How are van der Waals parameters determined experimentally? If the ideal gas law $PV=nRT$ is not enough, as a second approximation the van der Waals equation

$$\left(P+a\frac{n^2}{V}\right)\left(V-nb\right)=nRT$$

is applied. The constants $a$ and $b$ are van der Waals constants which take into account some of the attraction and repulsion  that the ideal gas law does not.

Solving for $P$, 

$$P=\frac{nRT}{V-nb}-a\frac{n^2}{V^2}$$

and plugging it into the definition formula of compression factor $Z$ yields
$$Z=\frac{V_{m,real}}{(RT)/P}=\frac{V_{real}}{(nRT)/P}=\frac{V_{real}}{V-nb}-a\frac{nV_{real}}{RTV^2}.$$

This result, supposedly, helps in the measuring of constants $a$ and $b$. But how is it really done? What are the more rigorous definitions of the constants?
At the critical point, the first and second derivatives of pressure with respect to volume are zero.  

This provides two equations to solve for the two unknowns (a and b) in terms of the critical temperature and critical volume (or pressure).  

See Critical Constants of the van der Waals Gas