Molecular model of a liquid mixture

We have seen that when two pure liquids mix to form an ideal liqiud mixture at the same T and p, the total volume and internal energy do not change. A simple molecular model of a binary liquid mixture will elucidate the energetic molecular properties that are consistent with this macroscopic behavior. The model assumes the excess molar entropy, but not necessarily the excess molar internal energy, is zero. The model is of the type sometimes called the qmsicrystalline lattice model, and the mixture it describes is sometimes called a simple mixture. Of course, a molecular model like this is outside the realm of classical thermodynamics. 

Similarly, the molar internal energy of pure liquid B is given by 

We assume that in a liquid mixture of A and B, the numbers of nearest-neighbor molecules of A and B surrounding any given molecule are in proportion to the mole fractions xa and xb Then the number of A-A interactions is proportional to haXa, the number of B-B 

The internal energy change for mixing amounts a of liquid A and b of liquid B is now 

If the internal energy change to form a mixture of any composition is to be zero, as it is for an ideal mixture, the quantity (2A ab — aa — bb) tnust be zero, which means A ab must equal ( aa + bb)/2. Thus, one requirement for an ideal mixture is that an A-B interaction equals the average of an A-A interaction and a B-B interaction. 

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