Entropy total configurational

In the theory, z is later related to the total configurational entropy because s JS = z /A, where s is the entropy of the unit of minimum or critical size, z (j c cannot be less than A ln 2). Further, the relaxation time. 

Next let s take a mineral such as garnet where substitution can occur on two different groups of sites instead of one. The general formula for garnet can be written X3Y2Si30i2 (remember that this choice for the formula now fixes our choice of n in equation (15.5)). The X sites are cubic (8-coordinated) and we will assume they accept Al " and Fe ions. The Y sites are octahedral (6-fold), and let s assume they accept only Mg , Fe , and The Si site is tetrahedral (4 fold), and it can also hold AP ions. The total configurational entropy for this mineral must be the sum of the entropies on all three of these sites, or... 

The result is Sn = FPaPb > so that the total configurational contribution to the entropy density is... 

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. 

Point defects are always present in every material in thermodynamic equilibrium. Considering the formation of n vacancies, the increase in configuration entropy is determined by the number of different possible ways of taking n atoms out of the crystal comprising N atoms in total. This number, c1, is given by... 

If each solvent molecule may occupy one of the remaining lattice sites, and in only one way, 0 represents also the total number of configurations for the solution, from which it follows that the configurational entropy of mixing the perfectly ordered pure polymer and the pure solvent is given hy Sc —k In Introduction of Stirling s approximations for the factorials occurring in Eq. (7) for fi, replacement of no with Ui+xn[Pg.501]

If the configurational entropy A>Sm is assumed to represent the total entropy change LSm on mixing, the free energy of mixing is simply obtained by combining Eqs. (10) and (20). That is,... 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

As a first approximation, assume that the total entropy change, ASy, is made up solely of configurational entropy. This can be determined by using the Boltzmann equation for the entropy of a disordered system ... 

From a thermodynamic perspective, Stillinger and Weber demonstrated that the total entropy of the liquid can similarly be divided into two additive terms, a configurational and a vibrational contribution.5,6 The configurational part Sc measures the number of structurally distinct basins of attraction on the PEL that the configuration point accesses at a given temperature, whereas the vibrational contribution Svib characterizes the number of states associated with intra-basin fluctuations. Thus, the AG relationship, when viewed from the PEL perspective, suggests that it is the thermodynamic availability of basins on the landscape that dominates the rate of liquid-state diffusive processes. 

These ideas have been employed to compute configurational entropy and hence test the AG relation via molecular simulation of several model systems.91-94,101,102 The approach used in those studies is conceptually simple. First, the total entropy of the fluid S is calculated by integration of standard thermodynamic relationships, for example, as discussed below. Then, the configurational contribution to the entropy Sc = S — Sv b, is approximated by subtracting from the total entropy an estimate for the vibrational contribution, Svib. 

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