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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:

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

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

For the van der Waals equation, the above yields:

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

${\displaystyle \displaystyle {V_{C}=3nb}}$

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

References

1. ^ Sommerfeld, A. (1956). Bopp, F.; Meixner, J. (eds.). Thermodynamics and Statistical Mechanics - Lectures on Theoretical Physics Volume V. Translated by Kestin, J. NY: Academic Press. pp. 56–57.

• Thermodynamic properties
• Conformal field theory