Entropy terms

The entropy term is a measure of the degree of freedom of the molecules and thus a measure of its flexibility. Measurement of the heats and entropies of fusion has provided interesting information on the relative importance of various factors... 

Since AG and AG are combinations of enthalpy and entropy terms, a linear free-energy relationship between two reaction series can result from one of three circumstances (1) AH is constant and the AS terms are proportional for the series, (2) AS is constant and the AH terms are proportional, or (3) AH and AS are linearly related. Dissection of the free-energy changes into enthalpy and entropy components has often shown the third case to be true. °... 

Finally, another important and interesting fact is established from the data treated in this manner. All the examples given confirm that the standard entropy term tends to increase with the standard enthalpy term. Consequently, the increase in retention is not as great as that which would be expected from the increase in standard enthalpy alone. 

We have seen that has to behave like a free energy in the MFA, and then in addition to the interaction an entropy term has to be introduced into Since the ideal entropy is a functional of the particle distributions we will assume that there is the same kind of functional in terms of fields. Thus... 

In addition to the entropy term we assume that there is an extra local coupling between the fields via and, in addition to the coulombic coupling which is long range, we assume the existence of a short-range non-local coupling via We can choose several functional forms to... 

Given the partition functions, the enthalpy and entropy terms may be calculated by carrying out the required differentiations in eq. (12.8). For one mole of molecules, the results for a non-linear system are (R being the gas constant)... 

In Eq. (12), the fourth term results from the increased volume available to the ends of the polymer chains on melting and the fifth term results mainly from the requirement that the ends of the molecules should stay out of the crystallites. Both terms are entropy terms giving the molecular weight dependence of the formation of bundle-like nucleus. Thus, the net transition rate J can be determined by the following equations ... 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

Approximate formulae for the point defect concentrations close (but not too close) to the stoichimetric composition in AB alloys have been derived. They show that the prefactors in the Arrhenius formulae are sensitive functions of the stoichiometry, besides representing the usual formation entropy term. 

The free-energy change AG is made up of two terms, an enthalpy term, AH, and a temperature-dependent entropy term, TAS. Of the two terms, the enthalpy term is often larger and more dominant. 

Neither the enthalpy nor the entropy of vaporization varies much with temperature so, for a given substance, ASvap° and AHtl )° can both be treated as approximately constant. It follows that the vapor pressures P, and P2 at any two temperatures T, and T2 are related by writing this equation for two temperatures and subtracting one from the other. In the process, the entropy terms cancel ... 

In terms of principal moments of inertia A, B and G, and the molecular mass M, the entropy term is then given by equation (14) (cf. also Leffek and Matheson, 1971). 

In each case, both the entropy and enthalpy terms favour the formation of the chelated complex, regardless of the t/-electron configuration. Note, however, that outside the d block, i.e. with alkaline earths and other main group metals, it is often found that the entropy term is dominant. 

In some cases, the enthalpy term may actually oppose the formation of the chelated complex, although the entropy term outweighs it to give an overall favourable free energy term. In general, this situation is the exception rather than the rule. 

In more mathematical language, the favourable entropy term is associated with the release of a large number of monodentate ligands upon the formation of the chelate. 

There is also an entropy term associated with the desolvation of the ligands. This is much more difficult to assess, and may make for either favourable or unfavourable contributions to the overall entropy changes. 

Although reactions in which molecules are cleaved into two or more pieces have favorable entropy effects, many potential cleavages do not take place because of large increases in enthalpy. An example is cleavage of ethane into two methyl radicals. In this case, a bond of 79 kcal mol (330 kJ mol ) is broken, and no new bond is formed to compensate for this enthalpy increase. However, ethane can be cleaved at very high temperatures, which illustrates the principle that entropy becomes more important as the temperature increases, as is obvious from the equation AG = AH — TAS. The enthalpy term is independent of temperature, while the entropy term is directly proportional to the absolute temperature. 

The decomposition of N2 O4 requires a bond to break. This is the reason why the decomposition has a positive A 77 °. At the same time, the number of molecules doubles during decomposition, which is the reason AS° has a positive value. The positive enthalpy change means that energy Is removed from the surroundings and constrained, whereas the positive entropy change means that matter is dispersed. At temperatures below 315 K, the enthalpy term dominates and decomposition is not spontaneous, but at temperatures above 315 K, the entropy term dominates and decomposition is spontaneous. 

The neutral situation, in which the entropy of the transition state does not notably differ from that of the reactant(s) in the ground state yields the standard pre-exponential factor hv = ek T/h which is of the order of 10 s . The influence of the entropy term gives rise to a range of possible pre-exponential factor from about 10 ... 

The rate sequence is determined by the entropy term and correlates with the oxidation potential of the chelate complex, indicating an outer-sphere electron transfer. 

In the Arrhenins eqnation, the real activation energy is combined with a real (measnrable) preexponential factor. According to Eqs. (14.1) and (14.10), this factor differs from the trne factor by the multiplicative entropy term exp( a AS /R). 

---