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Critical Point

The endpoint of a phase equilibrium curve, representing the conditions where distinct phases no longer exist in equilibrium.
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The statement of the theorem

Let F(T,V)F(T, V) be the Helmholtz free energy density of a system, where TT is temperature and VV is volume. The equation of state is defined by the pressure P=−(∂F∂V)TP = -\left(\frac{\partial F}{\partial V}\right)_{T}. The critical point (Tc,Pc,Vc)(T_c, P_c, V_c) is mathematically characterized by the simultaneous vanishing of the second and third derivatives of the free energy with respect to volume (or density ρ=1/V\rho = 1/V) at the point of maximum compressibility, such that the isothermal compressibility κT\kappa_T and the heat capacity CVC_V exhibit critical exponents. Specifically, the following conditions must hold at (Tc,Vc)(T_c, V_c): \begin{enumerate} \item ∂2P∂V2∣Tc=0\left.\frac{\partial^2 P}{\partial V^2}\right|_{T_c} = 0 \item ∂3P∂V3∣Tc=0\left.\frac{\partial^3 P}{\partial V^3}\right|_{T_c} = 0 \end{enumerate} Furthermore, the critical point is defined by the divergence of the generalized susceptibility χ\chi: \begin{equation} \chi^{-1} = \frac{\partial P}{\partial V} \bigg|_{(T, V)} \end{equation} \text{such that at } (T_c, V_c), \lim_{|(T, V) - (T_c, V_c)| \to 0} \chi^{-1} = 0.Thisimpliesthecriticalexponent. This implies the critical exponent \gamma = 0forthesusceptibility,orequivalently,thevanishingofthecoefficientofthequadratictermintheLandauexpansionofthefreeenergydensity for the susceptibility, or equivalently, the vanishing of the coefficient of the quadratic term in the Landau expansion of the free energy density f(\phi)$: \begin{equation} f(\phi) \approx \frac{1}{2} a(T-T_c) \phi^2 + \frac{1}{4} b \phi^4 + \frac{1}{6} c \phi^6 \end{equation} \text{where } a(T_c)=0 \text{ and } b>0 \text{ for the second-order transition.}