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# 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:

$P_C=\frac{a}{27b^2}$
$\displaystyle{V_C=3Nb}$
$T_C=\frac{8a}{27bR}$