Discontinuities in Some Thermodynamic Quantities at the Critical Point of an Analytical Fluid

For an analytical fluid it is shown that at the critical point the following discontinuities exist: $$C_V\left(\begin{array}{cc}2\mathrm{phases}& {\mathrm{n\hspace{0.17em}/\hspace{0.17em}V}}_T{\mathrm{\hspace{0.17em}=\hspace{0.17em}\rho }}_c\end{array}\right){\mathrm{\hspace{0.17em}-\hspace{0.17em}C}}_V{\mathrm{\hspace{0.17em}⧸\hspace{0.17em}RZ}}_c\mathrm{\hspace{0.17em}=\hspace{0.17em}}\frac{{\text{3pal}}^2}{\mathrm{paaa}}$$ , $$d^2{\mathrm{p\hspace{0.17em}/\hspace{0.17em}dt}}^2{\mathrm{\hspace{0.17em}-\hspace{0.17em}(\partial }}^2{\mathrm{p\hspace{0.17em}/\hspace{0.17em}\partial t}}^2{)}_a{\mathrm{\hspace{0.17em}=\hspace{0.17em}(P}}_{\mathrm{at}}{\text{\hspace{0.17em}/\hspace{0.17em}5p}}_{\mathrm{aaa}}{\text{)[16p}}_{\mathrm{at}}{\text{\hspace{0.17em}−\hspace{0.17em}10P}}_{\mathrm{aat}}{\text{\hspace{0.17em}+\hspace{0.17em}(3p}}_{\mathrm{at}}{P}_{\mathrm{aaaa}}{\mathrm{)\hspace{0.17em}/\hspace{0.17em}P}}_{\mathrm{aaa}}{\mathrm{],\; T}}_c{\mathrm{[(d}}^2{\mathrm{G\hspace{0.17em}/\hspace{0.17em}dT}}^2{\mathrm{)\hspace{0.17em}-\hspace{0.17em}(\partial }}^2{\mathrm{G\hspace{0.17em}/\hspace{0.17em}\partial T}}^2{)}_p{\mathrm{](RZ}}_c{)}^{\text{−1}}\mathrm{\hspace{0.17em}=\hspace{0.17em}}\frac{1}{5}{P}_{\mathrm{at}}{\mathrm{\hspace{0.17em}/\hspace{0.17em}P}}_{\mathrm{aaa}}{\mathrm{[P}}_{\mathrm{at}}{\text{\hspace{0.17em}−\hspace{0.17em}10p}}_{\mathrm{aat}}{\text{\hspace{0.17em}+\hspace{0.17em}(3P}}_{\mathrm{at}}{P}_{\mathrm{aaaa}}{\mathrm{)\hspace{0.17em}/\hspace{0.17em}P}}_{\mathrm{aaa}}]$$ , where ${\mathrm{a\hspace{0.17em}=\hspace{0.17em}(\rho /\rho }}_c{\text{)\hspace{0.17em}−\hspace{0.17em}1;p\hspace{0.17em}=\hspace{0.17em}(P\hspace{0.17em}/\hspace{0.17em}P}}_c{\text{)\hspace{0.17em}−\hspace{0.17em}1;t\hspace{0.17em}=\hspace{0.17em}(T\hspace{0.17em}/\hspace{0.17em}T}}_c\text{)\hspace{0.17em}−\hspace{0.17em}1;}$ G is the Gibbs free energy or chemical potential; $C_V$ is the heat capacity at constant volume; $n$ is the total number of moles in the calorimeter; $V_T$ is the total inside volume of the calorimeter; ${\rho }_c{\mathrm{,\; P}}_c{\mathrm{,\; T}}_c$ are the critical molal density, critical pressure, and critical temperature, respectively; $Z_c{\mathrm{\hspace{0.17em}=\hspace{0.17em}P}}_c{\mathrm{\hspace{0.17em}/\hspace{0.17em}\rho }}_c{\mathrm{RT}}_c;\mathrm{and}p$ with subscripts indicates partial derivatives of $p$ evaluated at the critical point.