The van der Waals theory of liquid-gas transitions

In Section 2.2 we introduced the van der Waals equation of state for a gas. This model, which provides one of the earliest explanations of critical phenomena, is also very suited for a qualitative explanation of the limits of mechanical stability of a homogeneous liquid. Following Stanley [17], we will apply the van der Waals equation of state to illustrate the limits of the stability of a liquid and a gas below the critical point. 

The van der Waals equation of state for one mole of gas is expressed in terms of the critical pressure, temperature and volume by eq. (2.40) as 

1 In practice the glasses are made by first quenching the liquid. The phase separation takes 

For sub-critical isotherms (T Tc), the parts of the isotherm where (dp/dV)T 0 become unphysical, since this implies that the thermodynamic system has negative compressibility. At the particular reduced volumes where (dp/dV)T =0, ( d2Gldp2 )T = 0 and we have spinodal points that correspond to those discussed for solutions in the previous section. This breakdown of the van der Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T TC, liquid and gaseous H2O, must have the same temperature and pressure in order to obey the equilibrium criteria. 

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