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

The van der Waals equation is a mathematical formula that describes the behavior of real gases. It is named after the Dutch physicist Johannes Diderik van der Waals. It is an equation of state that relates the pressure to temperature, volume and either mass or the number of particles (these are proportional, their ratio is the mass of one particle) in a fluid.

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 with no extension. Second, its particles interact with one other, whereas the particles of an ideal gas move as though they were alone in the volume.

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. Figures A and C are drawn with the same scales and limits; they also 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 molar volume. The surface is plotted using dimensionless variables that are formed by the ratio of each property to its respective critical value. This locates the critical point at the coordinates (1,1,1) of the space. When drawn using these dimensionless properties, this surface is, like that of the ideal gas, also universal. Moreover, it represents all real substances to a remarkably high degree of accuracy. This principle of corresponding states, developed by van der Waals from his equation, 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 marked, on each side of the critical point, by the spinodal curve, identified in Fig. B, and seen in Figs. B and C. However, this curve does not 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 additional condition needed for its specification was first provided by Maxwell. 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, which exist at the intersection of the mixture line and its isotherm. However, these mixture states are not part of the surface generated by the van der Waals equation.

At the time van der Waals created his equation, which he based on the idea that fluids are composed of discrete particles, there was a "movement in physics to replace atomic theories by ... theories based only on macroscopic variables.". (The debate was finally settled by 1909 with Robert Millikan's and Jean Perrin's experiments that accurately measured atomic properties.) 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. Van der Waals was awarded 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 the present.

Behavior of the equation

One way to write the van der Waals 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. In addition $N_{\text{A}}$ 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 (symbol mol) in the SI). Finally $R=N_{\text{A}}k$ is the universal gas constant, $k$ is the Boltzmann constant, and $a$ and $b$ are experimentally determinable, substance-specific constants.

As noted previously, when van der Waals created his equation the idea that fluids were composed of rapidly moving particles was believed by few scientists. Moreover, no one had knowledge of the atomic/molecular structure. Van der Waals used the simplest conception, a hard sphere. In that case two particles of diameter, $\sigma$, would come into contact when their centers were a distance $\sigma$ apart, hence the center of the one was excluded from a spherical volume equal to $4\pi \sigma ^{3}/3$ about the other. That is 8 times $V_{0}$, the volume of each particle of radius $\sigma /2$, but there are 2 particles which gives 4 times the volume per particle. The total excluded volume is then $B=4NV_{0}$, 4 times the volume of all the particles. Van der Waals and his contemporaries used an alternative, but equivalent, analysis based on the mean free path between molecular collisions that gave this result. From the fact that the volume fraction of particles, $f=(V-B)/V$ must be positive, van der Waals noted that as $N$ becomes larger the factor 4 must decrease (for spheres there is a known minimum $0<f_{\mbox{min}}<f\leq 1$), but he was never able to determine the nature of the decrease. The constant $b=N_{\text{A}}B/N$ in the equation above has dimension molar volume, [v]. The constant $a$ expresses the strength of the hypothesized interparticle attraction. Van der Waals only had as a model Newton's law of gravitation, in which two particles attracted in proportion to the product of their masses. Thus he argued that in his case the attractive pressure was proportional to the square of the density. The proportionality constant, $a$, when written in the form used above, has dimension pressure times molar volume squared, [pv²] which is also molar energy times molar volume.

The Sutherland potential (orange) represents two hard spheres that attract according to an inverse power law, and the Lennard-Jones potential (black) represents the induced-dipole--induced-dipole interaction of two non polar molecules. Both are simple realistic molecular models.

The intermolecular force was later conveniently described by the negative derivative of a pair potential function. For spherically symmetric particles this is most simply a function of separation distance with a single characteristic length, $\sigma$, and a minimum energy, $-\varepsilon$ (with $\varepsilon \geq 0$). Two of the many such functions that have been suggested are shown in the accompanying plot.

A modern theory based on statistical mechanics produces the same result for $b=4N_{\text{A}}[(4\pi /3)(\sigma /2)^{3}]$ obtained by van der Waals and his contemporaries. This result is valid for any pair potential for which the increase in $\varphi >0$ is sufficiently rapid. This includes the hard sphere model for which the increase is infinitely rapid and the result is exact. Indeed the Sutherland potential most accurately models van der Waals' conception of a molecule. It also includes potentials that do not represent hard sphere force interactions provided that the increase in $\varphi {\mbox{ for }}r<\sigma$ is fast enough, but then it is approximate; increasingly better the faster the increase. In that case $\sigma$ is only an "effective diameter" of the molecule. This theory also produces $a=IN_{\text{A}}\varepsilon b$ where $I$ is a number that depends on the shape of the potential function, $\varphi (r)/\varepsilon$. However, this result is only valid when the potential is weak, namely, when the minimum potential energy is very much smaller than the thermal energy, $\varepsilon \ll kT$.

In his book Boltzmann wrote equations using $V/M$ (specific volume) in place of $VN_{\text{A}}/N$ (molar volume) used here, 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 ${\bar {m}}=mN_{\text{A}}$ is the molar mass with $m$ the mass of a single particle), 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 Figs. B and C and 4 constant pressure planes. Each intersection produces a curve in the $T,v$ 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.

Relationship to the ideal gas

The ideal gas law follows from the van der Waals equation whenever $v$ is sufficiently large (or correspondingly whenever the molar density, $\rho =1/v$, is sufficiently small), Specifically

• when $v\gg b$, then $v-b$ is numerically indistinguishable from $v$,
• and when $v\gg (a/p)^{1/2}$, then $p+a/v^{2}$ is numerically indistinguishable from $p$.

Putting these two approximations into the van der Waals equation when $v$ is large enough that both inequalities are satisfied reduces it to

$$p=RT/v\quad {\mbox{or in terms of }}V{\mbox{ and }}N\quad pV=NkT$$

which is the ideal gas law. This is not surprising since the van der Waals equation was constructed by modifying the particles that had produced the ideal gas equation.

What is truly remarkable is the extent to which van der Waals succeeded. Indeed, Epstein in his classic thermodynamics textbook began his discussion of the van der Waals equation by writing, "In spite of its simplicity, it comprehends both the gaseous and the liquid state and brings out, in a most remarkable way, all the phenomena pertaining to the continuity of these two states". Also in Volume 5 of his Lectures on Theoretical Physics Sommerfeld, in addition to noting that "Boltzmann described van der Waals as the Newton of real gases", also wrote "It is very remarkable that the theory due to van der Waals is in a position to predict, at least qualitatively, the unstable [referring to superheated liquid, and subcooled vapor now called metastable] states" that are associated with the phase change process.

Utility of the equation

The equation has been, and remains very useful because:

• its specific heat at constant volume, $c_{v}$, can be shown to be a function of $T$ only, and its thermodynamic properties, internal energy $u$, entropy $s$, as well as the specific heat at constant pressure $c_{p}$ have simple analytic expressions [this is also true of enthalpy $h=u+pv$, Helmholtz free energy $f=u-Ts$, and Gibbs free energy $g=u+pv-Ts=f+pv=h-Ts$]

• Its coefficient of thermal expansion, $\alpha =(\partial _{T}v|_{p})/v$ has a simple analytic expression [this is also true of its isothermal compressibility, $\kappa _{T}=-(\partial _{p}v|_{T})/v$]
• it explains the existence of the critical point and the liquid–vapor phase transition including the observed metastable states
• it establishes the law of corresponding states
• its Joule–Thomson coefficient and associated inversion curve, which were instrumental in the development of the commercial liquefaction of gases, have simple analytic expressions.

In addition its vapor pressure curve (also called the coexistence, or saturation, curve) has a simple analytic solution. It depicts the liquid metals, Mercury and Cesium, quantitatively, and describes most real fluids qualitatively. Consequently it can be regarded as one member of a family of equations of state, that depend on a molecular parameter such as the critical compressibility factor, $Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})$, or the Pitzer (acentric) factor, $\omega =-\log[p_{s}(T/T_{\text{c}}=0.7)/p_{\text{c}}]-1$, where $p_{s}/p_{\text{c}}=p_{s}(T/T_{\text{c}},\omega )$ is a dimensionless saturation pressure, and log is the logarithm base 10. Consequently, the equation plays an important role in the modern theory of phase transitions. All this makes it a worthwhile pedagogical tool for physics, chemistry, and engineering lecturers, in addition to being a useful mathematical model which can aid student understanding.

History

In 1857 Rudolf Clausius published The Nature of the Motion which We Call Heat. In it he derived the relation $p=(N/V)m{\overline {c^{2}}}/3$ for the pressure, $p$, in a gas, composed of particles in motion, with number density $N/V$, mass $m$, and mean square speed ${\overline {c^{2}}}$. He then noted that using the classical laws of Boyle and Charles one could write $m{\overline {c^{2}}}/3=kT$ with $k$ a constant of proportionality. Hence temperature was proportional to the average kinetic energy of the particles. This article inspired further work based on the twin ideas that substances are composed of indivisible particles, and that heat is a consequence of the particle motion; movement that evolves in accordance with Newton's laws. The work, known as the kinetic theory of gases, was done principally by Clausius, James Clerk Maxwell, and Ludwig Boltzmann. At about the same time J. Willard Gibbs also contributed, and advanced it by converting it into statistical mechanics.

Van der Waals equation on a wall in Leiden

This environment influenced Johannes Diderik van der Waals. After initially pursuing a teaching credential, he was accepted for doctoral studies at the University of Leiden under Pieter Rijke. This led, in 1873, to a dissertation that provided a simple, particle based, equation that described the gas–liquid change of state, the origin of a critical temperature, and the concept of corresponding states. The equation is based on two premises, first that fluids are composed of particles with non-zero volumes, and second that at a large enough distance each particle exerts an attractive force on all other particles in its vicinity. These forces were called by Boltzmann van der Waals cohesive forces.

In 1869 Irish professor of chemistry Thomas Andrews at Queen's University Belfast in a paper entitled On the Continuity of the Gaseous and Liquid States of Matter, displayed an experimentally obtained set of isotherms of carbonic acid, H$_{2}$CO$_{3}$, that showed at low temperatures a jump in density at a certain pressure, while at higher temperatures there was no abrupt change; the figure can be seen here. Andrews called the isotherm at which the jump just disappeared the critical point. Given the similarity of the titles of this paper and van der Waals subsequent thesis one might think that van der Waals set out to develop a theoretical explanation of Andrews' experiments; however, this is not what happened. Van der Waals began work by trying to determine a mollecular attraction that appeared in Laplace's theory of capillarity, and only after establishing his equation he tested it using Andrews results.

By 1877 sprays of both liquid oxygen and liquid nitrogen had been produced, and a new field of research, low temperature physics, had been opened. The van der Waals equation played a part in all this especially with respect to the liquefaction of hydrogen and helium which was finally achieved in 1908. From measurements of $p_{1},T_{1}$ and $p_{2},T_{2}$ in two states with the same density, the van der Waals equation produces the values,

$$b=v-{\frac {R(T_{2}-T_{1})}{p_{2}-p_{1}}}\qquad {\mbox{and}}\qquad a=v^{2}{\frac {p_{2}T_{1}-p_{1}T_{2}}{T_{2}-T_{1}}}.$$

Thus from two such measurements of pressure and temperature one could determine $a$ and $b$, and from these values calculate the expected critical pressure, temperature, and molar volume. Goodstein summarized this contribution of the van der Waals equation as follows:

All this labor required considerable faith in the belief that gas–liquid systems were all basically the same, even if no one had ever seen the liquid phase. This faith arose out of the repeated success of the van der Waals theory, which is essentially a universal equation of state, independent of the details of any particular substance once it has been properly scaled. ... As a result, not only was it possible to believe that hydrogen could be liquefied. but it was even possible to predict the necessary temperature and pressure.

Van der Waals was awarded the Nobel Prize in 1910, in recognition of the contribution of his formulation of this "equation of state for gases and liquids".

As noted previously, modern day studies of first order phase changes make use of the van der Waals equation together with the Gibbs criterion, equal chemical potential of each phase, as a model of the phenomenon. This model has an analytic coexistence (saturation) curve expressed parametrically, $p_{s}=f_{p}(y),T_{s}=f_{T}(y)$ (the parameter $y$ is related to the entropy difference between the two phases), that was first obtained by Plank, was known to Gibbs and others, and was later derived in a beautifully simple and elegant manner by Lekner. A summary of Lekner's solution is presented in a subsequent section, and a more complete discussion in the Maxwell construction.

Critical point and corresponding states

Figure 1 shows four isotherms of the van der Waals equation (abbreviated as vdW) on a pressure, molar volume plane. The essential character of these curves is that:

Figure 1: Four isotherms of the van der Waals equation along with the black dash dot spinodal curve and the red dash dot coexistence (saturation) curve plotted using reduced (dimensionless) variables.

1. at some critical temperature, $T=T_{\text{c}}$ the slope is negative, $\partial p/\partial v|_{T}<0$, everywhere except at a single point, the critical point, $p=p_{\text{c}},v=v_{\text{c}}$, where both the slope and curvature are zero, $\partial p/\partial v|_{T}=\partial ^{2}p/\partial v^{2}|_{T}=0;$
2. at higher temperatures the slope of the isotherms is everywhere negative (values of $p,T$ for which the equation has 1 real root for $v$);
3. at lower temperatures there are two points on each isotherm where the slope is zero (values of $p$, $T$ for which the equation has 3 real roots for $v$)

Evaluating the two partial derivatives in 1) using the vdW equation and equating them to zero produces, $v_{\text{c}}=3b,T_{\text{c}}=8a/(27Rb)$, and using these in the equation gives $p_{\text{c}}=a/27b^{2}$.

This calculation can also be done algebraically by noting that the vdW equation can be written as a cubic in $v$, which at the critical point is,

$$p_{\text{c}}v^{3}-(p_{\text{c}}b+RT_{\text{c}})v^{2}+av-ab=0.$$

Moreover, at the critical point all three roots coalesce so it can also be written as

$$(v-v_{\text{c}})^{3}=v^{3}-3v_{\text{c}}v^{2}+3v_{\text{c}}^{2}v-v_{\text{c}}^{3}=0$$

Then dividing the first by $p_{\text{c}}$, and noting that these two cubic equations are the same when all their coefficients are equal gives three equations, $b+RT_{\text{c}}/p_{\text{c}}=3v_{\text{c}}\quad a/p_{\text{c}}=3v_{\text{c}}^{2}\quad ab/p_{\text{c}}=v_{\text{c}}^{3}$, whose solution produces the previous results for $p_{\text{c}},v_{\text{c}},T_{\text{c}}$.

Using these critical values to define reduced properties $p_{r}=p/p_{\text{c}},T_{r}=T/T_{\text{c}},v_{r}=v/v_{\text{c}}$ renders the equation in the dimensionless form used to construct Fig. 1

$$p_{r}={\frac {8T_{r}}{3v_{r}-1}}-{\frac {3}{v_{r}^{2}}}$$

This dimensionless form is a similarity relation; it indicates that all vdW fluids at the same $T_{r}$ will plot on the same curve. It expresses the law of corresponding states which Boltzmann described as follows:

All the constants characterizing the gas have dropped out of this equation. If one bases measurements on the van der Waals units [Boltzmann's name for the reduced quantities here], then he obtains the same equation of state for all gases. ... Only the values of the critical volume, pressure, and temperature depend on the nature of the particular substance; the numbers that express the actual volume, pressure, and temperature as multiples of the critical values satisfy the same equation for all substances. In other words, the same equation relates the reduced volume, reduced pressure, and reduced temperature for all substances.

Obviously such a broad general relation is unlikely to be correct; nevertheless, the fact that one can obtain from it an essentially correct description of actual phenomena is very remarkable.

This "law" is just a special case of dimensional analysis in which an equation containing 6 dimensional quantities, $p,v,T,a,b,R$, and 3 independent dimensions, [p], [v], [T] (independent means that "none of the dimensions of these quantities can be represented as a product of powers of the dimensions of the remaining quantities", and [R]=[pv/T]), must be expressible in terms of 6 − 3 = 3 dimensionless groups. Here $v^{*}=b$ is a characteristic molar volume, $p^{*}=a/b^{2}$ a characteristic pressure, and $T^{*}=a/(Rb)$ a characteristic temperature, and the 3 dimensionless groups are $p/p^{*},v/v^{*},T/T^{*}$. According to dimensional analysis the equation must then have the form $p/p^{*}=\Phi (v/v^{*},T/T^{*})$, a general similarity relation. In his discussion of the vdW equation Sommerfeld also mentioned this point. The reduced properties defined previously are $p_{r}=27(p/p^{*})$, $v_{r}=(1/3)(v/v^{*})$, and $T_{r}=(27/8)(T/T^{*})$. Recent research has suggested that there is a family of equations of state that depend on an additional dimensionless group, and this provides a more exact correlation of properties. Nevertheless, as Boltzmann observed, the van der Waals equation provides an essentially correct description.

The vdW equation produces $Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})=3/8$, while for most real fluids $0.23<Z_{\text{c}}<0.31$. Thus most real fluids do not satisfy this condition, and consequently their behavior is only described qualitatively by the vdW equation. However, the vdW equation of state is a member of a family of state equations based on the Pitzer (acentric) factor, $\omega$, and the liquid metals, Mercury and Cesium, are well approximated by it.

Thermodynamic properties

The properties molar internal energy, $u$, and entropy, $s$, defined by the first and second laws of thermodynamics, hence all thermodynamic properties of a simple compressible substance, can be specified, up to a constant of integration, by two measurable functions. These are a mechanical equation of state, $p=p(v,T)$, and a constant volume specific heat, $c_{v}(v,T)$.

Internal energy and specific heat at constant volume

The internal energy is given by the energetic equation of state,

$$u-C_{u}=\int \,c_{v}(v,T)\,dT+\int \,\left[T{\frac {\partial p}{\partial T}}-p(v,T)\right]\,dv=\int \,c_{v}(v,T)\,dT+\int \,\left[T^{2}{\frac {\partial (p/T)}{\partial T}}\right]\,dv$$

where $C_{u}$ is an arbitrary constant of integration.

Now in order for $du(v,T)$ to be an exact differential, namely that $u(v,T)$ be continuous with continuous partial derivatives, its second mixed partial derivatives must also be equal, $\partial _{v}\partial _{T}u=\partial _{T}\partial _{v}u$. Then with $c_{v}=\partial _{T}u$ this condition can be written simply as $\partial _{v}c(v,T)=\partial _{T}[T^{2}\partial _{T}(p/T)]$. Differentiating $p/T$ for the vdW equation gives $T^{2}\partial _{T}(p/T)]=a/v^{2}$, so $\partial _{v}c_{v}=0$. Consequently $c_{v}=c_{v}(T)$ for a vdW fluid exactly as it is for an ideal gas. To keep things simple it is regarded as a constant here, $c_{v}=cR$ with $c$ a number. Then both integrals can be easily evaluated and the result is

$$u-C_{u}=cRT-a/v$$

This is the energetic equation of state for a perfect vdW fluid. By making a dimensional analysis (what might be called extending the principle of corresponding states to other thermodynamic properties) it can be written simply in reduced form as,

$$u_{r}-{\mbox{C}}_{u}=cT_{r}-9/(8v_{r})$$

where $u_{r}=u/(RT_{\text{c}})$ and ${\mbox{C}}_{u}$ is a dimensionless constant.

Enthalpy

The enthalpy is $h=u+pv$, and the product $pv$ is just $pv=RTv/(v-b)-a/v$. Then

$h$ is simply$$h-C_{u}=RT[c+v/(v-b)]-2a/v$$

This is the enthalpic equation of state for a perfect vdW fluid, or in reduced form,

$$h_{r}-{\mbox{C}}_{u}=[c+3v_{r}/(3v_{r}-1)]T_{r}-9/(4v_{r})\quad {\mbox{where}}\quad h_{r}=h/(RT_{\text{c}}c)$$

Entropy

The entropy is given by the entropic equation of state:

$$s-C_{s}=\int \,c_{v}(T)\,{\frac {dT}{T}}+\int \,{\frac {\partial p}{\partial T}}\,dv$$

Using $c_{v}=cR$ as before, and integrating the second term using $\partial _{T}p=R/(v-b)$ we obtain simply

$$s-C_{s}=R\ln[T^{c}(v-b)]$$

This is the entropic equation of state for a perfect vdW fluid, or in reduced form,

$$s_{r}-{\mbox{C}}_{s}=\ln[T_{r}^{c}(3v_{r}-1)]$$

Helmholtz free energy

The Helmholtz free energy is $f=u-Ts$ so combining the previous results

$$f=C_{u}+cT-a/v-T\{C_{s}+R\ln[T^{c}(v-b)]\}$$

This is the Helmholtz free energy for a perfect vdw fluid, or in reduced form

$$f_{r}={\mbox{C}}_{u}+cT_{r}-9/(8v_{r})-T_{r}\{{\mbox{C}}_{s}+\ln[T_{r}^{c}(3v_{r}-1)]\}$$

Gibbs free energy

The Gibbs free energy is $g=h-Ts$ so combining the previous results gives

$$g-C_{u}=\{c+v/(v-b)-C_{s}-\ln[T^{c}(v-b)]\}RT-2a/v$$

This is the Gibbs free energy for a perfect vdW fluid, or in reduced form

$$g_{r}-{\mbox{C}}_{u}=\{c+3v_{r}/(3v_{r}-1)-{\mbox{C}}_{s}-\ln[T_{r}^{c}(3v_{r}-1)]\}T_{r}-9/(4v_{r})$$

Thermodynamic derivatives: α, κ_T and c_p

The two first partial derivatives of the vdW equation are

$$\left.{\frac {\partial p}{\partial T}}\right)_{v}={\frac {R}{v-b}}={\frac {\alpha }{\kappa _{T}}}\quad {\mbox{and}}\quad \left.{\frac {\partial p}{\partial v}}\right)_{T}=-{\frac {RT}{(v-b)^{2}}}+{\frac {2a}{v^{3}}}=-{\frac {1}{v\kappa _{T}}}$$

Here $\kappa _{T}=-v^{-1}\partial _{p}v$, the isothermal compressibility, is a measure of the relative increase of volume from an increase of pressure, at constant temperature, while $\alpha =v^{-1}\partial _{T}v_{p}$, the coefficient of thermal expansion, is a measure of the relative increase of volume from an increase of temperature, at constant pressure. Therefore,

$$\kappa _{T}={\frac {v^{2}(v-b)^{2}}{RTv^{3}-2a(v-b)^{2}}}\quad {\mbox{and}}\quad \alpha ={\frac {Rv^{2}(v-b)}{RTv^{3}-2a(v-b)^{2}}}$$

In the limit $v\rightarrow \infty$ $\alpha =1/T$ while $\kappa _{T}=v/(RT)$. Since the vdW equation in this limit becomes $p=RT/v$, finally $\kappa _{T}=1/p$. Both of these are the ideal gas values, which is consistent because, as noted earlier, the vdW fluid behaves like an ideal gas in this limit.

The specific heat at constant pressure, $c_{p}$ is defined as the partial derivative $c_{p}=\partial _{T}h|_{p}$. However, it is not independent of $c_{v}$, they are related by the Mayer equation, $c_{p}-c_{v}=-T(\partial _{T}p)^{2}/\partial _{v}p=Tv\alpha ^{2}/\kappa _{T}$. Then the two partials of the vdW equation can be used to express $c_{p}$ as,

$$c_{p}(v,T)-c_{v}(T)={\frac {R^{2}Tv^{3}}{RTv^{3}-2a(v-b)^{2}}}\geq R$$

Here in the limit $v\rightarrow \infty$, $c_{p}-c_{v}=R$, which is also the ideal gas result as expected; however the limit $v\rightarrow b$ gives the same result, which does not agree with experiments on liquids.

In this liquid limit we also find $\alpha =\kappa _{T}=0$, namely that the vdW liquid is incompressible. Moreover, since $\partial _{T}p=-\partial _{T}v/\partial _{p}v=\alpha /\kappa _{T}=\infty$, it is also mechanically incompressible, that is $\kappa _{T}\rightarrow 0$ faster than $\alpha$.

Finally $c_{p},\alpha$, and $\kappa _{T}$ are all infinite on the curve $T=2a(v-b)^{2}/(Rv^{3})=T_{\text{c}}(3v_{r}-1)^{2}/(4v_{r}^{3})$. This curve, called the spinodal curve, is defined by $\kappa _{T}^{-1}=0$, and is discussed at length in the next section.

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.