Residual properties hard sphere

By combining the Carnahan-Starling equation with the integral forms in 4.4.2, we can evaluate the residual properties of a pure hard-sphere fluid. The results are [15]... 

Other residual properties can be obtained via Legendre transforms. Note that in the zero-density limit, these residual properties all go to zero, as they should. Further, in the hard-sphere limit a = 0) these expressions revert to the Carnahan-Starling expressions (4.5.5)-(4.5.7), as they should. 

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