The Hard-Rod Approximations

The reason for studying the hard-rod approximations is to obtain some qualitative intuition of the consequences of various assumptions. By simplifying to one dimension, rather than three dimensions, the mathematics is simpler, albeit still messy in some places. More than one type of particle can also be included. To make the following discussion simple, only two molecular diameters will be assumed, and 02 for species 1 and 2, respectively. 

For the open system, one needs the grand canonical partition function or 

The unit step function, U, is inserted to account for the obvious fact that the total length of the rod cannot exceed the length of the box. For each individual and N2 the probability, P N jSl2 is 

The ID pressure, P, is obtained from S by analogy to the 3D case, that is the lengA, L, replaces the volume so 

The extension to more than two molecular species should be clear from the above by simply adding additional terms for species 3, 4, etc. It is easy to model the above on a spread sheet. To obtain an idea of what this would look like, consider the case of only one species. For the probabilities as a function of the distance, L, Eq. (258) becomes 

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