Critical variable

Critical variables are defined, for example in thermodynamics, in terms of the values of variables at the critical point.

On a PV diagram, the critical point is an inflection point. Thus:

$\left({\frac {\partial P}{\partial V}}\right)_{C}=0$

$\left({\frac {\partial ^{2}P}{\partial V^{2}}}\right)_{C}=0$

For the van der Waals equation, the above yields:

$P_{C}={\frac {a}{27b^{2}}}$

Failed to parse (syntax error): {\displaystyle V_C=\3Nb }

$T_{C}={\frac {8a}{27bR}}$