Minimum value of the free energy

The functional derivative in Eq. (60) represents deterministic relaxation of the system toward a minimum value of the free-energy functional E[< )(r, f)], which is usually taken to have the form of the coarse-grained Landau-Ginzburg free energy... 

The prevalence of the (S) or (R) antipode of the product gives a qualitative idea of the relative energy of the transition states controlling asymmetric induction. However, the values of the enantiomeric excess do not necessarily indicate the extent but only a minimum value of the free energy differences between the above transition states (e.g. the reaction product could racemize after its formation). On this basis the already discussed model for the transition state controlling asymmetric induction has been formulated (Sect. 4.). 

When the reaction quotient furnishes the minimum value of the free energy change for the reaction, which is zero (i.e., AG = 0), the reaction is in equilibrium and Q = K, where K is the equilibrium constant. Equation (1.133) becomes ... 

The stationary value of this integral results from the function Cg which satisfies Eq. (32). But a minimum value of the free energy in three dimensions is not obtained for a function of the form (33), but rather for a function which is asymptotically proportional to r. This will be true whenever m> 3. If the thermodynamic significance of the terms in Eq. (32) is to be preserved, tn must be odd, and the only exception to the previous statement comes for... 

The addition of the hard wall entropic confinement free energy to the interaction energy, as under (2), only raises somewhat the minimum, but almost does not modify its position. In contrast, in all the other cases, which take into account the other interactions in the calculation of entropy, considerable shifts of the equilibrium distance and of the value of the free energy at the minimum occur. This observation indicates that a common procedure, to add to the conventional interactions, calculated as for planar layers, the free energy due to the hard wall entropic confinement is inaccurate. 

Furthermore, unlike the free particle, no state exists for which = 0. Even the lowest energy quantum state, the ground state, has some kinetic energy. The difference between the energy of the ground state and the minimum value of the potential energy is called the zero-point energy,... 

We can now find the minimum value of the free activation energy AF, compatible with the conditions (3.11) and (3.12). Omitting the mathematical transformations, we can write the following final form of the result ... 

The assessment of k is of some importance since it relates to the question as to how much if any of the free energy of activation barrier is due to the spin-forbidden character of the transition. From the experimental point of view, Eq. (49) shows that the transmission coefficient k and the activation entropy AS appear in the temperature-independent part of the rate constant and thus cannot be separated without additional assumptions. Possible approaches to the partition of — TAS have been discussed in Sect. 4 for spin transition complexes of iron(II) and iron(III). If the assumption is made that the entropy of activation is completely due to k, minimum values between 10 and 10 are obtained for iron(II) and values between 10 and 10 for iron(III). There is an increase of entropy for the transition LS -+ HS and thus the above assumption implies that the transition state resembles the HS state. On the other hand, volumes of activation indicate that the transition state should be about midway between the LS and HS state. This appears indeed more reasonable and has the... 

In Fig. 1.21 the function G(S, 0.5, a) is shown for a given temperature a. On each curve the minimum of the free energy (shown by an arrow) gives the S value at equilibrium, which must be coincident with the result obtained from eqn (1.114). 

A system has the minimum value of the Gibbs free energy at equilibrium. Suppose we have a system which consists of three phases, namely gas, liquid and solid solutions, with a number of different species (1,2,3,...) ... 

Now, let us consider the fluctuations of a segment of M charges. The segment is located at least one Debye length away from the ends of the polyion. If Ag denotes the deviation of the free energy of the segment from its minimum value, then Ag = MAG. But, the thermal average of AG is, by definition, (AG) = kBT/2. Consequently, the mean-square number fluctuations of condensed counterions, ((A0)2), is [50]... 

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