Thermodynamic variables associated with critical points
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b949fc62aa3799881e7beb39abf328f6475872f8)
![{\displaystyle \left({\frac {\partial ^{2}P}{\partial V^{2}}}\right)_{C}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f241eea0860723a5f8af25cf8e6bb625579ad9db)
For the van der Waals equation, the above yields:
![{\displaystyle P_{C}={\frac {a}{27b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa91606e54b8e34b2f77acb7c1c4537edc1406bf)
![{\displaystyle \displaystyle {V_{C}=3nb}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00bd11734e22d6214a462d52162058acf1ed5a23)
![{\displaystyle T_{C}={\frac {8a}{27bR}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e56f8e41ea4be691c605a5c96a60ecbd6267023)
References