Entropic term

By following the standard procedures of statistical mechanics, one arrives at an equation which can be converted into the BET equation (2.12) by the simple substitution 9i/9i , ki = c. Thus parameter c acquires a significance different from that in the BET theory in essence it now involves entropic terms as well as energetic terms. 

It is sometimes informative to separate AG into hypothetical enthalpic and entropic terms, and then the Arrhenius factors may be related to the transition state activation parameters by Eqs 10.4 and 10.5. Thus, the Arrhenius activation energy can be approximately related to the potential energy of a transition state, and the preexponential A value includes probability factors. 

As noted in Chapter 2, the Gibbs free energy is composed of both an enthalpic and an entropic term. For reversible binding interactions, we can use the equality AG = AH - TAS, together with Equation (3.8) and a little algebra to obtain... 

The free energy of a polymer consists of the entropic term and the internal energy of the segmental interactions, U, which represents the excluded-volume effect, and can be expanded as a power series of the segment density p (Eq. 1) ... 

In summary, although the metal-ion selectivity of the cryptands is normally largely enthalpy-controlled, entropic terms may also be quite important. Once again, the factors underlying these respective terms may be quite variable and, as a consequence, a criterion for preferred complexation based solely on a match of the cavity for the cation radius may not always be appropriate. 

Since the absolute and the conventional electrode potentials differ only by an additive constant, the absolute potential depends on the concentration of the reactants through the familiar Nernst s equation. This dependence is implicitly contained in Eq. (2.6) the real Gibbs energies of solvation contain an entropic term, which depends on the concentration of the species in the solution. 

In summary, the foregoing examples show that for a given elementary reaction, the standard reaction enthalpy is derived from the difference between the enthalpies of activation of the forward and the reverse process. An identical conclusion is drawn for the entropic terms. If, in the cases of reactions 3.1 and 3.10, the rate constants k and k- are known as a function of temperature, those kinetic parameters may be determined by plotting In(k/T2) or In(k/T) versus l/T(k = k or k- ). This analysis is known as an Eyringplot, and the resulting activation enthalpies and entropies refer to the mean temperature of the experimental range. 

As a first approximation to entropic terms AS° in the respective solvent may be considered as constant E° for a given ion in different solvents is then determined by the enthalpy of solvation AH°. 

For the usual values of the physical parameters I, Vq and C, calculated values for Oq are between 10 and 10 Scm as found experimentally. This agreement means that there is little influence of the entropic term AS(. 

As we have seen, a given enthalpy term can be associated with each defect process (tables 4.2 and 4.3). The defect process generally also involves an entropic term, which may be quantified through... 

All of the theoretical methods that allow for geometry optimization predict the bowl-shaped conformation of 8 to predominate over the planar alternative. However, the calculated energy differences between the two conformations differ drastically and span the range of 0.8 to 84 kcal/mol. Moreover, the calculated energy difference between the plants and bowl-shaped conformers is quite important since it represents the barrier for bowl-to-bowl inversion if (a) planar corannulene represents the transition state (TS) for the process, and (b) the entropic term in AG is negligible. Condition (a) seems to be fulfilled, since vibrational frequency calculations give one and only one imaginary frequency for the planar structure... 

The calculation of the energy for the interaction between the solvent and solute is more complicated. Formally, the free energy for the process, AC/ini, is composed of two chemical parts and an essentially entropic term. The chemical terms are associated with van der Waals interactions AGvdw> electrostatic effects, AG, between solute and solvent. The entropic term measures the free volume i.e., the volume a molecule explores before encountering another, which is assumed to be proportional to the molar volume of the solvent. Thus, the free energy change associated with solute-solvent interactions at temperature T is given by... 

Two phenomena have to be taken into account in order to make the pre-vioiiK Irciitmcnl of an isolated solute applionhie to the present situation. First, one must consider the strength of the interaction between E and B in the absence of any solvent effects. Second, an additional entropic term must be introduced because B, and consequently EB, are bonded so that the complex has restricted translational freedom at the surface. We will neglect, however, the latter effect because its contribution to the free energy of binding is expected to be small, i.e., not more than a few hundred calories per mole. Moreover, the entropic contributions for both species B and EB should be nearly identical and cancel when the individual terms are summed. 

This positioning in the context of ring size and conformation may have major influence on enthalpic and entropic terms of the inhibitory interaction and, consequently, on the relative power of inhibition of a- and (3-specific enzymes, as distinguished by more or less subtle differences in the geometry of their active sites.83... 

One can interpret the log-uniform distribution through the Arrhenius law k — Aexp(—AG/fcT), where AG is the change of the Gibbs free energy inreaction (it includes both energetic and entropic terms AG = AH—TAS, where AH the enthalpy change and AS the entropy change in reaction, T the temperature). The log-uniform distribution of k corresponds to the uniform distribution of AG. 

For such cascades with intramolecular carbopalladations, smaller ring sizes (5 and 6) are preferred due to the entropic term thus, 1,1-disubstituted alkenes suitably located in the substrate preferably serve as the relay without formation of the larger ring size resulting from insertion into the terminal alkene moiety. 

These results clearly indicate that the chelate ligation is driven primarily by the enthalpic factor and the entropy plays merely a trivial role in determining the complex stability. This is quite reasonable since the structures of these chelate complexes are strictly defined by the number and direction of the coordination sites of given heavy/transition metal ions, and therefore there is little room for the entropic term to adjust flexibly the complex structure and stability. On the contrary, alkali and alkaline earth metal ions also have the formal coordination numbers, but the actual number and direction of ligand coordination are highly flexible in the weak interaction-driven ligation by hard donors like glyme and crown ether. 

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