Configurational entropies

The above treatment is one of the simplest available. The configurational entropy modeP is mgre rigorous, but conveniently gives the same form as free-volume theory for the temperature dependence of conductivity. However, the meaning of the parameters is different. Tq is the temperature at which the conformational entropy ( configurational entropy is actually a misnomer in a polymer context) becomes zero and... 

In the Maximum Entropy Method (MEM) which proceeds the maximization of the conditional probability P(fl p ) (6) yielding the most probable solution, the probability P(p) introducing the a priory knowledge is issued from so called ergodic situations in many applications for image restoration [1]. That means, that the a priori probabilities of all microscopic configurations p are all the same. It yields to the well known form of the functional 5(/2 ) [9] ... 

We can now proceed with various estimates of the entropy of adsorption Two extreme positions are sometimes taken (see Ref. 14). First, one assumes that for localized adsorption the only contribution is the configurational entropy. Thus... 

In general, it seems more reasonable to suppose that in chemisorption specific sites are involved and that therefore definite potential barriers to lateral motion should be present. The adsorption should therefore obey the statistical thermodynamics of a localized state. On the other hand, the kinetics of adsorption and of catalytic processes will depend greatly on the frequency and nature of such surface jumps as do occur. A film can be fairly mobile in this kinetic sense and yet not be expected to show any significant deviation from the configurational entropy of a localized state. 

Although the right-hand sides of Eqs. (8.27) and (8.28) are the same, the former applies to the mixture (subscript mix), while the latter applies to the mixing process (subscript m). The fact that these are identical emphasizes that in Eq. (8.27) we have calculated only that part of the total entropy of the mixture which arises from the mixing process itself. This is called the configurational entropy and is our only concern in mixing problems. The possibility that this mixing may involve other entropy effects—such as an entropy of solvation-is postponed until Sec. 8.12. 

In writing Eq. (8.41), we have clearly treated Aw as a contribution to enthalpy. This means we neglect volume changes (AHp jj. versus AUp jj.) and entropy changes beyond the configurational changes discussed in the last section (AGp jj. versus AH jj.). In a subsequent development it is... 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

The quantity in parentheses on the right-hand side is reminiscent of the expression AH - T AS, with the quantity 1/2R a contribution from the configurational entropy of the Flory-Huggins theory. Since our objective is to incorporate a solvation entropy into the discussion, we add the latter -in units of R for convenience-to 1/2R ... 

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