I have to add another answer because I cannot comment on morristtu's reply...

IMO it is important to understand that the critical point is characterized by certain properties. For example, the first and second partial derivative of the pressure with respect to the molar volume at constant temperature vanish.

To find the critical point of the Peng-Robinson equation of state one can calculate the first and second partial derivative with respect to the molar volume, and look for a given volume/Temperature pair where both partial derivatives vanish.

  Doing so I found that a pure substance described by the Peng-Robinson EOS has the same critical pressure and temperature as the real pure substance, but that the critical molar volume is slightly off. My guess is that the Peng-Robinson EoS was constructed in a way that it yields equal values for critical temperature and pressure at the expense of the critical molar volume.

I find the following points in morristtu's answer confusing:

• There is no reason for leaving the domain of real numbers, and just taking the average of the absolute values seems somewhat arbitrary to me.
• At the critical point, there are only two phases (liquid, vapour) and you cannot distinguish them. Three phases are only present at the triple point (solid, liquid, vapour).
• To me, the critical compressibility is no criterion to judge the quality of an EoS. More a means to classify it, as there are EoS's with constant or variable critical compressibility.

It is important to understand that the critical point of a pure substance is characterized by certain properties. For example, the first and second partial derivative of the pressure with respect to the molar volume at constant temperature vanish. Secondly, it is important to understand that the critical point given by an equation of state (EoS) is not necessarily the same point as determined by measurements.

To find the critical point of the Peng-Robinson (PR) EoS one can calculate the first and second partial derivative with respect to the molar volume, and look for a given volume/temperature pair where both partial derivatives vanish. Doing this with a solver for non-linear equation systems (e.g. a Levenberg-Marquardt method) I found the critical point of CO2 to be close to $1.0T_c$ and $0.89\rho_c$, giving a critical pressure of $1.0p_c$, which yields a critical compressibility of $Z_c=0.3074$.

To get good results I recommend the following:

• If you use iterative solvers be sure to check residuals and convergence of the method.
• Use the same constants and critical state variables everywhere, as the solution is quite sensitive.