICS; CRITICAL PHASES; CALCULUS OF VARIATIONS |
| Depositing User: | Dr. Gernot Deinzer |
| Last Modified: | 19 Oct 2022 08:47 |
| URI: | https://pred.uni-regensburg.de/id/eprint/55188 |
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