One-dimensional model for mixtures of hard spheres

The second example where we can write the exact expression for the chemical potential is a one-dimensional mixture of hard rods (i.e., hard spheres in onedimensional system). For a system of NA rods of length (diameter) aA and Nn rods of length aB, in a volume L at temperature T, the canonical partition function is well known1 (Ben-Naim 1992). 

1 This is obtained from eliminating pB from the equation of state (6.72) and substituting in (6.70). 

the equation of state, and the virial expansion can easily be written for this model  

Note that in this model we always have gab = (pA + crB)/2. Transforming into variables T, P, xA, we obtain the expression for the chemical potential 

We now use the last expression to derive the three deviations, or the excess chemical potential, with respect to the three ideal behaviors. As in the previous case, we shall rewrite expression (6.74) in three different forms, as follows  

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