Entropy calculated configurational

Differences in Afor different AB5Hn compounds compared with A for CeCosHs are listed in Table III. The values of these numbers (see Table III), calculated using the fractional site occupations for the 0 phase, can be compared with the experimentally determined entropy differences listed in Table I. The calculated configurational entropy differences (see Table III) agree satisfactorily with the experimental data (see Table I) currently available for seven ABsHn compounds. Structures of some ABsHn compounds deduced from neutron diffraction data (4) are listed in Table I. For compounds whose structures have not been determined, the occupation numbers listed in Table III are in best agreement with the thermodynamic data. 

Another example is that of lattice chain models. Simple square lattice models were established by Flory as a vehicle for calculating configurational entropies etc., and used later in the simulations of the qualitative behaviour, e.g. of block copolymer phase separation. More sophisticated models such as the bond fluctuation model, and the face centred cubic lattice chain modeP ... 

The calculation of the partition function can be done by the standard Flory-Huggins lattice method. The lattice model predicts the existence of a true second-order transition at a temperature T2. This is shown schematically in Figure 13 for the entropy-pressure-temperature equation of state. As can be seen, the transition occurs at a critical value of the entropy (zero configurational entropy) and the Kauzmann paradox is resolved for thermodynamic reasons rather than kinetic ones, i.e. one is simply not permitted to extrapolate high temperature behavior through the glass transition. Rather, as the material is cooled, a break in the S-T (or V-T) curves occurs because of a second-order transition. 

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. 

The critical size of the stable nucleus at any degree of under cooling can be calculated widr an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n atoms widr o atoms of the liquid per unit volume, is given by the Boltzmann equation... 

Block Entropies The definitions are the same as for finite lattices except that whereas before we could sum over all possible configurations, we now perform the same calculations for all possible sequences of sites within blocks of finite size B ... 

It is simplest to consider these factors as they are reflected in the entropy of the solution, because it is easy to subtract from the measured entropy of solution the configurational contribution. For the latter, one may use the ideal entropy of mixing, — In, since the correction arising from usual deviation of a solution (not a superlattice) from randomness is usually less than — 0.1 cal/deg-g atom. (In special cases, where the degree of short-range order is known from x-ray diffuse scattering, one may adequately calculate this correction from quasi-chemical theory.) Consequently, the excess entropy of solution, AS6, is a convenient measure of the sum of the nonconfigurational factors in the solution. 

In order to arrive ultimately at the entropy change accompanying deformation, we now proceed to calculate the configurational entropy change involved in the formation of a network structure in its deformed state as defined by a, ay, and (We shall avoid for the present the stipulation that the volume be constant, i.e., that axayag=l.) Then by subtracting the entropy of network formation when the sample is undeformed (ax = ay = az=l)j we shall have the desired entropy of deformation. As is obvious, explicit expressions will be required only for those terms in the entropy of network formation which are altered by deformation. 

Figure 3.2 Cyclic voltammograms for H adsorption on Pt(lll) and Pt(lOO). Two different methods have been applied. In (a) and (b), the H particles were assumed not to interact in the expression for the configurational entropy. In (c) and (d), the more elaborate model involving Metropolis Monte Carlo was applied. As can be seen, for these homogenous surfaces, the simple method suffices. The figure is adopted from [Karlberg et al., 2007a], where the full details of the calculations can also be found.

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