The exact solution for the Percus-Yevick integral equation for hard spheres
was derived by M. S. Wertheim in 1963 \cite{PRL_1963_10_000321} (See also \cite{JMP_1964_05_00643})
(and for mixtures by in Lebowitz 1964 \cite{PR_1964_133_00A895}).
The direct correlation function is given by (\cite{PRL_1963_10_000321} Eq. 6)
where
and $R$ is the hard sphere diameter.\\
The equation of state is (\cite{PRL_1963_10_000321} Eq. 7)
Everett Thiele (1963 \cite{JCP_1963_39_00474}) also studied this system,
resulting in (Eq. 23)
where (Eq. 24)
and
and
and where $x=\rho/4$.\\
The pressure via the pressure route (Eq.s 32 and 33) is
and the compressibility route is
References