Thermodynamics of ${P}$-${V}$ criticality in $d$-dimensional AdS black holes surrounded by a perfect fluid in Rastall theory

We study a $d$-dimensional Anti-de Sitter (AdS) black hole surrounded by a static anisotropic quintessence field within the framework of Rastall theory. The solution is characterized by several parameters including its mass ($M$), field structure parameter ($N_s$), Rastall coupling parameter ($ψ$), and the cosmological constant ($Λ$). Our objective here is to identify the $d$-dimensional black holes within the framework of Rastall theory, exploring special cases, e.g., a cosmological constant, dust, radiation, and quintessence fields. In addition to this, we derive the effective equation of state parameter, denoted as $ω_{eff}$. Next, we investigate the thermodynamics for $P$-$V$ criticality and phase transitions in the extended phase space of black hole thermodynamics. For a specific set of values for the Rastall coupling parameter, we numerically plot isothermal and isobaric curves in the reduced parameter space. We also compute the specific heat at constant pressure ($C_P$ ), volume expansion coefficient ($α$), and isothermal compressibility ($κ_T$) to deepen our understanding of the analogy between the thermodynamics of Rastall AdS black holes and that of a liquid-gas system. Our investigations indicate that the dimension of spacetime and the Rastall coupling parameter significantly influence the critical nature of phase transitions. By utilizing the expressions for $C_P$, $α$, and $κ_T$, we derive the Ehrenfest equations and perform an analytical investigation of phase transitions at their critical points. These results allow us to compare the thermodynamics of AdS black holes with liquid-gas systems, which closely mimics the behavior of van der Waals (vdWs) gases.