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:

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

The isothermal compressibility is given by:

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

where

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