Maxwell construction

In thermodynamic equilibrium, a necessary condition for stability is that pressure, ${\displaystyle p}$, does not increase with molar volume, ${\displaystyle v=V/N}$; this is expressed mathematically as ${\displaystyle \partial _{v}p|_{T}<0}$, where ${\displaystyle T}$ 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]

where ${\displaystyle a,b,R}$ are dimensional constants. This violation is not a defect, rather it is the origin of the observed discontinuity in properties that distinguish liquid from vapor, and defines a first order phase transition. Figure 1 shows an isotherm drawn, for ${\displaystyle v>b}$, as a continuously differentiable solid black, dotted black, and dashed gray curve. The decreasing right hand part of the curve in Fig. 1 describes a gas, while its left decreasing part describes a liquid. These two parts are separated by a region between the local minimum and local maximum on the curve with positive slope that violates the stability criterion. This mathematical criterion expresses a physical condition which Epstein^[4] described as follows:

"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 ${\displaystyle v_{\rm {min}}\leq v\leq v_{\rm {max}}}$ is a precursor of the actual phase change from liquid to vapor. The points ${\displaystyle (p_{\rm {min}},v_{\rm {min}})}$ and ${\displaystyle (p_{\rm {max}},v_{\rm {max}})}$, where ${\displaystyle \partial _{v}p|_{T}=0}$, 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, ${\displaystyle p_{s}(T)}$, vapor, (denoted by dots at points ${\displaystyle f}$ and ${\displaystyle g}$ 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, ${\displaystyle v_{f}=V_{f}/N_{f}}$, and a condensing (saturated) vapor, ${\displaystyle v_{g}=V_{g}/N_{g}}$ that coexist at the same saturation temperature and pressure. As the heating continues the amount of vapor, ${\displaystyle N_{g}}$ increases and that of the liquid, ${\displaystyle N_{f}=N-N_{g}}$, decreases. All the while the pressure, ${\displaystyle p_{s}}$ and temperature, ${\displaystyle T}$, remain constant and the volume ${\displaystyle V=V_{f}+V_{g}}$ increases. In this situation the molar volume of the mixture is a weighted average of its components

where ${\displaystyle x=N_{g}/N}$, the mole fraction of the vapor, ${\displaystyle 0\leq x\leq 1}$, increases continuously; however, the molar volume of the substance itself has only the largest possible stable value for its liquid state, and smallest possible stable value for its vapor state at the given ${\displaystyle p(T)}$. To repeat, although the mixture molar volume passes continuously from ${\displaystyle v_{f}}$ to ${\displaystyle v_{g}}$ (denoted by the dashed line in Fig. 1), the underlying fluid has a discontinuity in this property, and others as well. This equation of state of the mixture is called the lever rule.^[5][6]

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].