# (p.244) C Regular solution theory

# (p.244) C Regular solution theory

We consider a mixture of two molecules on a cubic lattice. Our goal is to calculate the entropy of mixing of this binary mixture. The approach here is in contrast to that of Section 7.5.1
, where we calculated the entropy per lattice site. Here we shall calculate the entropy of a lattice of n sites explicitly. For a lattice containing n sites, there are n! configurations for n molecules on this lattice. We have two components, with *n _{
A
}
* and

*n*molecules of each species, where

_{ B }*n*=

*n*+

_{A}*n*

_{B}, so ø =

*n*/

_{A}*n*and 1 − ø

*= n*/

_{B}*n.*The entropy of mixing is simply the (natural logarithm of the) number of arrangements of the two molecules compared to the number of possible configurations on the lattice. The number of configurations is simply

and is illustrated for a 12 × 12 lattice in Fig. C.1 . We recall Stirling's approximation

which is valid for large n, and use it to calculate the entropy of mixing, *∆Sl* for the whole lattice,

(The reader is reminded that Boltzmann's relation, *S* = *k _{B}
* ln Ω, links entropy,

*S*, to the number of configurations, Ω.) We can thus simplify eqn (C.3) to obtain

where ∆*S* = ∆*Sl*/*n.* By considering the whole lattice we nevertheless recover eqn (7.19)
from eqn (C.4)
when *N* = 1. In Section 7.5.1
we explicitly included the entropy of the pure states, whereas in the calculation outlined in this Appendix, the entropy of the pure states is implicitly included as *n*!.

We cannot easily extend this theory to a polymer in a solvent. Firstly, it is not possible to replace *n
_{A}
* by

*φn*/

*N*because

*C*(

*n*) requires

_{A}, n_{B}*n*+

_{A}*n*=

_{B}*n*, but

*φn*/

*N*+ (1 −

*φ*)

*n*≠

*n.*Secondly and related, a polymer molecule can have different configurations on the lattice, whereas a (p.245)