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 :

$Z={\frac {1+\eta +\eta ^{2}-{\frac {8}{13}}\eta ^{3}-\eta ^{4}+{\frac {1}{2}}\eta ^{5}}{(1-\eta )^{3}}}.$

The conjugate virial coefficient correlation is given by:

$B_{n}=0.9423n^{2}+1.28846n-1.84615,n>3.$

The excess Helmholtz free energy is given by:

$A^{ex}={\frac {A-A^{id}}{Nk_{b}}}={\frac {188\eta -126\eta ^{2}-13\eta ^{4}}{52(1-\eta )^{2}}}-{\frac {5}{13}}ln(1-\eta ).$

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

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 \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 }. }