Liu hard sphere equation of state

Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude :

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z={\frac {1+\eta +\eta ^{2}-{\frac {8}{13}}\eta ^{3}-\eta ^{4}+{\frac {1}{2}}\eta ^{5}}{(1-\eta )^{3}}}.}

The conjugate virial coefficient correlation is given by:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B_{n}=0.9423n^{2}+1.28846n-1.84615,n>3.}

The excess Helmholtz free energy is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{ex} = \frac{ A - A^{id}}{Nk_B}= \frac{ 188\eta - 126\eta^2 - 13\eta^4 }{52(1-\eta)^2} - \frac{5}{13} ln(1-\eta). }

The isothermal compressibility is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_T = (\eta\frac{ dZ}{d\eta} + Z)^{-1} \rho^{-1}. }

where

${\frac {dZ}{d\eta }}={\frac {4+4\eta -{\frac {11}{13}}\eta ^{2}-{\frac {52}{13}}\eta ^{3}+{\frac {7}{2}}\eta ^{4}-\eta ^{5}}{(1-\eta )^{4}}}.$