In thermodynamic equilibrium, a necessary condition for stability is that pressure, , does not increase with molar volume, ; this is expressed mathematically as , where is the temperature.[1]
This basic stability requirement, and similar ones for other conjugate pairs of variables, is violated in analytic models of first order phase transitions. The most famous case is the van der Waals equation,[2][3]
"It is obvious that this middle part, dotted in our curves [dashed in Fig.1 here], can have no physical reality. In fact, let us imagine the fluid in a state corresponding to this part of the curve contained in a heat conducting vertical cylinder whose top is formed by a piston. The piston can slide up and down in the cylinder, and we put on it a load exactly balancing the pressure of the gas. If we take a little weight off the piston, there will no longer be equilibrium and it will begin to move upward. However, as it moves the volume of the gas increases and with it its pressure. The resultant force on the piston gets larger, retaining its upward direction. The piston will, therefore, continue to move and the gas to expand until it reaches the state represented by the maximum of the isotherm. Vice versa, if we add ever so little to the load of the balanced piston, the gas will collapse to the state corresponding to the minimum of the isotherm."
This situation is similar to a body exactly balanced at the top of a smooth surface that, with the slightest disturbance will depart from its equilibrium position and continue until it reaches a local minimum. As they are described such states are dynamically unstable, and consequently they are not observed. The gap is a precursor of the actual phase change from liquid to vapor. The points and , where , that delimit the largest possible liquid and smallest possible vapor states are called spinodal points. Their locus forms a spinodal curve which bounds a region where no homogeneous stable states can exist.
Experiments show that if the volume of a vessel containing a fixed amount of liquid is heated and expands at constant temperature, at a certain pressure, , vapor, (denoted by dots at points and in Fig. 1) bubbles nucleate so the fluid is no longer homogeneous, but rather it has become heterogeneous. It is now a mixture of two separate components, a boiling (saturated) liquid, , and a condensing (saturated) vapor, that coexist at the same saturation temperature and pressure. As the heating continues the amount of vapor, increases and that of the liquid, , decreases. All the while the pressure, and temperature, , remain constant and the volume increases. In this situation the molar volume of the mixture is a weighted average of its components
The dotted parts of the curve in Fig. 1 are metastable states. For many years such states were an academic curiosity; Callen[7] gave as an example,
water that has been cooled below 0°C at a pressure of 1 atm. A tap on a beaker of water in this condition precipitates a sudden dramatic crystallization of the system.
However, studies of boiling heat transfer have made clear that metastable states occur routinely as an integral part of this process. In it the heating surface temperature is higher than the saturation temperature, often significantly so, hence the adjacent liquid must be superheated.[8] Further the advent of devices that operate with very high heat fluxes has created interest in the metastable states, and the thermodynamic properties associated with them, in particular the superheated liquid states.[9] Moreover, the fact that they are predicted by the van der Waals equation, and cubic equations in general, is compelling evidence of its efficacy in describing phase transitions; Sommerfeld[10] described this as follows:
It is very remarkable that the theory due to van der Waals is in a position to predict, at least qualitatively, the existence of the unstable [called metastable here] states along the branches AA` or BB` [BC and FE in Fig. 1 here].