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:

[math]\displaystyle{ \left(\frac{\partial P}{\partial V}\right)_{C}=0 }[/math]

[math]\displaystyle{ \left(\frac{\partial^2 P}{\partial V^2}\right)_{C}=0 }[/math]

For the van der Waals equation, the above yields:

[math]\displaystyle{ P_C=\frac{a}{27b^2} }[/math]

[math]\displaystyle{ \displaystyle{V_C=3nb} }[/math]

[math]\displaystyle{ T_C=\frac{8a}{27bR} }[/math]