Liu hard sphere equation of state

Hongqin Liu proposed a correction to the Carnahan-Starling equation of state which improved accuracy by almost two orders 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 energy function 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:

$k_{T}=(\eta {\frac {dZ}{d\eta }}+Z)^{-1}\rho ^{-1}.$

where

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