• We can find conditions for phase equilibrium by considering the total, isolated system to be composed of two open systems

$$ S(E,V,N)=S_1(E_1,V_1,N_1)+S_2(E_2,V_2,N_2) $$

Since the overall system is isolated it reaches equilibrium when entropy is maximised i.e. $\text dS=0$ so:

$$ \text dS=\left ( \frac{\partial S}{\partial E} \right ){V,N}\text dE+\left ( \frac{\partial S}{\partial V} \right ){E,N}\text dV+\left ( \frac{\partial S}{\partial N} \right )_{E,V}\text dN=0 $$

⚙️ Properties

• $T_1=T_2$ no heat flow between systems
• $P_1=P_2$ the systems are at mechanical equilibrium
• $\mu_1=\mu_2$ there is no preference to be in one state over the other

Phase Coexistence

The Clausius-Clapeyron relation related different point of phase coexistence

$$ \frac{\text d P}{\text d T}=\frac{\Delta S^m}{\Delta V^m}=\frac{\Delta H^m}{T\Delta V^m} $$

Where $S^m$ and $V^m$ refer to entropy and volume per mole, respectively, and $H^m$ is the enthalpy per mole(latent heat transformation)

Van der Waals Equation

$$ P=\frac{nRT}{V-nb}-a\frac{n^2}{V} $$

• The VdW equation is cubic for V

$$ PV^2(V-nb)-nRTV^2+an^2(V-nb)=0 $$

There is either one real root or three, this depends on temperature

• For $T>T_C=\frac{8a}{27bR}$ there is only one solution for $V$ where $P$ decreases with $V$
• For $T=T_C$, the isotherm has a point of inflexion where

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

at $V_C=3nb$

• For $T< T_C$ there are three solutions, however one is unphysical where $\left ( \frac{\partial P}{\partial V} \right )_{T_C}>0$