Van der Waals equation

The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them (both of which depend on the specific substance). As a result the equation is able to model the liquid–vapor phase change; it is the first equation that did this, and consequently it had a substantial impact on physics at that time. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is^[1][2][3]

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

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

The constant ${\displaystyle a}$ expresses the strength of the molecular interactions. It has dimension of pressure times molar volume squared [pv²], which is also molar energy times molar volume. The constant ${\displaystyle b}$ 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 ${\displaystyle \sigma }$ and a minimum energy ${\displaystyle -\varepsilon }$ (with ${\displaystyle \varepsilon \geq 0}$), as shown in the accompanying plot produces ${\displaystyle b=4N_{\text{A}}[(4\pi /3)(\sigma /2)^{3}]}$. Multiplying this by the number of moles, ${\displaystyle N/N_{\text{A}}}$, gives the excluded volume as 4 times the volume of all the molecules.^[4] This theory also produces ${\displaystyle a=IN_{\text{A}}\varepsilon b,}$ where ${\displaystyle I}$ is a number that depends on the shape of the potential function ${\displaystyle \varphi (r)/\varepsilon }$.^[5]

In his book Boltzmann wrote equations using ${\displaystyle V/M}$ (specific volume) in place of ${\displaystyle VN_{\text{A}}/N}$ (molar volume) used here;^[6] Gibbs did as well, so do most engineers. Also the property ${\displaystyle V/N=1/\rho _{N},}$ 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 ${\displaystyle R}$, while those using specific volume contain ${\displaystyle R/{\bar {m}}}$ (where ${\displaystyle {\bar {m}}=N_{\text{A}}m_{p}}$ is the molar mass of a substance whose particle mass is ${\displaystyle m_{p}}$), and those written with number density contain ${\displaystyle k}$.

Once ${\displaystyle a}$ and ${\displaystyle b}$ 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, ${\displaystyle p_{\text{c}}}$, ${\displaystyle T_{\text{c}}}$ such that the substance cannot be liquefied either when ${\displaystyle p>p_{\text{c}}}$ no matter how low the temperature is, or when ${\displaystyle T>T_{\text{c}}}$ no matter how high the pressure is), 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.

The graph on the right is a plot of ${\displaystyle T}$ vs ${\displaystyle v}$ calculated from the equation at four constant pressure values. On the red isobar, ${\displaystyle p=2p_{\text{c}}}$, the slope is positive over the entire range, ${\displaystyle b\leq v<\infty }$ (although the plot only shows a finite quadrant). This describes a fluid as a gas for all ${\displaystyle T}$, and is characteristic of all isobars ${\displaystyle p>p_{\text{c}}.}$ The green isobar, ${\displaystyle p=0.2\,p_{\text{c}}}$, has a physically unreal negative slope, hence shown dotted gray, between its local minimum, ${\displaystyle T_{\text{min}},v_{\text{min}}}$, and local maximum, ${\displaystyle T_{\text{max}},v_{\text{max}}}$. This describes the fluid as two disconnected branches; a gas for ${\displaystyle v\geq v_{\text{min}}}$, and a denser liquid for ${\displaystyle v\leq v_{\text{max}}\leq v_{\text{min}}}$.^[7]

The thermodynamic requirements of mechanical, thermal, and material equilibrium together with the equation specify two points on the curve, ${\displaystyle (T_{s},v_{f})}$, and ${\displaystyle (T_{s},v_{g})}$, shown as green circles that designate the coexisting boiling liquid and condensing gas respectively. Heating the fluid in this state increases the fraction of gas in the mixture; its ${\displaystyle v}$, an average of ${\displaystyle v_{f}}$ and ${\displaystyle v_{g}}$ weighted by this fraction, increases while ${\displaystyle T_{s}}$ 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 ${\displaystyle T_{s}}$, superheated liquid, and those below it, subcooled vapor, are metastable; a sufficiently strong disturbance causes them to transform to the stable alternative (like a ball trapped in a local minimum of a sloping curve that has a lower minimum; the ball has a higher energy than the minimum possible, but can only get there by a push that gets it over the local hill). Consequently they are shown dashed. Finally the points in the region of negative slope are unstable. All this describes a fluid as a stable gas for ${\displaystyle T>T_{\text{s}}}$, a stable liquid for ${\displaystyle T<T_{\text{s}}}$, and a mixture of liquid and gas at ${\displaystyle T=T_{\text{s}}}$, that also supports metastable states of subcooled gas and superheated liquid. It is characteristic of all isobars ${\displaystyle 0<p<p_{\text{c}}}$, where ${\displaystyle T_{\text{s}}}$ is a function of ${\displaystyle p}$.^[8] The orange isobar is the critical one on which the minimum and maximum are equal. 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. All this is a good explanation of the observed behavior of fluids.

The ideal gas law follows from the van der Waals equation whenever ${\displaystyle v}$ is sufficiently large (or correspondingly whenever the molar density, ${\displaystyle \rho =1/v}$, is sufficiently small), Specifically^[9]

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

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

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

which is the ideal gas law.^[9] This is not surprising since the van der Waals equation was constructed from the ideal gas equation in order to obtain an equation valid beyond the limit of ideal gas behavior.

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".^[9] Also in Volume 5 of his Lectures on Theoretical Physics Sommerfeld, in addition to noting that "Boltzmann^[10] described van der Waals as the Newton of real gases",^[11] 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.^[12]

The equation has been, and remains very useful because:^[13]

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

• Its coefficient of thermal expansion, ${\displaystyle \alpha =(\partial _{T}v|_{p})/v}$ has a simple analytic expression [this is also true of its isothermal compressibility, ${\displaystyle \kappa _{T}=-(\partial _{p}v|_{T})/v}$]
• 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 presure 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.^[14] Consequently it can be regarded as one member of a family of equations of state,^[15] that depend on a molecular parameter such as the critical compressibility factor, ${\displaystyle Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})}$, or the Pitzer (acentric) factor, ${\displaystyle \omega =-\log[p_{s}(T/T_{\text{c}}=0.7)/p_{\text{c}}]-1}$, where ${\displaystyle p_{s}/p_{\text{c}}=p_{s}(T/T_{\text{c}},\omega )}$ is a dimensionless saturation pressure, and log is the logarithm base 10.^[16] Consequently, the equation plays an important role in the modern theory of phase transitions.^[17]

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.

In 1857 Rudolf Clausius published The Nature of the Motion which We Call Heat. In it he derived the relation ${\displaystyle p=(N/V)m{\overline {c^{2}}}/3}$ for the pressure, ${\displaystyle p}$, in a gas, composed of particles in motion, with number density ${\displaystyle N/V}$, mass ${\displaystyle m}$, and mean square speed ${\displaystyle {\overline {c^{2}}}}$. He then noted that using the classical laws of Boyle and Charles one could write ${\displaystyle m{\overline {c^{2}}}/3=kT}$ with ${\displaystyle k}$ a constant of proportionality. Hence temperature was proportional to the average kinetic energy of the particles.^[18] 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.^[19]

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.^[20][21] 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.^[22]

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,^[23] displayed an experimentally obtained set of isotherms of carbonic acid, H${\displaystyle _{2}}$CO${\displaystyle _{3}}$, 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.^[24][25]

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.^[26] From measurements of ${\displaystyle p_{1},T_{1}}$ and ${\displaystyle p_{2},T_{2}}$ in two states with the same density, the van der Waals equation produces the values,^[27]

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

Thus from two such measurements of pressure and temperature one could determine ${\displaystyle a}$ and ${\displaystyle b}$, 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:^[28]

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, ${\displaystyle p_{s}=f_{p}(y),T_{s}=f_{T}(y)}$ (the parameter ${\displaystyle y}$ is related to the entropy difference between the two phases), that was first obtained by Plank,^[29] was known to Gibbs and others, and was later derived in a beautifully simple and elegant manner by Lekner.^[30] A summary of Lekner's solution is presented in a subsequent section, and a more complete discussion in the Maxwell construction.

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:

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

Evaluating the two partial derivatives in 1) using the vdW equation and equating them to zero produces, ${\displaystyle v_{\text{c}}=3b,T_{\text{c}}=8a/(27Rb)}$, and using these in the equation gives ${\displaystyle p_{\text{c}}=a/27b^{2}}$.^[31]

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

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

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

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

Then dividing the first by ${\displaystyle p_{\text{c}}}$, and noting that these two cubic equations are the same when all their coefficients are equal gives three equations, ${\displaystyle b+RT_{\text{c}}/p_{\text{c}}=3v_{\text{c}}\quad a/p_{\text{c}}=3v_{\text{c}}^{2}\quad ab/p_{\text{c}}=v_{\text{c}}^{3}}$, whose solution produces the previous results for ${\displaystyle p_{\text{c}},v_{\text{c}},T_{\text{c}}}$.^[32][33]

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

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

This dimensionless form is a similarity relation; it indicates that all vdW fluids at the same ${\displaystyle T_{r}}$ will plot on the same curve. It expresses the law of corresponding states which Boltzmann described as follows:^[34]

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, ${\displaystyle p,v,T,a,b,R}$, 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",^[35] and [R]=[pv/T]), must be expressible in terms of 6 − 3 = 3 dimensionless groups.^[36] Here ${\displaystyle v^{*}=b}$ is a characteristic molar volume, ${\displaystyle p^{*}=a/b^{2}}$ a characteristic pressure, and ${\displaystyle T^{*}=a/(Rb)}$ a characteristic temperature, and the 3 dimensionless groups are ${\displaystyle p/p^{*},v/v^{*},T/T^{*}}$. According to dimensional analysis the equation must then have the form ${\displaystyle p/p^{*}=\Phi (v/v^{*},T/T^{*})}$, a general similarity relation. In his discussion of the vdW equation Sommerfeld also mentioned this point.^[37] The reduced properties defined previously are ${\displaystyle p_{r}=27(p/p^{*})}$, ${\displaystyle v_{r}=(1/3)(v/v^{*})}$, and ${\displaystyle T_{r}=(27/8)(T/T^{*})}$. 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 ${\displaystyle Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})=3/8}$, while for most real fluids ${\displaystyle 0.23<Z_{\text{c}}<0.31}$.^[38] 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, ${\displaystyle \omega }$, and the liquid metals, Mercury and Cesium, are well approximated by it.^[14][39]

The properties molar internal energy, ${\displaystyle u}$, and entropy, ${\displaystyle s}$, 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, a mechanical equation of state, ${\displaystyle p=p(v,T)}$, and a constant volume specific heat, ${\displaystyle c_{v}(v,T)}$.

The internal energy is given by the energetic equation of state,^[40][41]

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

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

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

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

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, ^[42]

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

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

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

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

This is the enthalpic equation of state for a perfect vdW fluid, or in reduced form,^[43]

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

The entropy is given by the entropic equation of state:^[44][41]

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

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

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

This is the entropic equation of state for a perfect vdW fluid, or in reduced form,^[43]

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

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

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

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

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

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

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

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

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

The two first partial derivatives of the vdW equation are

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

Here ${\displaystyle \kappa _{T}=-v^{-1}\partial _{p}v}$, the isothermal compressibility, is a measure of the relative increase of volume from an increase of pressure, at constant temperature, while ${\displaystyle \alpha =v^{-1}\partial _{T}v_{p}}$, the coefficient of thermal expansion, is a measure of the relative increase of volume from an increase of temperature, at constant pressure. Therefore,^[45][43]

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

In the limit ${\displaystyle v\rightarrow \infty }$ ${\displaystyle \alpha =1/T}$ while ${\displaystyle \kappa _{T}=v/(RT)}$. Since the vdW equation in this limit becomes ${\displaystyle p=RT/v}$, finally ${\displaystyle \kappa _{T}=1/p}$. 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, ${\displaystyle c_{p}}$ is defined as the partial derivative ${\displaystyle c_{p}=\partial _{T}h|_{p}}$. However, it is not independent of ${\displaystyle c_{v}}$, they are related by the Mayer equation, ${\displaystyle c_{p}-c_{v}=-T(\partial _{T}p)^{2}/\partial _{v}p=Tv\alpha ^{2}/\kappa _{T}}$.^[46][47][48] Then the two partials of the vdW equation can be used to express ${\displaystyle c_{p}}$ as,^[49]

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

Here in the limit ${\displaystyle v\rightarrow \infty }$, ${\displaystyle c_{p}-c_{v}=R}$, which is also the ideal gas result as expected;^[49] however the limit ${\displaystyle v\rightarrow b}$ gives the same result, which does not agree with experiments on liquids.

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

Finally ${\displaystyle c_{p},\alpha }$, and ${\displaystyle \kappa _{T}}$ are all infinite on the curve ${\displaystyle T=2a(v-b)^{2}/(Rv^{3})=T_{\text{c}}(3v_{r}-1)^{2}/(4v_{r}^{3})}$.^[49] This curve, called the spinodal curve, is defined by ${\displaystyle \kappa _{T}^{-1}=0}$, and is discussed at length in the next section.

According to the extremum principle of thermodynamics ${\displaystyle dS=0}$ and ${\displaystyle d^{2}S<0}$, namely that at equilibrium the entropy is a maximum. This leads to a requirement that ${\displaystyle \partial p/\partial v|_{T}<0}$.^[50] This mathematical criterion expresses a physical condition which Epstein described as follows:^[9]

"It is obvious that this middle part, dotted in our curves [the place where the requirement is violated, dashed gray in Fig. 1 and repeated 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"

While on an isotherm ${\displaystyle T>T_{\text{c}}}$ this requirement is satisfied everywhere so all states are gas, those states on an isotherm, ${\displaystyle T<T_{\text{c}}}$ which lie between the local minimum, ${\displaystyle v_{\rm {min}}}$, and local maximum, ${\displaystyle v_{\rm {max}}}$, for which ${\displaystyle \partial p/\partial v|_{T}>0}$ (shown dashed gray in Fig. 1), are unstable and thus not observed. This is the genesis of the phase change; there is a range ${\displaystyle v_{\rm {min}}\leq v\leq v_{\rm {max}}}$, for which no observable states exist. The states for ${\displaystyle v<v_{\rm {min}}}$ are liquid, and for ${\displaystyle v>v_{\rm {max}}>v_{\rm {min}}}$ are vapor; the denser liquid lies below the vapor due to gravity. The transition points, states with zero slope, are called spinodal points.^[51] Their locus is the spinodal curve that separates the regions of the plane for which liquid, vapor, and gas exist from a region where no observable homogeneous states exist. This spinodal curve is obtained here from the vdW equation by differentiation (or equivalently from ${\displaystyle \kappa _{T}=\infty }$) as

$${\displaystyle T_{\rm {sp}}=2a{\frac {(v-b)^{2}}{Rv^{3}}}=T_{\text{c}}{\frac {(3v_{r}-1)^{2}}{4v_{r}^{3}}}\qquad p_{\rm {sp}}={\frac {a(v-2b)}{v^{3}}}=p_{\text{c}}{\frac {(3v_{r}-2)}{v_{r}^{3}}}}$$

A projection of this space curve is plotted in Fig. 1 as the black dash dot curve. It passes through the critical point which is also a spinodal point.

Although the gap in ${\displaystyle v}$ delimited by the two spinodal points on an isotherm (e.g. ${\displaystyle T_{r}=7/8}$ shown in Fig. 1) is the origin of the phase change, the spinodal points do not represent its full extent, because both states, saturated liquid and saturated vapor coexist in equlilbrium; they both must have the same pressure as well as the same temperature.^[52] Thus the phase change is characterized, at temperature ${\displaystyle T_{s}}$, by a pressure ${\displaystyle p_{\rm {min}}<p_{s}<p_{\rm {max}}}$ that lies between that of the minimum and maximum spinodal points, and with molar volumes of liquid, ${\displaystyle v_{f}<v_{\rm {min}}}$ and vapor ${\displaystyle v_{g}>v_{\rm {max}}}$. Then from the vdW equation applied to these saturated liquid and vapor states

$${\displaystyle p_{s}={\frac {RT_{s}}{v_{f}-b}}-{\frac {a}{v_{f}^{2}}}\quad {\mbox{and}}\quad p_{s}={\frac {RT_{s}}{v_{g}-b}}-{\frac {a}{v_{g}^{2}}}}$$

These two vdW equations contain 4 variables, ${\displaystyle p_{\text{s}},T_{\text{s}},v_{f},v_{g}}$, so another equation is required in order to specify the values of 3 of these variables uniquely in terms of a fourth. Such an equation is provided here by the equality of the Gibbs free energy in the saturated liquid and vapor states, ${\displaystyle g_{f}=g_{g}}$.^[53] This condition of material equilibrium can be obtained from a simple physical argument as follows: the energy required to vaporize a mole is from the second law at constant temperature ${\displaystyle q_{\rm {vap}}=T(s_{g}-s_{f})}$, and from the first law at constant pressure ${\displaystyle q_{\rm {vap}}=h_{g}-h_{f}}$. Equating these two, rearranging, and recalling that ${\displaystyle g=h-Ts}$ produces the result.

The Gibbs free energy is one of the 4 thermodynamic potentials whose partial derivatives produce all other thermodynamics state properties;^[54] its differential is ${\displaystyle dg=\partial _{p}g\,dp+\partial _{T}g\,dT=v\,dp-s\,dT}$. Integrating this over an isotherm from ${\displaystyle p_{s},v_{f}}$ to ${\displaystyle p_{s},v_{g}}$, noting that the pressure is the same at each endpoint, and setting the result to zero yields

$${\displaystyle g_{g}-g_{f}=\sum _{j=1}^{3}\int _{p_{j}}^{p_{j}+1}\,v_{j}\,dp=-\int _{v_{f}}^{v_{g}}\,p\,dv+p_{s}(v_{g}-v_{f})=0}$$

Here because ${\displaystyle v}$ is a multivalued function, the ${\displaystyle p}$ integral must be divided into 3 parts corresponding to the 3 real roots of the vdW equation in the form, ${\displaystyle v(p,T)}$ (this can be visualized most easily by imagining Fig. 1 rotated ${\displaystyle 90^{\circ }}$); the result is a special case of material equilibrium.^[55] The last equality, which follows from integrating ${\displaystyle v\,dp=d(pv)-p\,dv}$, is the Maxwell equal area rule which requires that the upper area between the vdW curve and the horizontal through ${\displaystyle p_{\text{s}}}$ be equal to the lower one.^[56] This form means that the thermodynamic restriction that fixes ${\displaystyle p_{s}}$ is specified by the equation of state itself, ${\displaystyle p=p(v,T)}$. Using the equation for the Gibbs free energy obtained previously for the vdW equation applied to the saturated vapor state and subtracting the result applied to the saturated liquid state produces,

$${\displaystyle RT_{s}\left[{\frac {v_{g}}{v_{g}-b}}-{\frac {v_{f}}{v_{f}-b}}-\ln \left({\frac {v_{g}-b}{v_{f}-b}}\right)\right]-2a\left({\frac {1}{v_{g}}}-{\frac {1}{v_{f}}}\right)=0}$$

This is a third equation that along with the two vdW equations above can be solved numerically. This has been done given a value for either ${\displaystyle T_{s}}$ or ${\displaystyle p_{s}}$, and tabular results presented;^[57][58] however, the equations also admit an analytic parametric solution obtained most simply and elegantly, by Lekner.^[30] Details of this solution may be found in the Maxwell Construction; the results are

$${\displaystyle T_{rs}(y)=\left({\frac {27}{8}}\right){\frac {2f(y)[\cosh y+f(y)]}{g(y)^{2}}}\quad p_{rs}=27{\frac {f(y)^{2}[1-f(y)^{2}]}{g(y)^{2}}}}$$

$${\displaystyle v_{rf}=\left({\frac {1}{3}}\right){\frac {1+f(y)e^{y}}{f(y)e^{y}}}\qquad \qquad \qquad \quad v_{rg}=\left({\frac {1}{3}}\right){\frac {1+f(y)e^{-y}}{f(y)e^{-y}}}}$$ where

$${\displaystyle f(y)={\frac {y\cosh y-\sinh y}{\sinh y\cosh y-y}}\qquad \qquad g(y)=1+2f(y)\cosh y+f(y)^{2}}$$

and the parameter ${\displaystyle 0\leq y<\infty }$ is given physically by ${\displaystyle y=(s_{g}-s_{f})/(2R)}$. The values of all other property discontinuities across the saturation curve also follow from this solution.^[59] These functions define the coexistence curve which is the locus of the saturated liquid and saturated vapor states of the vdW fluid. The curve is plotted in Fig. 1 and Fig. 2, two projections of the state surface. These curves and the numerical results referenced earlier agree exactly, as they must.

Referring back to Fig. 1 the isotherms for ${\displaystyle T_{r}<1}$ are discontinuous. Considering ${\displaystyle T_{r}=7/8}$ as an example, it consists of the two separate green segments. The solid segment above the green circle on the left, and below the one on the right correspond to stable states, the dots represent the saturated liquid and vapor states that comprise the phase change, and the two green dotted segments below and above the dots are metastable states, superheated liquid and subcooled vapor, that are created in the process of phase transition, have a short lifetime, then devolve into their lower energy stable alternative.

In his treatise of 1898 in which he described the van der Waals equation in great detail Boltzmann discussed these states in a section titled "Undercooling, Delayed evaporation";^[60] they are now denoted subcooled vapor, and superheated liquid. Moreover, it has now become clear that these metastable states occur regularly in the phase transition process. In particular processes that involve very high heat fluxes create large numbers of these states, and transition to their stable alternative with a corresponding release of energy can be dangerous. Consequently there is a pressing need to study their thermal properties.^[61]

In the same section Boltzmann also addressed and explained the negative pressures which some liquid metastable states exhibit (for example ${\displaystyle T_{r}=0.8}$ of Fig. 1). He concluded that such liquid states of tensile stresses were real, as did Tien and Lienhard many years later who wrote "The van der Waals equation predicts that at low temperatures liquids sustain enormous tension...In recent years measurements have been made that reveal this to be entirely correct."^[62]

Even though the phase change produces a mathematical discontinuity in the homogeneous fluid properties, for example ${\displaystyle v}$, there is no physical discontinuity.^[55] As the liquid begins to vaporize the fluid becomes a heterogeneous mixture of liquid and vapor whose molar volume varies continuously from ${\displaystyle v_{f}}$ to ${\displaystyle v_{g}}$ according to the equation of state

$${\displaystyle v=v_{f}+x(v_{g}-v_{f})\qquad x=N_{g}/(N_{f}+N_{g})}$$

where ${\displaystyle 0\leq x\leq 1}$ is the mole fraction of the vapor. This equation is called the lever rule and applies to other properties as well.^[12][55] The states it represents form a horizontal line connecting the same colored dots on an isotherm, but not shown in Fig. 1 as noted already since it is a distinct equation of state for the heterogeneous combination of liquid and vapor components.

The idea of corresponding states originated when van der Waals cast his equation in the dimensionless form, ${\displaystyle p_{r}=p(v_{r},T_{r})}$. However, as Boltzmann noted, such a simple representation could not correctly describe all substances. Indeed, the saturation analysis of this form produces ${\displaystyle p_{rs}=p_{s}(T_{r})}$, namely all substances have the same dimensionless coexistence curve.^[63] In order to avoid this paradox an extended principle of corresponding states has been suggested in which ${\displaystyle p_{r}=p(v_{r},T_{r},\phi )}$ where ${\displaystyle \phi }$ is a substance dependent dimensionless parameter related to the only physical feature associated with an individual substance, its critical point.

The most obvious candidate for ${\displaystyle \phi }$ is the critical compressibility factor ${\displaystyle Z_{\text{c}}=p_{\text{c}}v_{\text{c}}/(RT_{\text{c}})}$, but because ${\displaystyle v_{c}}$ is difficult to measure accurately, the acentric factor developed by Kenneth Pitzer,^[16] ${\displaystyle \omega =-{\mbox{log}}_{10}[p_{r}(T_{r}=0.7)]-1}$, is more useful. The saturation pressure in this situation is represented by a one parameter family of curves, ${\displaystyle p_{rs}=p_{s}(T_{r},\omega )}$. Several investigators have produced correlations of saturation data for a number of substances, the best is that of Dong and Lienhard,^[39]

${\displaystyle \ln p_{rs}=5.37270(1-1/T_{r})+\omega (7.49408-}$ $${\displaystyle \qquad 11.181777T_{r}^{3}+3.68769T_{r}^{6}+17.92998\,\ln T_{r})}$$

which has an rms error of ${\displaystyle \pm 0.42}$ over the range ${\displaystyle 1\leq T_{r}\leq 0.3}$

Figure 3 is a plot of ${\displaystyle p_{rs}}$ vs ${\displaystyle T_{r}}$. for various values of ${\displaystyle \omega }$ as given by this equation. The ordinate is logarithmic in order to show the behavior at pressures far below the critical where differences among the various substances (indicated by varying values of ${\displaystyle \omega }$) are more pronounced.

Figure 4 is another plot of the same equation showing ${\displaystyle T_{r}}$ as a function of ${\displaystyle \omega }$ for various values of ${\displaystyle p_{rs}}$. It includes data from 51 substances, including the vdW fluid, over the range ${\displaystyle -0.4<\omega <0.9}$. This plot shows clearly that the vdW fluid (${\displaystyle \omega =-0.302}$) is a member of the class of real fluids; indeed it quantitatively describes the behavior of the liquid metals cesium (${\displaystyle \omega =-0.267}$) and mercury (${\displaystyle \omega =-0.21}$) whose values of ${\displaystyle \omega }$ are close to the vdW value. However, it describes the behavior of other fluids only qualitatively, because specific numerical values are modified by differing values of their Pitzer factor, ${\displaystyle \omega }$.

The Joule–Thomson coefficient, ${\displaystyle \mu _{J}=\partial _{p}T|_{h}}$, is of practical importance because the two end states of a throttling process (${\displaystyle h_{2}=h_{1}}$) lie on a constant enthalpy curve. Although ideal gases, for which ${\displaystyle h=h(T)}$, do not change temperature in such a process, real gases do, and it is important in applications to know whether they heat up or cool down.^[64]

This coefficient can be found in terms of the previously described derivatives as,^[65]

$${\displaystyle \mu _{J}={\frac {v(\alpha T-1)}{c_{p}}}}$$

so when ${\displaystyle \mu _{J}}$ is positive the gas temperature decreases when it passes through a throttle, and if it is negative the temperature increases. Therefore the condition ${\displaystyle \mu _{J}=0}$ defines a curve that separates the region of the ${\displaystyle T,p}$ plane where ${\displaystyle \mu _{J}>0}$ from the region where it is less than zero. This curve is called the inversion curve, and its equation is ${\displaystyle \alpha T-1=0}$. Using the expression for ${\displaystyle \alpha }$ derived previously for the van der Waals equation this is

$${\displaystyle {\frac {2a(v-b)^{2}-RTv^{2}b}{RTv^{3}-2a(v-b)^{2}}}=0\quad {\mbox{or}}\quad 2a(v-b)^{2}-RTv^{2}b=0}$$

Note that for ${\displaystyle v\gg b}$ there will be cooling for ${\displaystyle 2a>RTb}$ or in terms of the critical temperature ${\displaystyle T<27T_{\text{c}}/4}$. As Sommerfeld noted, "This is the case with air and with most other gases. Air can be cooled at will by repeated expansion and can finally be liquified."^[66]

In terms of ${\displaystyle b/v}$ the equation has a simple positive solution ${\displaystyle b/v=1-{\sqrt {RTb/(2a)}}}$ which, for ${\displaystyle b/v=0}$ produces, ${\displaystyle T=2a/(Rb)=27T_{c}/4}$. Using this to eliminate ${\displaystyle v}$ from the vdW equation then gives the inversion curve as

$${\displaystyle {\frac {p}{p^{*}}}=-1+4\,\left({\frac {T}{2T^{*}}}\right)^{1/2}-3\,\left({\frac {T}{2T^{*}}}\right),}$$

where, for simplicity, ${\displaystyle a,b,R}$ have been replaced by ${\displaystyle p^{*},T^{*}}$.

The maximum of this, quadratic, curve occurs, with ${\displaystyle z^{2}=T/(2T^{*})}$, for

$${\displaystyle D_{z}p/p^{*}=4-6(T/2T^{*})^{1/2}=0}$$

which gives ${\displaystyle (T/2T^{*})_{\rm {max}}^{1/2}=2/3}$, or ${\displaystyle T_{\rm {max}}=8T^{*}/9}$, and the corresponding ${\displaystyle p_{\rm {max}}=p^{*}/3}$. The zeros of the curve ${\displaystyle 3z^{2}-4z+1=0}$, are, making use of the quadratic formula, ${\displaystyle z=(4\pm {\sqrt {16-12}}\,)/6}$, or ${\displaystyle z=1/3}$ and ${\displaystyle 1}$ (${\displaystyle T/T^{*}=2/9=0.{\overline {2}}}$ and ${\displaystyle 2}$). In terms of the dimensionless variables, ${\displaystyle T_{r},p_{r}}$ the zeros are at ${\displaystyle T_{r}=3/4}$ and ${\displaystyle 27/4}$, while the maximum is ${\displaystyle p_{r{\rm {max}}}=9}$, and occurs at ${\displaystyle T_{r{\rm {max}}}=3}$. A plot of the curve is shown in green in Fig. 5. Sommerfeld also displays this plot,^[67] together with a curve drawn using experimental data from H₂. The two curves agree qualitatively, but not quantitatively. For example the maximum on these two curves differ by about 40% in both magnitude and location.

Figure 5 shows an overlap between the saturation curve and the inversion curve plotted there. This region is shown enlarged in the right hand graph of the figure. Thus a van der Waals gas can be liquified by passing it through a throttle under the proper conditions; real gases are liquified in this way.

Real gases are characterized by their difference from ideal by writing ${\displaystyle pv=ZRT}$. Here ${\displaystyle Z}$, called the compressibility factor, is expressed either as ${\displaystyle Z(p,T)}$ or ${\displaystyle Z(\rho ,T)}$. In either case

$${\displaystyle \lim _{p\rightarrow 0}Z=1,\quad \lim _{\rho \rightarrow 0}Z=1;}$$

${\displaystyle Z}$ takes the ideal gas value. In the second case ${\displaystyle Z(\rho ,T)=p(\rho ,T)/\rho RT}$,^[68] so for a van der Waals fluid the compressibility factor is simply ${\displaystyle Z=1/(1-b\rho )-a\rho /(RT)}$, or in terms of reduced variables

$${\displaystyle Z={\frac {3}{3-\rho _{r}}}-{\frac {9\rho _{r}}{8T_{r}}}}$$

where ${\displaystyle 0\leq \rho _{r}=1/v_{r}\leq 3}$. At the critical point, ${\displaystyle T_{r}=\rho _{r}=1}$, ${\displaystyle Z=Z_{\text{c}}=3/2-9/8=3/8}$.

In the limit ${\displaystyle \rho \rightarrow 0}$, ${\displaystyle Z=1}$; the fluid behaves like an ideal gas, a point noted several times earlier. The derivative ${\displaystyle \partial _{\rho }Z|_{T}=b[(1-b\rho )^{-2}-a/bRT]}$ is never negative when ${\displaystyle a/bRT=T^{*}/T\leq 1}$, namely when ${\displaystyle T/T^{*}\geq 1}$ (${\displaystyle T_{r}\geq 27/8}$). Alternatively when ${\displaystyle T/T^{*}<1}$ the initial slope is negative, it becomes zero at ${\displaystyle b\rho =1-(T/T^{*})^{1/2}}$, and is positive for larger ${\displaystyle b\rho \leq 1}$ (see Fig. 6). In this case the value of ${\displaystyle Z}$ passes through ${\displaystyle 1}$ when ${\displaystyle b\rho _{B}=1-T_{B}/T^{*}}$. Here ${\displaystyle T_{B}=(27T_{\text{c}}/8)(1-b\rho _{B})}$ is called the Boyle temperature. It varies between ${\displaystyle 27T_{\text{c}}/8\geq T_{B}\geq 0}$, and denotes a point in ${\displaystyle T,\rho }$ space where the equation of state reduces to the ideal gas law. However the fluid does not behave like an ideal gas there, because neither its derivatives ${\displaystyle (\alpha ,\kappa _{T}){\mbox{ nor }}c_{p}}$ reduce to their ideal gas values, other than where ${\displaystyle b\rho _{B}\ll 1,\,T_{B}\sim 27T_{\text{c}}/8}$ the actual ideal gas region.^[69]

Figure 6 shows a plot of various isotherms of ${\displaystyle Z(\rho ,T_{r})}$ vs ${\displaystyle \rho _{r}}$. Also shown are the spinodal and coexistence curves described previously. The subcritical isotherm consists of stable, metastable, and unstable segments, and are identified the same as they were in Fig. 1. Also included are the zero initial slope isotherm and the one corresponding to infinite temperature.

By plotting ${\displaystyle Z(\rho _{r},T_{r})}$ vs ${\displaystyle p_{r}(\rho _{r},T_{r})}$ using ${\displaystyle \rho _{r}}$ as a parameter, one obtains the generalized compressibility chart for a vdW gas, which is shown in Fig. 7. Like all other vdW properties, this is not quantitatively correct for most gases but it has the correct qualitative features as can be seen by comparison with this figure which was produced from data using real gases.^[70][71] The two graphs are similar, including the caustic generated by the crossing isotherms; they are qualitatively very much alike.

Statistical mechanics suggests that ${\displaystyle Z}$ can be expressed by a power series called a virial expansion,^[72]

$${\displaystyle Z(\rho ,T)=1+\sum _{k=2}^{\infty }\,B_{k}(T)(\rho )^{k-1}}$$

The functions ${\displaystyle B_{k}(T)}$ are the virial coefficients; the ${\displaystyle k}$th term represents a ${\displaystyle k}$ particle interaction.

Expanding the term ${\displaystyle (1-b\rho )^{-1}}$ in the compressibility factor of the vdW equation in its infinite series, convergent for ${\displaystyle b\rho <1}$, produces

$${\displaystyle Z(\rho ,T)=1+[1-a/(bRT)]b\rho +\sum _{k=3}^{\infty }\,B_{k}(T)(b\rho )^{k-1}.}$$

The corresponding expression for ${\displaystyle Z(\rho _{r},T_{r})}$ when ${\displaystyle \rho _{r}<3}$ is

$${\displaystyle Z(\rho _{r},T_{r})=1+[1-27/(8T_{r})]^{-1}](\rho _{r}/3)+\sum _{k=2}^{\infty }\,(\rho _{r}/3)^{k-1}.}$$

These are the virial expansions, one dimensional and one dimensionless, for the van der Waals fluid. The second virial coefficient is the slope of ${\displaystyle Z(\rho _{r},T_{r})}$ at ${\displaystyle \rho _{r}=0}$. Notice that it can be positive or negative depending on whether or not ${\displaystyle T_{r}>{\mbox{ or}}<27/8}$, which agrees with the result found previously by differentiation.

For molecules that are non attracting hard spheres, ${\displaystyle a=0}$, the vdW virial expansion becomes simply

$${\displaystyle Z(\rho )=(1-b\rho )^{-1}=1+\sum _{k=2}^{\infty }(b\rho )^{k-1},}$$

which illustrates the effect of the excluded volume alone. It was recognized early on that this was in error beginning with the term ${\displaystyle (b\rho )^{2}}$. Boltzmann calculated its correct value as ${\displaystyle (5/8)(b\rho )^{2}}$, and used the result to propose an enhanced version of the vdW equation

$${\displaystyle (p+a/v^{2})(v-b/3)=RT[1+2b/(3v)+7b^{2}/(24v^{2})].}$$

On expanding ${\displaystyle (v-b/3)^{-1}}$, this produced the correct coefficients thru ${\displaystyle (b/v)^{2}}$ and also gave infinite pressure at ${\displaystyle b/3}$, which is approximately the close packing distance for hard spheres.^[73] This was one of the first of many equations of state proposed over the years that attempted to make quantitative improvements to the remarkably accurate explanations of real gas behavior produced by the vdW equation.^[74]

In 1890 van der Waals published an article that initiated the study of fluid mixtures. It was subsequently included as Part III of a later published version of his thesis.^[75] His essential idea was that in a binary mixture of vdw fluids described by the equations

$${\displaystyle p_{1}={\frac {RT}{v-b_{11}}}-{\frac {a_{11}}{v^{2}}}\quad {\mbox{and}}\quad p_{2}={\frac {RT}{v-b_{22}}}-{\frac {a_{22}}{v^{2}}}}$$

the mixture is also a vdW fluid given by

$${\displaystyle p={\frac {RT}{v-b_{x}}}-{\frac {a_{x}}{v^{2}}}}$$ where $${\displaystyle a_{x}=a_{11}x_{1}^{2}+2a_{12}x_{1}x_{2}+a_{22}x_{2}^{2}\quad {\mbox{and}}\quad b_{x}=b_{11}x_{1}^{2}+2b_{12}x_{1}x_{2}+b_{22}x_{2}^{2}}$$

Here ${\displaystyle x_{1}=N_{1}/N}$, and ${\displaystyle x_{2}=N_{2}/N}$, with ${\displaystyle N=N_{1}+N_{2}}$ (so that ${\displaystyle x_{1}+x_{2}=1}$) are the mole fractions of the two fluid substances. Adding the equations for the two fluids shows that ${\displaystyle p\neq p_{1}+p_{2}}$, although for ${\displaystyle v}$ sufficiently large ${\displaystyle p\approx p_{1}+p_{2}}$ with equality holding in the ideal gas limit. The quadratic forms for ${\displaystyle a_{x}}$ and ${\displaystyle b_{x}}$ are a consequence of the forces between molecules. This was first shown by Lorentz,^[76] and was credited to him by van der Waals. The quantities ${\displaystyle a_{11},\,a_{22}}$ and ${\displaystyle b_{11},\,b_{22}}$ in these expressions characterize collisions between two molecules of the same fluid component while ${\displaystyle a_{12}=a_{21}}$ and ${\displaystyle b_{12}=b_{21}}$ represent collisions between one molecule of each of the two different component fluids. This idea of van der Waals was later called a one fluid model of mixture behavior.^[77]

Assuming that ${\displaystyle b_{12}}$ is the arithmetic mean of ${\displaystyle b_{11}}$ and ${\displaystyle b_{22}}$, ${\displaystyle b_{12}=(b_{11}+b_{22})/2}$, substituting into the quadratic form, and noting that ${\displaystyle x_{1}+x_{2}=1}$ produces

$${\displaystyle b=b_{11}x_{1}+b_{22}x_{2}}$$

Van der Waals wrote this relation, but did not make use of it initially.^[78] However, it has been used frequently in subsequent studies, and its use is said to produce good agreement with experimental results at high pressure.^[79]

In this article van der Waals used the Helmholtz Potential Minimum Principle to establish the conditions of stability. This principle states that in a system in diathermal contact with a heat reservoir ${\displaystyle T=T_{R}}$, ${\displaystyle DF=0}$ and ${\displaystyle D^{2}F>0}$, namely at equilibrium the Helmholtz potential is a minimimum.^[80] Since, like ${\displaystyle g(p,T)}$, the molar Helmholtz function ${\displaystyle f(v,T)}$ is also a potential function whose differential is

$${\displaystyle df=\partial _{v}f|_{T}\,dv+\partial _{T}f|_{v}\,dT=-p\,dv-s\,dT,}$$

this minimum principle leads to the stability condition ${\displaystyle \partial ^{2}f/\partial v^{2}|_{T}=-\partial p/\partial v|_{T}>0}$. This condition means that the function, ${\displaystyle f}$, is convex at all stable states of the system. Moreover, for those states the previous stability condition for the pressure is necessarily satisfied as well.

For a single substance the definition of the molar Gibbs free energy can be written in the form ${\displaystyle f=g-pv}$. Thus when ${\displaystyle p}$ and ${\displaystyle g}$ are constant along with temperature the function ${\displaystyle f(T_{R},v)}$ represents a straight line with slope ${\displaystyle -p}$, and intercept ${\displaystyle g}$. Since the curve, ${\displaystyle f(T_{R},v)}$, has positive curvature everywhere when ${\displaystyle T_{R}\geq T_{c}}$, the curve and the straight line will be have a single tangent. However, for a subcritical ${\displaystyle T_{R},\,f(T_{R},v)}$ is not everwhere convex. With ${\displaystyle p=p_{s}(T_{R})}$ and a suitable value of ${\displaystyle g}$ the line will be tangent to ${\displaystyle f(T_{R},v)}$ at the molar volume of each coexisting phase, saturated liquid, ${\displaystyle v_{f}(T_{R})}$, and saturated vapor, ${\displaystyle v_{g}(T_{R})}$; there will be a double tangent. Furthermore, each of these points is characterized by the same value of ${\displaystyle g}$ as well as the same values of ${\displaystyle p}$ and ${\displaystyle T_{R}.}$ These are the same three specifications for coexistence that were used previously.

As depicted in Fig. 8, the region on the green curve ${\displaystyle f(T_{R},v)}$ for ${\displaystyle v\leq v_{f}}$ (${\displaystyle v_{f}}$ is designated by the left green circle) is the liquid. As ${\displaystyle v}$ increases past ${\displaystyle v_{f}}$ the curvature of ${\displaystyle f}$ (proportional to ${\displaystyle \partial _{v}\partial _{v}f=-\partial _{v}p}$) continually decreases. The point characterized by ${\displaystyle \partial _{v}\partial _{v}f=-\partial _{v}p=0}$, is a spinodal point, and between these two points is the metastable superheated liquid. For further increases in ${\displaystyle v}$ the curvature decreases to a minimum then increases to another spinodal point; between these two spinodal points is the unstable region in which the fluid cannot exist in a homogeneous equilibrium state. With a further increase in ${\displaystyle v}$ the curvature increases to a maximum at ${\displaystyle v_{g}}$, where the slope is ${\displaystyle p_{s}}$; the region between this point and the second spinodal point is the metastable subcooled vapor. Finally, the region ${\displaystyle v\geq v_{g}}$ is the vapor. In this region the curvature continually decreases until it is zero at infinitely large ${\displaystyle v}$. The double tangent line is rendered solid between its saturated liquid and vapor values to indicate that states on it are stable, as opposed to the metastable and unstable states, above it (with larger Helmholtz free energy), but black, not green, to indicate that these states are heterogeneous, not homogeneous solutions of the vdW equation.^[81] The combined green black curve in Fig. 8 is the convex envelope of ${\displaystyle f(T_{R},v)}$, which is defined as the largest convex curve that is less than or equal to the function.^[82]

For a vdW fluid the molar Helmholtz potential is

$${\displaystyle f_{r}=u_{r}-T_{r}s_{r}={\mbox{C}}_{u}+T_{r}\{c-{\mbox{C}}_{s}-\ln[T_{r}^{c}(3v_{r}-1)]\}-9/(8v_{r})}$$ where ${\displaystyle f_{r}=f/(RT_{\text{c}})}$. Its derivative is $${\displaystyle \partial _{v_{r}}f_{r}=-3T_{r}/(3v_{r}-1)+9/(8v_{r})^{2}=-p_{r}}$$

which is the vdW equation, as it must be. A plot of this function ${\displaystyle f_{r}}$, whose slope at each point is specified by the vdW equation, for the subcritical isotherm ${\displaystyle T_{r}=7/8}$ is shown in Fig. 8 along with the line tangent to it at its two coexisting saturation points. The data illustrated in Fig. 8 is exactly the same as that shown in Fig.1 for this isotherm. This double tangent construction thus provides a simple graphical aternative to the Maxwell construction to establish the saturated liquid and vapor points on an isotherm.

Van der Waals used the Helmholtz function because its properties could be easily extended to the binary fluid situation. In a binary mixture of vdW fluids the Helmholtz potential is a function of 2 variables, ${\displaystyle f(T_{R},v,x)}$, where ${\displaystyle x}$ is a composition variable, for example ${\displaystyle x=x_{2}}$ so ${\displaystyle x_{1}=1-x}$. In this case there are three stability conditions

$${\displaystyle {\frac {\partial ^{2}f}{\partial v^{2}}}>0\qquad {\frac {\partial ^{2}f}{\partial x^{2}}}>0\qquad {\frac {\partial ^{2}f}{\partial v^{2}}}{\frac {\partial ^{2}f}{\partial x^{2}}}-\left({\frac {\partial ^{2}f}{\partial x\partial v}}\right)^{2}>0}$$

and the Helmholtz potential is a surface (of physical interest in the region ${\displaystyle 0\leq x\leq 1}$). The first two stability conditions show that the curvature in each of the directions ${\displaystyle v}$ and ${\displaystyle x}$ are both non negative for stable states while the third condition indicates that stable states correspond to elliptic points on this surface.^[83] Moreover its limit,

$${\displaystyle {\frac {\partial ^{2}f}{\partial v^{2}}}{\frac {\partial ^{2}f}{\partial x^{2}}}-{\frac {\partial ^{2}f}{\partial x\partial v}}=0,}$$ specifies the spinodal curves on the surface.

For a binary mixture the Euler equation,^[84] can be written in the form

$${\displaystyle f=-pv+\mu _{1}x_{1}+\mu _{2}x_{2}=-pv+(\mu _{2}-\mu _{1})x+\mu _{1}}$$

Here ${\displaystyle \mu _{j}=\partial _{x_{j}}f}$ are the molar chemical potentials of each substance, ${\displaystyle j=1,2}$. For ${\displaystyle p}$, ${\displaystyle \mu _{1}}$ and ${\displaystyle \mu _{2}}$, all constant this is the equation of a plane with slopes ${\displaystyle -p}$ in the ${\displaystyle v}$ direction, ${\displaystyle \mu _{2}-\mu _{1}}$ in the ${\displaystyle x}$ direction, and intercept ${\displaystyle \mu _{1}}$. As in the case of a single substance, here the plane and the surface can have a double tangent and the locus of the coexisting phase points forms a curve on each surface. The coexistence conditions are that the two phases have the same ${\displaystyle T}$, ${\displaystyle p}$, ${\displaystyle \mu _{2}-\mu _{1}}$, and ${\displaystyle \mu _{1}}$; the last two are equivalent to having the same ${\displaystyle \mu _{1}}$ and ${\displaystyle \mu _{2}}$ individually, which are just the Gibbs conditions for material equilibrium in this situation. The two methods of producing the coexistence surface are equivalent

Although this case is similar to the previous one of a single component, here the geometry can be much more complex. The surface can develop a wave (called a plait or fold in the literature) in the ${\displaystyle x}$ direction as well as the one in the ${\displaystyle v}$ direction. Therefore, there can be two liquid phases that can be either miscible, or wholly or partially immiscible, as well as a vapor phase.^[85][86] Despite a great deal of both theoretical and experimental work on this problem by van der Waals and his successors, work which produced much useful knowledge about the various types of phase equilibria that are possible in fluid mixtures,^[87] complete solutions to the problem were only obtained after 1967, when the availability of modern computers made calculations of mathematical problems of this complexity feasible for the first time.^[88] The results obtained were, in Rowlinson's words,^[89]

a spectacular vindication of the essential physical correctness of the ideas behind the van der Waals equation, for almost every kind of critical behavior found in practice can be reproduced by the calculations, and the range of parameters that correlate with the different kinds of behavior are intelligible in terms of the expected effects of size and energy.

In order to obtain these numerical results the values of the constants of the individual component fluids ${\displaystyle a_{11},a_{22},b_{11},b_{22}}$ must be known. In addition, the effect of collisions between molecules of the different components, given by ${\displaystyle a_{12}}$ and ${\displaystyle b_{12}}$, must also be specified. In he absence of experimental data, or computer modelling results to estimate their value the empirical combining rules,

$${\displaystyle a_{12}=(a_{11}a_{22})^{1/2}\qquad {\mbox{and}}\qquad b_{12}^{1/3}=(b_{11}^{1/3}+b_{22}^{1/3})/2,}$$

the geometric and algebraic means respectively can be used.^[90] These relations correspond to the empirical combining rules for the intermolecular force constants,

$${\displaystyle \epsilon _{12}=(\epsilon _{11}\epsilon _{22})^{1/2}\qquad {\mbox{and}}\qquad \sigma _{12}=(\sigma _{11}+\sigma _{22})/2,}$$

the first of which follows from a simple interpretation of the dispersion forces in terms of polarizabilities of the individual molecules while the second is exact for rigid molecules.^[91] Then, generalizing for ${\displaystyle n}$ fluid components, and using these empirical combinig laws, the quadradic mixing rules for the material constants are:^[79]

$${\displaystyle a_{x}=\sum _{i=1}^{n}\sum _{j=1}^{n}(a_{ii}a_{jj})^{1/2}x_{i}x_{j}\quad {\mbox{which can be written as}}\quad a_{x}={\Big (}\sum _{i=1}^{n}a_{ii}^{1/2}x_{i}{\Big )}^{2}}$$ $${\displaystyle b_{x}=(1/8)\sum _{i=1}^{n}\sum _{j=1}^{n}(b_{ii}^{1/3}+b_{jj}^{1/3})^{3}x_{i}x_{j}}$$

Using similar expressions in the vdW equation is apparently helpful for divers.^[92] They are also important for physical scientists, and engineers in their study and management of the various phase equilibria and critical behavior observed in fluid mixtures. However more sophisticated mixing rules have often been found to be necessary, in order to obtain satisfactory agreement with reality over the wide variety of mixtures encountered in practice.^[93][94]

Another method of specifying the vdW constants pioneered by W.B. Kay, and known as Kay's rule. ^[95] specifies the effective critical temperature and pressure of the fluid mixture by

$${\displaystyle T_{{\text{c}}x}=\sum _{i=1}^{n}T_{{\text{c}}i}x_{i}\qquad {\mbox{and}}\qquad p_{{\text{c}}x}=\sum _{i=1}^{n}\,p_{{\text{c}}i}x_{i}}$$

In terms of these quantities the vdW mixture constants are then,

$${\displaystyle a_{x}=\left({\frac {3}{4}}\right)^{3}{\frac {(RT_{{\text{c}}x})^{2}}{p_{{\text{c}}x}}}\qquad \qquad b_{x}=\left({\frac {1}{2}}\right)^{3}{\frac {RT_{{\text{c}}x}}{p_{{\text{c}}x}}}}$$

and Kay used these specifications of the mixture critical constants as the basis for calculations of the thermodynamic properties of mixtures.^[96]

Kay's idea was adopted by T. W. Leland, who applied it to the molecular parameters, ${\displaystyle \epsilon ,\sigma }$, which are related to ${\displaystyle a,b}$ through ${\displaystyle T_{\text{c}},p_{\text{c}}}$ by ${\displaystyle a\propto \epsilon \sigma ^{3}}$ and ${\displaystyle b\propto \sigma ^{3}}$. Using these together with the quadratic mixing rules for ${\displaystyle a,b}$ produces

$${\displaystyle \sigma _{x}^{3}=\sum _{i=i}^{n}\sum _{j=1}^{n}\,\sigma _{ij}^{3}x_{i}x_{j}\qquad {\mbox{and}}\qquad \epsilon _{x}=\left[\sum _{i=1}^{n}\sum _{j=1}^{n}\epsilon _{ij}\sigma _{ij}^{3}x_{i}x_{j}\right]\left[\sum _{i=i}^{n}\sum _{j=1}^{n}\,\sigma _{ij}^{3}x_{i}x_{j}\right]^{-1}}$$

which is the van der Waals approximation expressed in terms of the intermolecular constants.^{[97] [98]} This approximation, when compared with computer simulations for mixtures, are in good agreement over the range ${\displaystyle 1/2<(\sigma _{11}/\sigma _{22})^{3}<2}$, namely for molecules of not too different diameters. In fact Rowlinson said of this approximation, "It was, and indeed still is, hard to improve on the original van der Waals recipe when expressed in [this] form".^[99]

Since van der Waals presented his thesis, "[m]any derivations, pseudo-derivations, and plausibility arguments have been given" for it.^[100] However, no mathematically rigorous derivation of the equation over its entire range of molar volume that begins from a statistical mechanical principle exists. Indeed, such a proof is not possible, even for hard spheres.^{[101][102][103]} Goodstein put it this way, "Obviously the value of the van der Waals equation rests principally on its empirical behavior rather than its theoretical foundation."^[104]

Nevertheless a review of the work that has been done is useful in order to better understand where and when the equation is valid mathematically, and where and why it fails.

The classical canonical partition function, ${\displaystyle Q(V,T,N)}$, of statistical mechanics for a three dimensional ${\displaystyle N}$ particle macroscopic system is, $${\displaystyle Q_{N}=\Lambda ^{-3N}N!^{-1}{\cal {Z}}_{N}\qquad {\mbox{with}}\qquad {\cal {Z}}_{N}=\int _{V^{\bf {N}}}\,\exp[-\Phi ({\bf {r^{N}}})/kT]\,d{\bf {r^{N}}}}$$ Here ${\displaystyle Q_{N}(V,T)\equiv Q(V,T,N)}$, ${\displaystyle \Lambda =h(2\pi mkT)^{-1/2}}$ is the DeBroglie wavelength (alternatively ${\displaystyle \Lambda ^{-3}\equiv n_{Q}}$ is the quantum concentration), ${\displaystyle {\cal {Z}}_{N}(V,T)\equiv {\cal {Z}}(V,T,N)}$ is the ${\displaystyle N}$ particle configuration integral, and ${\displaystyle \Phi }$ is the intermolecular potential energy, which is a function of the ${\displaystyle N}$ particle position vectors ${\displaystyle {\bf {r^{N}}}={\bf {r}}_{1},{\bf {r}}_{2}\ldots {\bf {r}}_{N}}$. Lastly ${\displaystyle d{\bf {r^{N}}}=d{\bf {r}}_{1}d{\bf {r}}_{2}\ldots d{\bf {r}}_{N}}$ is the volume element of ${\displaystyle V^{\bf {N}}}$, which is a ${\displaystyle 3N}$ dimensional space.^{[105][106][107][108]}

The connection of ${\displaystyle Q_{N}}$ with thermodynamics is made through the Helmholtz free energy, ${\displaystyle F=-kT\ln Q_{N}}$ from which all other properties can be found; in particular ${\displaystyle p=-\partial _{V}F|_{T}=kT\partial _{V}\ln {\cal {{Z}_{N}}}}$. For point particles that have no force interactions, ${\displaystyle \Phi =0}$, all ${\displaystyle 3N}$ integrals of ${\displaystyle {\cal {Z}}}$ can be evaluated producing ${\displaystyle Q_{N}=(n_{Q}V)^{N}/N!}$. In the thermodynamic limit, ${\displaystyle N\rightarrow \infty ,\,V\rightarrow \infty }$ with ${\displaystyle V/N}$ finite, the Helmholtz free energy per particle (or per mole, or per unit mass) is finite, for example per mole it is ${\displaystyle N_{\text{A}}F/N=f=-RT[\ln(n_{Q}v/N_{\text{A}})+1]}$. The thermodynamic state equations in this case are those of a monatomic ideal gas, specifically ${\displaystyle -\partial _{v}f|_{T}=p=RT/v.}$^[109]

Early derivations of the vdW equation were criticized mainly on two grounds;^[110] 1) a rigorous derivation from the partition function should produce an equation that does not include unstable states for which, ${\displaystyle \partial _{v}p|_{T}>0}$; 2) the constant ${\displaystyle b=4N_{\text{A}}V_{0}}$ in the vdw equation (here ${\displaystyle V_{0}}$ is the volume of a single molecule) gives the maximum possible number of molecules as ${\displaystyle 4N_{\text{max}}V_{0}=V}$, or a close packing density of 1/4=0.25, whereas the known close packing density of spheres is ${\displaystyle \pi /(3{\sqrt {2}})\approx 0.740}$.^[111] Thus a single value of ${\displaystyle b}$ cannot describe both gas and liquid states.

The second criticism is an indication that the vdW equation cannot be valid over the entire range of molar volume. Van der Waals was well aware of this problem; he devoted about 30% of his Nobel lecture to it, and also said that it is^[112]

... the weak point in the study of the equation of state. I still wonder whether there is a better way. In fact this question continually obsesses me, I can never free myself from it, it is with me even in my dreams.

In 1949 the first criticism was proved by van Hove when he showed that in the thermodynamic limit hard spheres with finite range attractive forces have a finite Helmholtz free energy per particle. Furthermore this free energy is a continuously decreasing function of the volume per particle, (see Fig. 8 where ${\displaystyle f,v}$ are molar quantities). In addition its derivative exists and defines the pressure, which is a non increasing function of the volume per particle.^[113] Since the vdW equation has states for which the pressure increases with increasing volume per particle, this proof means it cannot be derived from the partition function, without an additional constraint that precludes those states.

In 1891 Korteweg showed using kinetic theory ideas,^[114] that a system of ${\displaystyle N}$ hard rods of length ${\displaystyle \delta }$, constrained to move along a straight line of length ${\displaystyle L>N\delta ,}$, and exerting only direct contact forces on one another satisfy a vdW equation with ${\displaystyle a=0}$; Rayleigh also knew this.^[115] Later Tonks, by evaluating the configuration integral,^[116] showed that the force exerted on a wall by this system is given by, ${\displaystyle F_{W}=NkT/(L-N\delta )=kT/(l-\delta )\,{\mbox{with}}\,l=L/N.}$ This can be put in a more recognizable, molar, form by dividing by the rod cross sectional area ${\displaystyle A}$, and defining ${\displaystyle N_{\text{A}}AL/N=v>b=N_{\text{A}}A\delta }$. This produces ${\displaystyle p=RT/(v-b)}$; clearly there is no condensation, ${\displaystyle \partial _{v}p|_{T}<0}$ for all ${\displaystyle b\leq v<\infty }$. This simple result is obtained because in one dimension particles cannot pass by one another as they can in higher dimensions; their mass center coordinates, ${\displaystyle x_{i},\,i=1,N}$ satisfy the relations ${\displaystyle \delta /2\leq x_{1}\leq x_{2}-\delta ,\cdots ,\leq x_{N}\leq L-\delta /2}$. As a result the configuration integral is simply ${\displaystyle {\cal {Z}}_{N}=(L-N\delta )^{N}}$.^[117]

In 1959 this one-dimensional gas model was extended by Kac to include particle pair interactions through an attractive potential, ${\displaystyle \varphi (x)=-\varepsilon \exp(-x/\ell ),\,x\geq \delta }$. This specific form allowed evaluation of the grand partition function,

$${\displaystyle Q_{G}(T,V,\mu )=\sum _{{\cal {N}}\geq 0}\,Q_{\cal {N}}(T,V)e^{\mu {\cal {N}}/kT},}$$

in the thermodynamic limit, in terms of the eigenfunctions and eigenvalues of a homogeneous integral equation.^[118] Although an explicit equation of state was not obtained, it was proved that the pressure was a strictly decreasing function of the volume per particle, hence condensation did not occur.

Four years later, in 1963, Kac together with Uhlenbeck and Hemmer modified the pair potential of Kac's previous work as ${\displaystyle \varphi (x)=-\varepsilon (\delta /\ell )\exp(-x/\ell ),\,x\geq \delta }$, so that

$${\displaystyle a_{N}=-\int _{0}^{\infty }\,\varphi (x)\,dx=\varepsilon \delta }$$

was independent of ${\displaystyle \ell }$.^[119] They found, that a second limiting process they called the van der Waals limit, ${\displaystyle \ell \rightarrow \infty }$ (in which the pair potential becomes both infinitely long range and infinity weak) and performed after the thermodynamic limit, produced the one-dimensional vdW equation (here rendered in molar form)

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

in which ${\displaystyle a=N_{\text{A}}^{2}Aa_{N}=N_{\text{A}}\varepsilon b}$ and ${\displaystyle b=N_{\text{A}}V_{0}=N_{\text{A}}A\delta }$, together with the Gibbs criterion, ${\displaystyle \mu _{f}=\mu _{g}}$ (equivalently the Maxwell construction). As a result all isotherms satisfy the condition ${\displaystyle \partial _{v}p|_{T}\leq 0}$ as shown in Fig. 9, and hence the first criticism of the vdW equation is not as serious as originally thought.^[120]

Then, in 1966, Lebowitz and Penrose generalized what they called the Kac potential to apply to a non specific function in an arbitrary number, ${\displaystyle \nu }$, of dimensions, ${\displaystyle \varphi =-\varepsilon (\sigma /\ell )^{\nu }{\bar {\varphi }}(x/\ell )}$. For ${\displaystyle \nu =1}$ and ${\displaystyle {\bar {\varphi }}=\exp(-x/\ell )}$ this reduces to the specific one-dimensional function considered by Kac, et. al. and for ${\displaystyle \nu =3}$ it is an arbitrary function (although subject to specific requirements) in physical three dimensional space. In fact the function ${\displaystyle {\bar {\varphi }}(x/\ell )}$ must be bounded, non-negative, and one whose integral

$${\displaystyle a_{N}=-{\frac {1}{2}}\int _{V^{\nu }}\,\varphi \left({\frac {r}{\ell }}\right)\,d{\bf {r}}^{\nu }={\frac {\varepsilon \sigma ^{\nu }}{2}}\int _{V^{\nu }}\,{\bar {\varphi }}(x)\,d{\bf {x}}^{\nu }}$$

is finite, independent of ${\displaystyle \ell }$.^[121][122] By obtaining upper and lower bounds on ${\displaystyle {\cal {Z}}(V,T,N,\ell )}$ and hence on ${\displaystyle F}$, taking the thermodynamic limit (${\displaystyle \lim N,V\rightarrow \infty {\mbox{ with }}N/V{\mbox{ finite}}}$) to obtain upper and lower bounds on the function ${\displaystyle F(V,T,N/V,\ell )/V}$, then subsequently taking the van der Waals limit, they found that the two bounds coalesced and thereby produced a unique limit, here written in terms of the free energy per mole and the molar volume,

$${\displaystyle f(v,T)={\mbox{CE}}\{f^{0}(v,T)-a/v\}.}$$

The abbreviation CE stands for convex envelope; this is a function which is the largest convex function that is less than or equal to the original function. The function ${\displaystyle f^{0}(v,T)}$ is the limit function when ${\displaystyle \varphi =0}$; also here ${\displaystyle a=N_{A}^{2}a_{N}}$. This result is illustrated in the present context by the solid green curves and black line in Fig. 8, which is the convex envelope of ${\displaystyle f(T_{R},v)}$ also shown there.

The corresponding limit for the pressure is a generalized form of the vdW equation

$${\displaystyle p(v,T)=p^{0}(v,T)-a/v^{2}}$$

together with the Gibbs criterion, ${\displaystyle \mu _{f}=\mu _{g}}$ (equivalently the Maxwell construction). Here ${\displaystyle p^{0}=-\partial _{v}f^{0}|_{T}}$ is the pressure when attractive molecular forces are absent.

The conclusion from all this work is that a rigorous mathematical derivation from the partition function produces a generalization of the vdW equation together with the Gibbs criterion if the attractive force is infinitely weak with an infinitely long range. In that case ${\displaystyle p^{0},}$ the pressure that results from direct particle collisions (or more accurately the core repulsive forces), replaces ${\displaystyle RT/(v-b)}$. This is consistent with the second criticism that can be stated as ${\displaystyle p^{0}\neq RT/(v-b){\mbox{ for all }}b\leq v<\infty }$. Consequently the vdW equation cannot be rigorously derived from the configuration integral over the entire range of ${\displaystyle v}$.

Nevertheless, it is possible to rigorously show that the vdW equation is equivalent to a two term approximation of the virial equation, hence it can be rigorously derived from the partition function as a two term approximation in the additional limit ${\displaystyle \lim b/v=\rho b\rightarrow 0}$.

This derivation is simplest when begun from the grand partition function, ${\displaystyle Q_{G}(V,T,\mu )}$ (see above for its definition),^[123]

In this case the connection with thermodynamics is through ${\displaystyle pV=kT\ln Q_{G}}$, together with the number of particles ${\displaystyle N=kT\partial _{\mu }\ln Q_{G}|_{T,V}}$. Substituting the expression for ${\displaystyle Q_{N}}$ written above in the series for ${\displaystyle Q_{G}}$ produces

$${\displaystyle Q_{G}=e^{pV/kT}=1+\sum _{{\cal {N}}\geq 1}{\frac {{\cal {Z}}_{\cal {N}}}{{\cal {N}}!}}z^{\cal {N}}\quad {\mbox{where}}\quad z={\frac {\exp(\mu /kT)}{\Lambda ^{3}}}.\quad {\mbox{Writing}}\quad p/kT=\sum _{j\geq 1}\,b_{j}(T)z^{j},}$$

expanding ${\displaystyle \exp(pV/kT)}$ in its convergent power series, using the series for ${\displaystyle p/kT}$ in each term, and equating powers of ${\displaystyle z}$ produces relations that can be solved for the ${\displaystyle b_{j}}$ in terms of the ${\displaystyle {\cal {Z}}_{j}}$. For example ${\displaystyle b_{1}(T)={\cal {Z}}_{1}/V=1}$, ${\displaystyle b_{2}(T)=({\cal {Z}}_{2}-{\cal {Z}}_{1}^{2})/(2!V)}$, and ${\displaystyle b_{3}=({\cal {Z}}_{3}-3{\cal {Z}}{\cal {Z}}_{2}+2{\cal {Z}}_{1}^{2})/(3!V)}$. From ${\displaystyle N=kT\partial _{\mu }\ln Q_{G}|_{T,V}=kTV\partial _{\mu }(p/kT)=Vz\partial _{z}(p/kT)}$, the number density, ${\displaystyle \rho _{N}=N/V}$, is expressed as the series

$${\displaystyle \rho _{N}=\sum _{j\geq 1}\,jb_{j}(T)z^{j}\quad {\mbox{which can be inverted to give}}\quad z=\sum _{i\geq 1}\,a_{i}\rho _{N}^{i}.}$$

The coefficients ${\displaystyle a_{i}}$ are given in terms of ${\displaystyle b_{j}}$ by a known formula, or determined simply by substituting ${\displaystyle z}$ into the series for ${\displaystyle \rho _{N}}$, and equating powers of ${\displaystyle \rho _{N}}$; thus ${\displaystyle a_{1}=1/b_{1}=1,a_{2}=-2b_{2},a_{3}=-3b_{3}+8b_{2}^{2}}$, etc. Finally, using this series in the series for ${\displaystyle p/kT}$ produces the virial expansion,^[124] or virial equation of state

$${\displaystyle p/\rho _{N}kT=Z(\rho _{N},T)=\sum _{i\geq 1}\,B_{k}(T)\rho _{N}^{k-1}\quad {\mbox{where}}\quad B_{1}(T)=1.}$$

This conditionally convergent series is also an asymptotic power series for the limit ${\displaystyle \rho _{N}\rightarrow 0}$, and a finite number of terms is an asymptotic approximation to ${\displaystyle Z(\rho _{N},T)}$.^[125] The dominant order approximation in this limit is ${\displaystyle Z\sim 1}$, which is the ideal gas law. It can be written as an equality using order symbols,^[126] for example ${\displaystyle Z=1+o(1)}$, which states that the remaining terms approach zero in the limit, or ${\displaystyle Z=1+O(\rho _{N})}$, which states, more accurately, that they approach zero in proportion to ${\displaystyle \rho _{N}}$. The two term approximation is ${\displaystyle Z=1+B_{2}(T)\rho _{N}+O(\rho _{N}^{2})}$, and the expression for ${\displaystyle B_{2}(T)}$ is

$${\displaystyle B_{2}=-b_{2}=-({\cal {Z}}_{2}-{\cal {Z}}_{1}^{2})/(2V)=-(2V)^{-1}\int _{V_{1}}\int _{V_{2}}\,f({\bf {r}}_{12})\,d{\bf {r}}_{1}d{\bf {r}}_{2},}$$

where ${\displaystyle f({\bf {r}}_{12})=\exp[-(\varepsilon /kT){\bar {\varphi }}({\bf {r}}_{12})]-1}$ and ${\displaystyle {\bar {\varphi }}=\varphi /\varepsilon }$ is a dimensionless two particle potential function. For spherically symetric molecules this function can be represented most simply with two parameters, ${\displaystyle \sigma ,\varepsilon \geq 0}$, a characteristic molecular diameter, and binding energy respectively as shown in the accompanying plot in which ${\displaystyle r=|{\bf {r}}_{12}|}$. Also for spherically symetric molecules 5 of the 6 integrals in the expression for ${\displaystyle B_{2}(T)}$ can be done with the result

$${\displaystyle B_{2}(T)=-2\pi \int _{0}^{\infty }\,f(r)r^{2}\,dr}$$

From its definition ${\displaystyle \varphi (r)}$ is positive for ${\displaystyle r<\sigma }$, and negative for ${\displaystyle r>\sigma }$ with a minimum of ${\displaystyle \varphi (r_{0})=-\varepsilon }$ at some ${\displaystyle r_{0}>\sigma }$. Furthermore ${\displaystyle \varphi }$ increases so rapidly that whenever ${\displaystyle r<\sigma }$ then ${\displaystyle \exp[-\varphi (r)/(kT)]\approx 0}$. In addition in the limit ${\displaystyle \tau =\varepsilon /kT\rightarrow 0}$ (${\displaystyle \tau }$ is a dimensionless coldness, and the quantity ${\displaystyle \varepsilon /k}$ is a characteristic molecular temperature) the exponential can be approximated for ${\displaystyle r>\sigma }$ by two terms of its power series expansion. In these circumstances ${\displaystyle f(r)}$ can be approximated as

$${\displaystyle f(r)=\left\{{\begin{array}{lll}-1&&r<\sigma \\-\tau {\bar {\varphi }}(r)+O[(\tau ^{2}]&&r>\sigma \end{array}}\right.}$$

where ${\displaystyle {\bar {\varphi }}=\varphi /\varepsilon }$ has the minimum value of ${\displaystyle -1}$. On splitting the interval of integration into 2 parts, one less than and the other greater than ${\displaystyle \sigma }$, evaluating the first integral, and making the second integration variable dimensionless using ${\displaystyle \sigma }$ produces,^{[127] [128]}

$${\displaystyle B_{2}(T)=2\pi \sigma ^{3}/3+2\pi \sigma ^{3}\tau \int _{1}^{\infty }\,{\bar {\varphi }}(x)x^{2}\,dx+O(\tau ^{2})\sim b_{N}-a_{N}/kT,}$$

where ${\displaystyle b_{N}=2\pi \sigma ^{3}/3=4[4\pi /3(\sigma /2)^{3}]}$ and ${\displaystyle a_{N}=\varepsilon b_{N}I}$ with ${\displaystyle I}$ a numerical factor whose value depends on the specific dimensionless intermolecular pair potential

$${\displaystyle I=-3\int _{1}^{\infty }\,{\bar {\varphi }}(x)x^{2}\,dx.}$$

Here ${\displaystyle b_{N}=b/N_{\text{A}},a_{N}=a/N_{\text{A}}^{2}}$ where ${\displaystyle a,b}$ are the constants given in the introduction. The condition that ${\displaystyle I}$ be finite requires that ${\displaystyle {\bar {\varphi }}}$ be integrable over the range [1,${\displaystyle \infty }$). This result indicates that a dimensionless ${\displaystyle B_{2}(\tau )/\sigma ^{3}}$ that is a function of a dimensionless molecular temperature ${\displaystyle \tau }$ is a universal function for all real gases with an intermolecular pair potential of the form ${\displaystyle \varphi =\varepsilon {\bar {\varphi }}(r/\sigma )}$; this is an example of the principle of corresponding states on the molecular level.^[129] Moreover this is true in general and has been developed extensively both theoretically and experimentally.^[130][131]

Substituting the (approximate in ${\displaystyle \tau }$) expression for ${\displaystyle B_{2}(T)}$ into the two term virial approximation produces

${\displaystyle Z=1+[b_{N}-a_{N}/kT+O(\tau ^{2})]\rho _{N}+O(\rho _{N}^{2}b_{N}^{2})\sim 1+(1-a/bRT)\rho b+O(\rho ^{2}b^{2})+}$$${\displaystyle O(\tau ^{2}\rho b)=1+\rho b-\rho a/RT+O(\rho ^{2}b^{2})+O(\tau ^{2}\rho b).}$$

Here the approximation is written in terms of molar quantities; its first two terms are the same as the first two terms of the vdW virial equation. The Taylor expansion of ${\displaystyle (1-\rho b)^{-1}}$, uniformly convergent for ${\displaystyle \rho b\ll 1}$, can be written as ${\displaystyle (1-\rho b)^{-1}=1+\rho b+O(\rho ^{2}b^{2})}$, so substituting for ${\displaystyle 1+\rho b}$ produces

$${\displaystyle Z=p/\rho RT=(1-\rho b)^{-1}-a\rho /RT+O(\rho ^{2}b^{2})+O(\tau ^{2}\rho b)}$$. Alternatively this is

$${\displaystyle p\sim RT\rho /(1+\rho b)-\rho ^{2}a=RT/(v-b)-a/v^{2},}$$ the vdW equation.^[132]

According to this derivation the vdW equation is an equivalent of the two term approximation of the virial equation of statistical mechanics in the limits ${\displaystyle \tau ,\rho b\rightarrow 0}$. Consequently the equation produces an accurate approximation in a region defined by ${\displaystyle v\gg b,\,T\gg \varepsilon /k}$ (on a molecular basis ${\displaystyle \rho _{N}\sigma ^{3}\ll 1}$), which corresponds to a dilute gas. But as the density becomes larger the behavior of the vdW approximation and the 2 term virial expansion differ markedly. Whereas the virial approximation in this instance either increases or decreases continuously, the vdW approximation together with the Maxwell construction expresses physical reality in the form of a phase change, while also indicating the existence of metastable states. This difference in behaviors was pointed out long ago by Korteweg,^[133] and Rayleigh (see Rowlinson^[134]) in the course of their dispute with Tait about the vdW equation.

In this extended region, use of the vdW equation is not justified mathematically, however it has empirical validity. Its various applications in this region that attest to this, both qualitative and quantitative, have been described previously in this article. This point was also made by Alder, et. al. who, at a conference marking the 100th anniverary of van der Waals thesis, noted that:^[135]

It is doubtful whether we would celebrate the centenial of the Van der Waals equation if it were applicable only under circumstances where it has been proven to be rigorously valid. It is empirically well established that many systems whose molecules have attractive potentials that are neither long-range nor weak conform nearly quantatively to the Van der Waals model. An example is the theoretically much studied system of Argon, where the attractive potential has only a range half as large as the repulsive core.

They continued by saying that this model has "validity down to temperatures below the critical temperature, where the attractive potential is not weak at all but, in fact, comparable to the thermal energy." They also described its application to mixtures "where the Van der Waals model has also been applied with great success. In fact, its success has been so great that not a single other model of the many proposed since, has equalled its quantitative predictions,^[136] let alone its simplicity."^[137]

Engineers have made extensive use of this empirical validity, modifying the equation in numerous ways (by one account there have been some 400 cubic equations of state produced^[138]) in order to manage the liquids,^[139] and gases of pure substances and mixtures,^[140] they encounter in practice.

This situation has been described by Boltzmann most aptly as follows:^[141]

...van der Waals has given us such a valuable tool that it would cost us much trouble to obtain by the subtlest deliberations a formula that would really be more useful than the one that van der Waals found by inspiration, as it were.

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• Ideal gas
• Inversion temperature
• Iteration
• Maxwell construction
• Real gas
• Theorem of corresponding states
• Van der Waals constants (data page)
• Redlich–Kwong equation of state

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