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.