Maxwellʼs equal-area law for charged Anti-de Sitter black holes

In this Letter we present the construction of the Maxwell equal-area law in the Hawking temperature graph for a charged black hole in Anti-de Sitter background. We are able to find exact solution for the corresponding isotherm and entropies for “gaseous” (large) black holes and “liquid” (near-extremal) black holes. Isothermal construction removes the unphysical, negative heat capacity, regions. Furthermore, extremal black holes turn out to be dual to un-shrinkable molecules of Van der Waals real fluid, which may explain their thermodynamical stability.