Calculation of Mixing Entropy

The mixing entropy normally contains the contributions of translation, rotation, vibration and combination, as 

Since the fourth term is obtained from the combination of chain units and solvent molecules, the first three terms for the whole chain are far less than the fourth one if considering their global contributions in the lattice space. Therefore, only the combinatirm entropy Scombinate is calculated. According to the Boltzmann s relation Scombinate = k aQ, we only need to calculate the total amount of arrangement of molecules in the lattice space. 

It should be noted that the probability of a vacant coordination site surrounding 7th monomer is pij = 1 —(77 + i)/N, which is on the basis of assumpticm two, i.e. the so-called random-mixing approximation. On this point, FlOTy and Huggins made different treatments. Let s consider the 7th monomer that has been put into a previously vacant site, the (7 + l)th monomer has to be put into a vacant coordination site surrounding the previously vacant site. Therefore, should be the fraction of two consecutively connected vacant sites in the total pairs of two neighboring sites containing one vacant site. The total vacant sites are N—rj—i, and their total coordination number is q N—rj—i), each with the vacant probability 1 (ry + i) N, so the total number of two consecutively connected vacant sites is 

Here 1/2 is the symmetric factor for estimating the vacant site pairs twice. Similarly, the total amount of coordination site pairs in the lattice space is qNH. In 1942, Flory did not consider the consecutive occupation of vacant site pairs (Flory 1942), and calculated only those neighboring site pairs containing one vacant site, as 

Also in 1942, Huggins considered the doubly vacant case in the total amount of coordination pairs qN/2 (Huggins 1942), as 

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Equation (4.434) permits the calculation of mixing entropy (change of entropy at mixing of pure constituents to mixmre) defined as the difference between the entropy of the mixture and the sum of entropies of pure constituents molar mixing entropy (related to one mole of mixture, is therefore (using molar quantities in (4.434), (4.91), (4.292) at the same T, P of pure constituents and in mixture) ... 

We note that the calculation of mixing entropy transcends the microemulsion field and is relevant also to phenomenological theories of nucleation. In that field the length / appears in the replacement free energy problem (see Ref 33 and, more general. Ref 34). 

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