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Figure A: The surface generated by the ideal gas equation.

The van der Waals equation is named for its originator the Dutch physicist Johannes Diderik van der Waals. It is an equation of state that gives the pressure as a function of temperature and volume fraction. This last property is the volume divided by the number of particles (or an equivalent property, e.g. mass), which is the reciprocal of number density (if mass is used the reciprocal is mass density). The equation modifies the ideal gas law in two ways. First its particles have a finite diameter, whereas the ideal gas consists of point particles that have zero diameter. This change makes the minimum value of the volume fraction greater than zero instead of the ideal gas value, zero. Second, it provides for particle interactions with one other, whereas the particles of an ideal gas move as though they were alone.

The surface calculated from the ideal gas equation of state is drawn in Fig. A. This universal (all ideal gases are represented by it) surface is normalized so that the black dot, with coordinates $p_{0},v_{0},T_{0}$, appears at the location (1,1,1) of the 3 dimensional plot space. This device makes it easy to compare this surface with the one generated by the van der Waals equation in Fig. C. Also Figs. A and C present the two surfaces from the same viewpoint to make the comparison easier. Whereas the ideal gas surface is rather plain, the van der Waals surface has an interesting fold.

Figure B: The surface generated by the van der Waals equation

Figures B and C show different views of the surface calculated from the van der Waals equation. The fold seen on this surface is what enables the equation to predict the phenomenon of liquid--vapor phase change. This fold develops from a critical point defined by specific, critical, values of pressure, temperature, and volume fraction. The surface is plotted using dimensionless variables that are formed by the ratio of pressure, temperature, and volume fraction to their respective critical values. This locates the critical point at the coordinates (1,1,1) of the space. When drawn using these dimensionless properties, this surface is also universal. Moreover, it represents all real substances to a remarkably high degree of accuracy. This principle of corresponding states has become one of the fundamental ideas in the thermodynamics of fluids.

Figure C: Another view of the folded surface generated by the van der Waals equation.

The boundary of the fold on the surface is the spinodal curve. However, this curve does not also define the location of the phase change. That place is given by the saturation curve, a curve that is not specified by the properties of the surface alone. The saturation curve is the locus of saturated liquid and vapor states which, being in equilibrium with each other, can coexist. The saturated liquid and vapor curves are identified in Fig. B. Together they comprise the saturation (or coexistence) curve seen in Figs. B and C. Also the inset in Fig. B shows the mixture states that can exist between these two saturated states.

Although it is hard to appreciate today when atomic physics is so highly developed, at the time van der Waals created his equation based on the particle nature of gases few scientists believed such particles really existed. However, the theoretical explanation of the critical point, which had been discovered a few years earlier, and later its qualitative and quantitative agreement with experiments cemented its acceptance in the scientific community. Ultimately these accomplishments won him the 1910 Nobel prize in physics. Today the equation is recognized as an important model of phase change processes. Van der Waals also adapted his equation so that it applied to a binary mixture of fluids. He, and others, then used the modified equation to discover a host of important facts about the phase equilibria of such fluids. This application, expanded to treat multi-component mixtures, has extended the predictive ability of the equation to fluids of industrial and commercial importance. In this arena it has spawned many similar equations in a continuing attempt by engineers to improve their ability to understand and manage these fluids. Indeed, it remains relevant to this day.

Behavior of the equation

One way to write this equation is:

$$p={\frac {RT}{v-b}}-{\frac {a}{v^{2}}}$$

where $p$ is pressure, $T$ is temperature, and $v=VN_{\text{A}}/N$ is molar volume, $N_{\text{A}}$ is the Avogadro constant, $V$ is the volume, and $N$ is the number of molecules (the ratio $N/N_{\text{A}}$ is a physical quantity with base unit mole in the SI). In addition $R=N_{\text{A}}k$ is the universal gas constant, $k$ is the Boltzmann constant, and $a$ and $b$ are experimentally determinable, substance-specific constants.

The force exerted by a molecule on another at a distance $r$ is the negative of the slope of this curve at $r$. The force is repulsive, and large, for $r<\sigma$, and attractive when $r>r_{0}>\sigma$.

The constant $a$ expresses the strength of the molecular interactions. It has dimension: pressure times molar volume squared, [pv²] which is also molar energy times molar volume. The constant $b$ denotes an excluded molar volume; it is some multiple of the molecular volume, because the centers of two hard spheres can never be closer than their diameter. It has dimension molar volume, [v].

A theoretical calculation of these constants at low density for spherical molecules with an interparticle potential characterized by a length, $\sigma$, and a minimum energy, $-\varepsilon$ (with $\varepsilon \geq 0$), as shown in the accompanying plot produces $b=4N_{\text{A}}[(4\pi /3)(\sigma /2)^{3}]$. Multiplying this by the number of moles, $N/N_{\text{A}}$, gives the excluded volume as 4 times the volume of all the molecules. This theory also produces $a=IN_{\text{A}}\varepsilon b$ where $I$ is a number that depends on the shape of the potential function, $\varphi (r)/\varepsilon$.

In his book (see references [9] and [10]) Boltzmann wrote equations using $V/M$ (specific volume) in place of $VN_{\text{A}}/N$ (molar volume) used here, Gibbs did as well, so do most engineers. Also the property, $V/N=1/\rho _{N}$ the reciprocal of number density, is used by physicists, but there is no essential difference between equations written with any of these properties. Equations of state written using molar volume contain $R$, those using specific volume contain $R/{\bar {m}}$ (the substance specific ${\bar {m}}$ is the molar mass), and those written with number density contain $k$.

Once $a$ and $b$ are experimentally determined for a given substance, the van der Waals equation can be used to predict the boiling point at any given pressure, the critical point (defined by pressure and temperature values, $p_{\text{c}}$, $T_{\text{c}}$ such that the substance cannot be liquefied either when $p>p_{\text{c}}$ no matter how low the temperature, or when $T>T_{\text{c}}$ no matter how high the pressure; $p_{c},T_{c}$ uniquely define $v_{c}$), and other attributes. These predictions are accurate for only a few substances. For most simple fluids they are only a valuable approximation. The equation also explains why superheated liquids can exist above their boiling point and subcooled vapors can exist below their condensation point.

Constant pressure curves of the van der Waals equation. The green circles are the saturated liquid, $v_{f}$, and vapor, $v_{g}$, at $p$, and $T_{s}$. The locus of the saturated liquid and vapor states--the green dots in this figure--for all subcritical isobars form the saturation curve on the $p,v,T$ surface.

The graph on the right follows from the intersection of the surface shown in the introduction and 4 constant pressure planes. Each intersection produces a curve in the $T$ vs $v$ plane corresponding to the value of the pressure chosen. On the red isobar (another name for a constant pressure curve), $p=2p_{\text{c}}$, the slope is positive over the entire range, $b\leq v<\infty$ (although the plot only shows a finite quadrant). This describes a fluid as homogeneous for all $T$, and is characteristic of all supercritical isobars $p>p_{\text{c}}.$

The green isobar, $p=0.2\,p_{\text{c}}$, has a region of negative slope. This region consists of states that are unstable and therefore never observed (for this reason this region is shown dotted gray). The green curve thus consists of two disconnected branches; a vapor on the right, and a denser liquid on the left. For this pressure, at a temperature, $T_{s}$, specified by mechanical, thermal, and material equilibrium, and shown as green circles on the curve, the boiling (saturated) liquid, $v_{f}$, (the left circle) and condensing (saturated) vapor, $v_{g}$, (the right circle) coexist. Due to gravity the denser liquid appears below the vapor, and a meniscus is seen between them. This heterogeneous combination of coexisting liquid and vapor is the phase change. Heating the fluid in this state increases the fraction of vapor in the mixture; its $v$, an average of $v_{f}$ and $v_{g}$ weighted by this fraction, increases while $T_{s}$ remains the same. This is shown as the dotted gray line, because it does not represent a solution of the equation; however, it does describe the observed behavior. The points above $T_{s}$, superheated liquid, and those below it, subcooled vapor, are metastable; a sufficiently strong disturbance causes them to transform to the stable alternative. Consequently they are shown dashed.

All this describes a fluid as a stable vapor for $T>T_{\text{s}}$, a stable liquid for $T<T_{\text{s}}$, and a mixture of liquid and vapor at $T=T_{\text{s}}$, that also supports metastable states of subcooled vapor and superheated liquid. It is characteristic of all subcritical isobars $0<p<p_{\text{c}}$, where $T_{\text{s}}$ is a function of $p$.

The orange isobar is the critical one on which the minimum and maximum are equal. The critical point lies on this isobar.

The black isobar is the limit of positive pressures, although drawn solid none of its points represent stable solutions, they are either metastable (positive or zero slope) or unstable (negative slope). Interestingly, states of negative pressure (tension) exist. They lie below the black isobar, and although they are not not drawn in this figure, they form those parts of the surfaces seen in Figs. B and C that lie below the zero pressure plane. In this, $T,v$, plane they have a parabola like shape and like the zero pressure isobar their states are all either metastable (positive or zero slope) or unstable (negative slope).

All this is a good explanation of the observed behavior of fluids.

Reference

• Valderrama, Jose O. (2010). "The legacy of Johannes Diderik van der Waals, a hundred years after his Nobel Prize for physics". Jour Supercrit Fluids. 55: 415–420.