Internal energy minimum principle

The entropy maximum principle can be stated equivalently in terms of the internal energy minimum principle. In the following, we present the mathematical proof. 

Next, we derive the relation between partial derivatives that was used in the mathematical derivation of the internal energy minimum principle. 

The macroconformation of chainlike macro molecules in crystalline polymers is principally determined by two factors, which are inter- and intracatenary forces. Calculation of potential barriers of isolated molecules, that is, in a vacuum, is based exclusively on intracatenarily effective forces (see also Section 4.1.2). Microconformations calculated in this way correspond to an internal energy minimum. According to the equivalence principle, all structural units should adopt geometrically equivalent positions in relation to the crystallographic axes, whereby a monomeric unit, for example, may serve as a structural unit. Thus the regular sequence of microconformations should lead to a regular macroconformation. 

An extremum principle minimizes or maximizes a fundamental equation subject to certain constraints. For example, the principle of maximum entropy (dS)v = 0 and, (d2S)rj < 0, and the principle of minimum internal energy (dU)s = 0 and (d2U)s>0, are the fundamental principles of equilibrium, and can be associated with thermodynamic stability. The conditions of equilibrium can be established in terms of extensive parameters U and. S, or in terms of intensive parameters. Consider a composite system with two simple subsystems of A and B having a single species. Then the condition of equilibrium is... 

At threshold (i.e. when = o), N 0) = 1. Thus, the minimum rate is in principle equal to llh p(Eq), However, when the internal energy is equal to, or even is slightly less than the barrier, then the reaction proceeds via tunnelling and a transmission probability K is then inserted into Equation [1]. Isotope effects are then to be expected. When the shape of the barrier can be described by an inverted parabola, a simple expression can be derived for the transmission... 

Principle of Minimum Internal Conformational Energy. The conformation of a polymer chain in a crystal approaches one of the minima of the internal conformational energy, which would be taken by an isolated chain subjected to the restrictions imposed by the equivalence principle. 

For mixtures of substances of markedly different surface tensions also we have noted that over a considerable range of concentration the Gibbs film appears to behave as if it were unimolecular in character, but for strong solutions of these substances as well as for mixtures of liquids of similar surface activities the evidence for such a restricted film thickness is by no means so conclusive. It must indeed rather be assumed that in these cases the application of the principle of minimum surface energy to mixtures somewhat similar in internal pressure leads to the formation of a diffuse layer in which the composition varies possibly in an exponential manner with the depth. The top layer alone may be said to be formed by the operation of chemical forces. 

The maximum hardness principle also demands that hardness will be minimum at the transition state. This has been found to be true for different processes including inversion of NH3 [147] and PH3 [148], intramolecular proton transfer [147], internal rotations [149], dissociation reactions for diatomics [150,151], and hydrogen-bonded complexes [152]. In all these processes, chemical potential remains either constant or passes through an extremum at the transition state. The maximum hardness principle has also been found to be true (a local maximum in hardness profile) for stable intermediate, which shows a local minimum on the potential energy surface [150]. The energy change in the dissociation reaction of diatomic molecules does not pass through a... 

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