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 phase change, liquid $$\leftrightarrow$$ vapor. 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: $$p=\frac{RT}{v-b}-\frac {a}{v^2}$$ where $$p$$ is pressure, $$T$$ is temperature, and $$v=VN_\text{A}/N$$ is molar volume, $$N_\text{A}$$ is the Avogadro constant, $$V$$ is the volume, and $$N$$ is the number of molecules (the ratio $$N/N_\text{A}$$ is a physical quantity with base unit mole in the SI). In addition $$R=N_\text{A}k$$ is the universal gas constant, $$k$$ is the Boltzmann constant, and $$a$$ and $$b$$ are experimentally determinable, substance-specific constants. The constant $$a$$ expresses the strength of the molecular interactions. It has dimension pressure times molar volume squared, [pv2] which is also molar energy times molar volume. The constant $$b$$ denotes an excluded molar volume due to a finite molecule size, since the centers of two molecules cannot be closer than their diameter, $$\sigma$$. A theoretical calculation of these constants for spherical molecules with an interparticle potential characterized by, $$\sigma$$, and a minimum energy, $$-\varepsilon$$, as shown in the accompanying plot produces $$b=4[4\pi(\sigma/2)^3/3]N_\text{A}$$. Multiplying this by the number of moles, $$N/N_\text{A}$$, gives the excluded volume as 4 times the volume of all the molecules. This theory also produces $$a=I\varepsilon N_\text{A}b$$ where $$I$$ is a number that depends on the shape of the potential function, $$\varphi(r)$$.

In his book (see references [3] and [4]) Boltzmann wrote equations using $$V/M$$ (specific volume) in place of $$VN_\text{A}/N$$ (molar volume) used here, Gibbs did as well, so do most engineers. Also the property, $$V/N$$ the reciprocal of number density, is used by physicists, but there is no essential difference between equations written with any of these properties. Equations of state written using molar volume contain $$R$$, those using specific volume contain $$R/\bar{m}$$ (the substance specific $$\bar{m}$$ is the molar mass), and those written with number density contain $$k$$.

Once $$a$$ and $$b$$ are experimentally determined for a given substance, the van der Waals equation can be used to predict the boiling point at any given pressure, the critical point (defined by pressure and temperature values, $$p_\text{c}$$, $$T_\text{c}$$ such that the substance cannot be liquefied either when $$p>p_\text{c}$$ no matter how low the temperature, or when $$T>T_\text{c}$$ no matter how high the pressure), 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 $$T$$ vs $$v$$ calculated from the equation at four constant pressure values. On the red isobar, $$p=2p_\text{c}$$, the slope is positive over the entire range, $$b\le v<\infty$$ (although the plot only shows a finite quadrant). This describes a fluid as a gas for all $$T$$, and is characteristic of all isobars $$p>p_\text{c}.$$ The green isobar, $$p=0.2\,p_\text{c}$$, has a physically unreal negative slope, hence shown dotted gray, between its local minimum, $$T_{\rm min}, v_{\rm min}$$, and local maximum, $$T_{\rm max}, v_{\rm max}$$. This describes the fluid as two disconnected branches; a gas for $$v\ge v_{\rm min}$$, and a denser liquid for $$v\le v_{\rm max}\le v_{\rm min}$$.

The thermodynamic requirements of mechanical, thermal, and material equilibrium together with the equation specify two points on the curve, $$(T_s,v_f)$$, and $$(T_s,v_g)$$, 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 $$v$$, 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. Finally the points in the region of negative slope are unstable. All this describes a fluid as a stable gas for $$T>T_\text{s}$$, a stable liquid for $$T<T_\text{s}$$, and a mixture of liquid and gas at $$T=T_\text{s}$$, that also supports metastable states of subcooled gas and superheated liquid. It is characteristic of all 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 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.

Relationship to the ideal gas law
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 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 from the ideal gas equation in order to obtain an equation valid beyond the limit of ideal gas behavior.
 * 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 $$p$$.

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 coefficient of thermal expansion, $$\alpha=(\partial_Tv|_p)/v$$ has a simple analytic expression [this is also true of its isothermal compressibility, $$\kappa_T=-(\partial_pv|_T)/v$$]
 * 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 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 Joule–Thomson coefficient and associated inversion curve, which were instrumental in the commercial liquefaction of gases have simple analytic expressions.

The equation also plays an important role in the modern theory of phase transitions. 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.

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 $$p=nm\overline{v^2}/3$$ for the pressure, $$p$$, in a gas with $$n$$ particles per unit volume (number density), mass $$m$$, and mean square speed $$\overline{v^2}$$. He then noted that using the classical laws of Boyle and Charles one could write $$m\overline{v^2}/3=kT$$ 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 as governed by 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.

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 the opposite is true, 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 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 $$b=v-R(T_2-T_1)/(p_2-p_1)]$$ and $$a=v^2(p_2T_1-p_1T_2)/(T_2-T_1)$$. 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, $$p_s=f_p(y), T_s=f_T(y)$$, that was first obtained by Plank, known to Gibbs, and 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:
 * 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, $$\partial p/\partial v|_{T}=\partial^2p/\partial v^2|_{T}=0$$;
 * 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 be done algebraically by noting that the vdW equation can be written as a cubic in $$v$$, which at the critical point is, $$p_\text{c}cv^3-(p_\text{c}b+RT_\text{c})v^2+av-ab=0$$. Moreover, at the critical point all three roots coalesce so it can also be written as $$(v-v_\text{c})^3=v^3-3v_\text{c}v^2+3v_\text{c}^2v-v_\text{c}^3=0$$ 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 dimensions, [p], [v], [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 $$p/p^*, v/v^*,T/T^*$$. 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, 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, $$\partial^2_{vT}u=\partial^2_{Tv}u$$. Then with $$c_v=\partial_Tu$$ this condition can be written simply as $$\partial_vc(v,T)= \partial_T[T^2\partial_T(p/T)]$$. Differentiating $$p/T$$ for the vdW equation gives $$T^2\partial_T(p/T)]=a/v^2$$, so $$\partial_vc_v=0$$. 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 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 $$\partial_Tp=R/(v-b)$$ 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 $$s_r-\mbox{C}_s=\ln[T_r^c(3v_r-1)]$$

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, κT and cp
The two first partial derivatives of the vdW equation are $$\frac{\partial p}{\partial T}=\frac{R}{v-b}\quad\mbox{and}\quad\frac{\partial p}{\partial v}=-\frac{RT}{(v-b)^2}+\frac{2a}{v^3}$$ The first equation is $$\alpha/\kappa_T$$, while the second is $$-(v\kappa_T)^{-1}$$, where $$\kappa_T=-v^{-1}\partial_pv$$, the isothermal compressibility, is a measure of the relative increase of volume from an increase of pressure, at constant temperature, while $$\alpha=v^{-1}\partial_Tv_p$$, 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 $$c_p=\partial_Th|_p$$. However, it is not independent of $$c_v$$, they are related by the Mayer equation, $$c_p-c_v=-T(\partial_Tp)^2/\partial_vp=Tv\alpha^2/\kappa_T$$. Then the two partials of the vdW equation can be used to express $$c_p$$ as $$c_p(v,T)-c_v(T)=\frac{R^2Tv^3}{RTv^3-2a(v-b)^2}\ge R$$ 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 $$\partial_Tp=-\partial_Tv/\partial_pv=\alpha/\kappa_T=\infty$$, 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.

Stability
According to the extremum principle of thermodynamics $$dS=0$$ and $$d^2S<0$$, namely that at equilibrium the entropy is a maximum. This leads to a requirement that $$\partial p/\partial v|_{T}<0$$. This mathematical criterion expresses a physical condition which Epstein described as follows: "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 $$T> T_\text{c}c$$ this requirement is satisfied everywhere so all states are gas, those states on an isotherm, $$T0$$ (shown dashed gray in Fig. 1), are unstable and thus not observed. This is the genesis of the phase change; there is a range $$v_{\rm min}\le v\le v_{\rm max}$$, for which no observable states exist. The states for $$vv_{\rm max}>v_{\rm min}$$ are vapor; due to gravity the denser liquid lies below the vapor. The transition points, states with zero slope, are called spinodal points. 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 states exist. This spinodal curve is obtained here from the vdW equation by differentiation (or equivalently from $$\kappa_T=\infty$$) as $$T_{\rm sp}=2a\frac{(v-b)^2}{Rv^3}=T_\text{c}\frac{(3v_r-1)^2}{4v_r^3}\qquad p_{\rm sp}= \frac{a(v-2b)}{v^3}=p_\text{c}\frac{(3v_r-2)}{v_r^3}$$ 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.

Saturation
Although the gap in $$v$$ delimited by the two spinodal points on an isotherm (e.g. $$T_r=7/8$$ 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. Thus the phase change is characterized, at temperature $$T_s$$, by a pressure $$p_{\rm min}v_{\rm max}$$. Then from the vdW equation applied to these saturated liquid and vapor states $$p_s=\frac{RT_s}{v_f-b}-\frac{a}{v_f^2}\quad\mbox{and}\quad p_s=\frac{RT_s}{v_g-b}-\frac{a}{v_g^2}$$ These two vdW equations contain 4 variables, $$p_\text{s},T_\text{s},v_f,v_g$$, 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, $$g_f=g_g$$. 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 $$q_{\rm vap}=T(s_g-s_f)$$, and from the first law at constant pressure $$q_{\rm vap}=h_g-h_f$$. Equating these two, rearranging, and recalling that $$g=h-Ts$$ produces the result.

The Gibbs free energy is one of the 4 thermodynamic potentials whose partial derivatives produce all other thermodynamics state properties; its differential is $$dg=\partial_pg\,dp+\partial_Tg\,dT =v\,dp-s\,dT$$. Integrating this over an isotherm from $$p_s, v_f$$ to $$p_s, v_g$$, noting that the pressure is the same at each endpoint, and setting the result to zero yields $$g_g-g_f=\sum_{j=1}^3\int_{p_j}^{p_j+1}\,v_j\,dp=-\int_{v_f}^{v_g}\,p\,dv+p_s(v_g-v_f)=0$$ Here because $$v$$ is a multivalued function, the $$p$$ integral must be divided into 3 parts corresponding to the 3 real roots of the vdW equation in the form, $$v(p,T)$$ (this can be visualized most easily by imagining Fig. 1 rotated 90$$^\circ$$); the result is a special case of material equilibrium. The last equality, which follows from integrating $$v\,dp=d(pv)-p\,dv$$, is the Maxwell equal area rule which requires that the upper area between the vdW curve and the horizontal through $$p_\text{s}$$ be equal to the lower one. This form means that the thermodynamic restriction that fixes $$p_s$$ is specified by the equation of state itself, $$p=p(v,T)$$. 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, $$RT_s\left[\frac{v_g}{v_g-b}-\frac{v_f}{v_f-b}-\ln\left(\frac{v_g-b}{v_f-b}\right)\right] -2a\left(\frac{1}{v_g}-\frac{1}{v_f}\right)=0$$ 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 $$T_s$$ or $$p_s$$, and tabular results presented; however, the equations also admit an analytic parametric solution obtained most simply and elegantly, by Lekner. Details of this solution may be found in the Maxwell Construction; the results are $$T_{rs}(y)=\left(\frac{27}{8}\right)\frac{2f(y)[\cosh y+f(y)]}{g(y)^2}\quad p_{rs}=27\frac{f(y)^2[1-f(y)^2]}{g(y)^2}$$ $$v_{rf}=\left(\frac{1}{3}\right)\frac{1+f(y)e^y}{f(y)e^y} \qquad\qquad\qquad\quad v_{rg}=\left(\frac{1}{3}\right)\frac{1+f(y)e^{-y}}{f(y)e^{-y}}$$ where $$f(y)=\frac{y\cosh y-\sinh y}{\sinh y\cosh y-y}\qquad\qquad g(y)=1+2f(y)\cosh y+f(y)^2$$ and the parameter $$0\le y<\infty$$ is given physically by $$y=(s_g-s_f)/(2R)$$. The values of all other property discontinuities across the saturation curve also follow from this solution. 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 agree exactly with the numerical results referenced earlier.

Referring back to Fig. 1 the isotherms for $$T_r<1$$ are discontinuous. Considering $$T_r=7/8$$ 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"; 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.

In the same section Boltzmann also addressed and explained the negative pressures which some liquid metastable states exhibit (for example $$T_r=0.8$$ 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."

Even though the phase change produces a mathematical discontinuity in the homogeneous fluid properties, for example $$v$$, there is no physical discontinuity. As the liquid begins to vaporize the fluid becomes a heterogeneous mixture of liquid and vapor whose molar volume varies continuously from $$v_f$$ to $$v_g$$ according to the equation of state $$v=v_f+x(v_g-v_f)\qquad x=N_g/(N_f+N_g)$$ where $$0\le x\le 1$$ is the mole fraction of the vapor. This equation is called the lever rule and applies to other properties as well. 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.

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

The first candidate for $$\phi$$ was the critical compressibility factor $$Z_\text{c}=p_\text{c}v_\text{c}/(RT_\text{c})$$, but because that quantity is difficult to measure accurately, the acentric factor developed by Kenneth Pitzer, $$\omega=-\mbox{log}_{10}[p_r(T_r=0.7)]-1$$, is more useful. The saturation pressure in this situation is represented by a one parameter family of curves, $$p_{rs}=p_s(T_r,\omega)$$. Several investigators have produced correlations of saturation data for a number of substances, the best is that of Dong and Lienhard, $$\ln p_{rs}=5.37270(1-1/T_r)+\omega(7.49408-11.181777T_r^3+3.68769T_r^6+17.92998\ln T_r)$$ which has an rms error of $$\pm 0.42$$ over the range $$1\le T_r\le0.3$$

Figure 3 is a plot of $$p_{rs}$$ vs $$T_r$$. for various values of $$\omega$$ 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 $$\omega$$) are more pronounced.

Figure 4 is another plot of the same equation showing $$T_r$$ as a function of $$\omega$$ for various values of $$p_{rs}$$. It includes data from 51 substances including the vdW fluid over the range $$-0.4<\omega<0.9$$. This plot shows clearly that the vdW fluid ($$\omega=-0.302$$) is a member of the class of real fluids; indeed it quantitatively describes the behavior of the liquid metals cesium ($$\omega=-0.267$$) and mercury ($$\omega=-0.21$$) whose values of $$\omega$$ 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, $$\omega$$.

Joule–Thomson coefficient and the inversion curve
The Joule–Thomson coefficient, $$\mu_J=\partial_pT|_h$$, is of practical importance because the two end states of a throttling process ($$h_2=h_1$$) lie on a constant enthalpy curve. Although ideal gases, for which $$h=h(T)$$, 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.

This coefficient can be found in terms of the previously described derivatives as, $$\mu_J=\frac{v(\alpha T-1)}{c_p}$$ so when $$\mu_J$$ is positive the gas temperature decreases when it passes through a throttle, and if it is negative the temperature increases. Therefore the condition $$\mu_J=0$$ defines a curve that separates the region of the $$T,p$$ plane where $$\mu_J>0$$ from the region where it is less than zero. This curve is called the inversion curve, and its equation is $$\alpha T-1=0$$. Using the expression for $$\alpha$$ derived previously for the van der Waals equation this is $$\frac{2a(v-b)^2-RTv^2b}{RTv^3-2a(v-b)^2}=0\quad\mbox{or}\quad 2a(v-b)^2-RTv^2b=0$$ Note that for $$v\gg b$$ there will be cooling for $$2a>RTb$$ or in terms of the critical temperature $$T<27T_\text{c}/4$$. 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."

The inversion curve can be found by solving its equation for $$b/v$$, and substituting into the vdW equation. This produces $$p/p^*=-1+4(T/2T^*)^{1/2}-3(T/2T^*)$$, where, for simplicity, $$a,b,R$$ have been replaced by $$p^*,T^*$$. The maximum of this, quadratic, curve occurs, with $$z^2=T/(2T^*)$$, for $$D_zp/p^*=4-6(T/2T^*)^{1/2}=0$$ which gives $$(T/2T^*)^{1/2}_{\rm max}=2/3$$, or $$T_{\rm max}=8T^*/9$$, and the corresponding $$p_{\rm max}=p^*/3$$. The zeros of the inversion curve $$3z^2- 4z+1=0$$, are, making use of the quadratic formula, $$z=(4\pm\sqrt{16-12}\,)/6$$, or $$z=1/3$$ and $$1$$ ($$T/T^*=2/9=0.\overline{2}$$ and $$2$$). In terms of the dimensionless variables, $$T_r, p_r$$ the zeros are at $$T_r=3/4$$ and $$27/4$$, while the maximum is $$p_{r\rm max}=9$$, and occurs at $$T_{r\rm max}=3$$. Note from Fig. 5 that there is 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. Therefore 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.

Compressibility factor
Real gases can be characterized by their difference from ideal. This is done by writing the mechanical equation of state in the form $$pv=ZRT$$ where $$Z$$, called the compressibility factor, is usually expressed either as a function of pressure and temperature, or density and temperature, and in each case in the limit $$p$$ or $$\rho=1/v\rightarrow 0$$, $$Z= 1$$, the ideal gas value. In the second case $$p=Z(\rho,T)\rho RT$$. Thus for a van der Waals fluid the compressibility factor is $$Z=1/(1-b\rho)-a\rho/(RT)$$, or in terms of reduced variables $$Z=\frac{3}{3-\rho_r}-\frac{9\rho_r}{8T_r}$$ where $$0\le\rho_r=1/v_r\le 3$$. At the critical point, $$T_r=\rho_r=1$$, $$Z=Z_\text{c}=3/2-9/8=3/8$$.

In the limit $$\rho_r\rightarrow 0$$, $$Z=1$$ (for finite $$T$$); the fluid behaves like an ideal gas, a point noted several times earlier. Note additionally that the derivative $$D_{\rho_r}Z = 3/(3-\rho_r)^2-9/(8T_r)=0$$ for $$T_r=3(3-\rho_r)^2/8$$, and when $$\rho_r=0$$ this is $$T_r=27/8$$. The slope is positive or negative depending on whether $$T_r$$ is greater than or less than $$27/8$$, and becomes infinitely large negative as $$T$$ approaches $$0$$.

Figure 6 shows a plot of various isotherms of $$Z(\rho,T_r)$$ vs $$\rho_r$$. 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 $$Z(\rho_r,T_r)$$ vs $$p_r(\rho_r,T_r)$$ using $$\rho_r$$ 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. The two graphs are similar, including the caustic generated by the crossing isotherms; they are qualitatively very much alike.

Virial expansion
Statistical mechanics suggests that $$Z$$ can be expressed by a power series called a virial expansion, $$Z(\rho,T)=1+\sum_{k=1}^\infty\,V_k(T)(\rho)^k$$ The functions $$V_k(T)$$ are the virial coefficients; the $$k$$th term represents a $$k+1$$ particle interaction.

If we note in the expression for $$Z(\rho_r, T_r)$$ that for $$\rho_r<3$$, the term $$(3-\rho_r)^{-1}$$ can be expanded in an absolutely convergent series; this yields $$Z(\rho_r,T_r)=1+[1-27/(8T_r)]^{-1}](\rho_r/3)+\sum_{k=2}^\infty\,(\rho_r/3)^k$$ This is the virial expansion for the van der Waals fluid. The first virial coefficient is the slope of $$Z(\rho_r,T_r)$$ at $$\rho_r=0$$. Notice that it can be positive or negative depending on whether or not $$T_r>\mbox{ or}<27/8$$, which agrees with the result found previously by differentiation.

For molecules that are non attracting hard spheres, $$a=0$$, the vdW virial expansion becomes simply $$Z=(1-b\rho)^{-1}=\sum_{i=0}^\infty (b\rho)^i$$, which illustrates the effect of the excluded volume alone. It was recognized early on that this was in error beginning with the term $$(b\rho)^2$$. Boltzmann calculated its correct value as $$(5/8)(b\rho)^2$$, and used the result to propose an enhanced version of the vdW equation $$(p+a/v^2)(v-b/3)=RT[1+2b/(3v)+7b^2/(24v^2)].$$ On expanding $$(v-b/3)^{-1}$$, this produced the correct coefficients thru $$(b/v)^2$$ and also gave infinite pressure at $$b/3$$, which is approximately the close packing distance for hard spheres. 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.

Mixtures
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. His essential idea was that in a binary mixture of vdw fluids described by the equations $$ p_1=\frac{RT}{v-b_{11}}-\frac{a_{11}}{v^2}\quad\mbox{and}\quad p_2=\frac{RT}{v-b_{22}}-\frac{a_{22}}{v^2}$$ the mixture is also a vdW fluid given by $$p=\frac{RT}{v-b_x}-\frac{a_x}{v^2}$$ where $$a_x=a_{11}x_1^2+2a_{12}x_1x_2+a_{22}x_2^2\quad\mbox{and}\quad b_x=b_{11}x_1^2+2b_{12}x_1x_2+b_{22}x_2^2$$ Here $$x_1=N_1/N$$, and $$x_2=N_2/N$$, with $$N=N_1+N_2$$ (so that $$x_1+x_2=1$$) are the mole fractions of the two fluid substances. Adding the equations for the two fluids shows that $$p\ne p_1+p_2$$, although for $$v$$ sufficiently large $$p\approx p_1+p_2$$ with equality holding in the ideal gas limit. The quadratic forms for $$a_x$$ and $$b_x$$ are a consequence of the forces between molecules. This was first shown by Lorentz, and was credited to him by van der Waals. The quantities $$a_{11},\,a_{22}$$ and $$b_{11},\,b_{22}$$ in these expressions characterize collisions between two molecules of the same fluid component while $$a_{12}=a_{21}$$ and $$b_{12}=b_{21}$$ 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.

Assuming that $$b_{12}$$ is the arithmetic mean of $$b_{11}$$ and $$b_{22}$$, $$b_{12}=(b_{11}+b_{22})/2$$, substituting into the quadratic form, and noting that $$x_1+x_2=1$$ produces $$b=b_{11}x_1+b_{22}x_2$$ Van der Waals wrote this relation, but did not make use of it initially. However, it has been used frequently in subsequent studies, and its use is said to produce good agreement with experimental results at high pressure.

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 $$T=T_R$$, $$DF=0$$ and $$D^2F>0$$, namely at equilibrium the Helmholtz potential is a minimimum. This leads to the requirement $$\partial^2f/\partial v^2|_T=-\partial p/\partial v|_T>0$$, which is the previous stability condition for the pressure, but in addition requires that the function, $$f=FN_{\mbox{A}}/N$$, is convex at all $$v$$ that describe stable states.

For a single substance the definition of the molar Gibbs free energy can be written in the form $$f=g-pv$$. Thus when $$p$$ and $$g$$ are constant along with temperature the function $$f(T_R, v)$$ represents a straight line with slope $$-p$$, and intercept $$g$$. Since the curve, $$f(T_R, v)$$, has positive curvature everywhere when $$T_R\ge T_c$$, the curve and the straight line  will be have a single tangent. However, for a subcritical, $$T_R$$, with $$p=p_s(T_R)$$ and a suitable value of $$g$$ the line will be tangent to $$f(T_R,v)$$ at the molar volume of each coexisting phase, saturated liquid, $$v_f(T_R)$$, and saturated vapor, $$v_g(T_R)$$; there will be a double tangent. Furthermore, each of these points is characterized by the same value of $$g$$ as well as the same values of $$p$$ and $$T_R.$$ These are the same three specifications for coexistence that were used previously.

As depicted in Fig. 8, the region on the green curve $$f(T_R,v)$$ for $$v\le v_f$$ ($$v_f$$ is designated by the left green circle) is the liquid. As $$v$$ increases past $$v_f$$ the curvature of $$f$$ (proportional to $$\partial^2_{v^2}f=-\partial_vp$$) continually decreases. The point characterized by $$\partial^2_{v^2}f=-\partial_vp=0$$, is a spinodal point, and between these two points is the metastable superheated liquid. For further increases in $$v$$ 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 $$v$$ the curvature increases to a maximum at $$v_g$$, where the slope is $$p_s$$; the region between this point and the second spinodal point is the metastable subcooled vapor. Finally, the region $$v\ge v_g$$ is the vapor. In this region the curvature continually decreases until it is zero at infinitely large $$v$$. 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.

For a vdW fluid the molar Helmholtz potential is simply $$f_r=u_r-T_rs_r=\mbox{C}_u+T_r\{c-\mbox{C}_s-\ln[T_r^c(3v_r-1)]\}-9/(8v_r)$$ where $$f_r=f/(RT_c)$$. A plot of this function for the suncritical isotherm $$T_r=7/8$$ is the one 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.

Van der Waals introduced 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, $$f(T_R,v,x)$$, where $$x$$ is a composition variable, for example $$x=x_2$$ so $$x_1=1-x$$. In this case there are three stability conditions $$\frac{\partial^2f}{\partial v^2}>0\qquad\frac{\partial^2f}{\partial x^2}>0\qquad\frac{\partial^2f}{\partial v^2}\frac{\partial^2f}{\partial x^2}-\left(\frac{\partial^2f}{\partial x\partial v}\right)^2>0$$ and the Helmholtz potential is a surface (of physical interest in the region $$0\le x\le 1$$). The first two stability conditions show that the curvature in each of the directions $$v$$ and $$x$$ are both positive for stable states while the third condition indicates that stable states correspond to elliptic points on this surface. Moreover its limit, $$\partial^2_{x^2}f\partial^2_{v^2}f-(\partial^2_{xv}f)^2=0$$, is in this case the specification of spinodal points.

For a binary mixture the Euler equation, can be written in the form $$f=-pv+\mu_1x_1+\mu_2x_2=-pv+(\mu_2-\mu_1)x+\mu_1$$ Here $$\mu_j=\partial_{x_j}f$$ are the molar chemical potentials of each substance, $$j=1,2$$. For $$p$$, $$\mu_1$$ and $$\mu_2$$, all constant this is the equation of a plane with slopes $$-p$$ in the $$v$$ direction, $$\mu_2-\mu_1$$ in the $$x$$ direction, and intercept $$\mu_1$$. 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 $$T$$, $$p$$, $$\mu_2-\mu_1$$, and $$\mu_1$$; the last two are equivalent to having the same $$\mu_1$$ and $$\mu_2$$ individually, which are just the Gibbs conditions for material equilibrium in this situation. 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 $$x$$ direction as well as the one in the $$v$$ direction. Therefore, there can be two liquid phases that can be either miscible, or wholly or partially immiscible, as well as a vapor phase. 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, 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. The results obtained were, in Rowlinson's words, "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 $$a_{11}, a_{22},b_{11}, b_{22}$$ must be known. In addition, the effect of collisions between molecules of the different components, given by $$a_{12}$$ and $$b_{12}$$, must also be specified. In he absence of experimental data, or computer modelling results to estimate their value, the empirical combining laws, $$a_{12}=(a_{11}a_{22})^{1/2}\qquad\mbox{and}\qquad b_{12}^{1/3}=(b_{11}^{1/3}+b_{22}^{1/3})/2,$$ the geometric and algebraic means respectively can be used. These relations correspond to the empirical combining laws for the intermolecular force constants, $$\epsilon_{12}=(\epsilon_{11}\epsilon_{22})^{1/2}\qquad\mbox{and}\qquad\sigma_{12}=(\sigma_{11}+\sigma_{22})/2,$$ 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. Then, generalizing for $$n$$ fluid components, and using these empirical combinig laws, the expressions for the material constants are: $$a_x = \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{ii}a_{jj})^{1/2}x_ix_j\quad\mbox{which can be written as}\quad a_x=\Big(\sum_{i=1}^{n}a_{ii}^{1/2}x_i \Big)^2$$ $$b_x=(1/8)\sum_{i=1}^{n}\sum_{j=1}^{n}(b_{ii}^{1/3} +b_{jj}^{1/3})^3 x_ix_j$$ Using these expressions in the vdW equation is apparently helpful for divers, as well as being important for physicists, physical chemists, and chemical engineers in their study and management of the various phase equilibria and critical behavior observed in fluid mixtures.

Another method of specifying the vdW constants pioneered by W.B. Kay, and known as Kay's rule. specifies the effective critical temperature and pressure of the fluid mixture by $$T_{cx}=\sum_{i=1}^nT_{ci}x_i\qquad\mbox{and}\qquad p_{cx}=\sum_{i=1}^n\,p_{ci}x_i$$ In terms of these quantities the vdW mixture constants are then, $$a_x=\frac{27}{64}\frac{(RT_{cx})^2}{p_{cx}}\qquad\qquad b_x=\frac{1}{8}\frac{RT_{cx}}{p_{cx}}$$ and Kay used these specifications of the mixture critical constants as the basis for calculations of the thermodynamic properties of mixtures.

Kay's idea was adopted by T. W. Leland, who applied it to the molecular parameters, $$\epsilon, \sigma$$, which are related to $$a, b$$ through $$T_c, p_c$$ by $$a\propto\epsilon\sigma^3$$ and $$ b\propto\sigma^3$$ (see the introduction to this article). Using these together with the quadratic form of $$a, b$$ for mixtures produces $$\sigma_x^3=\sum_{i=i}^n\sum_{j=1}^n\, \sigma_{ij}^3x_ix_j\qquad\mbox{and}\qquad \epsilon_x=\left[\sum_{i=1}^n\sum_{j=1}^n\epsilon_{ij}\sigma_{ij}^3x_ix_j\right]\left[\sum_{i=i}^n\sum_{j=1}^n\, \sigma_{ij}^3x_ix_j\right]^{-1}$$ which is the van der Waals approximation expressed in terms of the intermolecular constants. This approximation, when compared with computer simulations for mixtures, are in good agreement over the range $$1/2< (\sigma_{11}/\sigma_{22})^3<2$$, 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".

Derivation
Textbooks in physical chemistry generally give two derivations of the title equation. One is the conventional derivation that goes back to Van der Waals, a mechanical equation of state that cannot be used to specify all thermodynamic functions; the other is a statistical mechanics derivation that makes explicit the intermolecular potential neglected in the first derivation. A particular advantage of the statistical mechanical derivation is that it yields the partition function for the system, and allows all thermodynamic functions to be specified (including the mechanical equation of state).

Conventional derivation
Consider one mole of gas composed of non-interacting point particles that satisfy the ideal gas law: (see any standard Physical Chemistry text, op. cit.)
 * $$ p = \frac{RT}{V_\mathrm{m}}.$$

Next, assume that all particles are hard spheres of the same finite radius r (the Van der Waals radius). The effect of the finite volume of the particles is to decrease the available void space in which the particles are free to move. We must replace V by V − b, where b is called the excluded volume (per mole) or "co-volume". The corrected equation becomes
 * $$ p = \frac{RT}{V_\mathrm{m}-b}.$$

The excluded volume $$b$$ is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. To see this, we must realize that a particle is surrounded by a sphere of radius 2r (two times the original radius) that is forbidden for the centers of the other particles. If the distance between two particle centers were to be smaller than 2r, it would mean that the two particles penetrate each other, which, by definition, hard spheres are unable to do.

The excluded volume for the two particles (of average diameter d or radius r) is
 * $$b'_2 = 4\pi d^3/3 = 8\times (4\pi r^3/3)$$,

which, divided by two (the number of colliding particles), gives the excluded volume per particle:
 * $$b' = b'_2/2 \quad \rightarrow \quad b'=4\times (4\pi r^3/3)$$,

So  is four times the proper volume of the particle. It was a point of concern to Van der Waals that the factor four yields an upper bound; empirical values for  are usually lower. Of course, molecules are not infinitely hard, as Van der Waals thought, and are often fairly soft. To obtain the excluded volume per mole we just need to multiply by the number of molecules in a mole, i.e. by the Avogadro number:
 * $$b = N_\text{A} b' $$.

Next, we introduce a (not necessarily pairwise) attractive force between the particles. Van der Waals assumed that, notwithstanding the existence of this force, the density of the fluid is homogeneous; furthermore, he assumed that the range of the attractive force is so small that the great majority of the particles do not feel that the container is of finite size. Given the homogeneity of the fluid, the bulk of the particles do not experience a net force pulling them to the right or to the left. This is different for the particles in surface layers directly adjacent to the walls. They feel a net force from the bulk particles pulling them into the container, because this force is not compensated by particles on the side where the wall is (another assumption here is that there is no interaction between walls and particles, which is not true, as can be seen from the phenomenon of droplet formation; most types of liquid show adhesion). This net force decreases the force exerted onto the wall by the particles in the surface layer. The net force on a surface particle, pulling it into the container, is proportional to the number density. On considering one mole of gas, the number of particles will be NA
 * $$C=N_\mathrm{A}/V_\mathrm{m}$$.

The number of particles in the surface layers is, again by assuming homogeneity, also proportional to the density. In total, the force on the walls is decreased by a factor proportional to the square of the density, and the pressure (force per unit surface) is decreased by:
 * $$a'C^2= a' \left(\frac{N_\mathrm{A}}{V_\mathrm{m}}\right)^2 = \frac{a}{V_\mathrm{m}^2} $$,

so that:
 * $$ p = \frac{RT}{V_\mathrm{m}-b}-\frac{a}{V_\mathrm{m}^2} \Rightarrow

\left(p + \frac{a}{V_\mathrm{m}^2}\right)(V_\mathrm{m}-b) = RT. $$ Upon writing n for the number of moles and nVm = V, the equation obtains the second form given above:
 * $$ \left(p + \frac{n^2 a}{V^2}\right)(V-nb) = nRT.

$$

It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density.

Statistical thermodynamics derivation
The canonical partition function Z of an ideal gas consisting of N = nNA identical (non-interacting) particles, is:

Z = \frac{z^N}{N!}\quad \hbox{with}\quad z = \frac{V}{\Lambda^3} $$ where $$\Lambda$$ is the thermal de Broglie wavelength:

\Lambda = \sqrt{\frac{h^2}{2\pi m k T}} $$ with the usual definitions: h is the Planck constant, m the mass of a particle, k the Boltzmann constant and T the absolute temperature. In an ideal gas z is the partition function of a single particle in a container of volume V. In order to derive the Van der Waals equation we assume now that each particle moves independently in an average potential field offered by the other particles. The averaging over the particles is easy because we will assume that the particle density of the Van der Waals fluid is homogeneous. The interaction between a pair of particles, which are hard spheres, is taken to be:

u(r) = \begin{cases} \infty                 &\hbox{when}\quad r < d, \\ -\epsilon \left(\frac{d}{r}\right)^6 & \hbox{when}\quad r \ge d, \end{cases} $$ r is the distance between the centers of the spheres and d is the distance where the hard spheres touch each other (twice the Van der Waals radius). The depth of the Van der Waals well is $$\epsilon$$.

Because the particles are not coupled under the mean field Hamiltonian, the mean field approximation of the total partition function still factorizes:
 * $$Z= z^N/N!$$,

but the intermolecular potential necessitates two modifications to z. First, because of the finite size of the particles, not all of V is available, but only V − Nb′, where (just as in the conventional derivation above):
 * $$ b' = 2\pi d^3/3 $$.

Second, we insert a Boltzmann factor exp[−Φ/2kT] to take care of the average intermolecular potential. We divide here the potential by two because this interaction energy is shared between two particles. Thus:

z= \frac{(V-Nb') \, e^{-\phi/(2kT)}}{\Lambda^3}. $$

The total attraction felt by a single particle is:

\phi = \int_d^{\infty} u(r) \frac{N}{V} 4\pi r^2 dr , $$ where we assumed that in a shell of thickness dr there are N/V 4π r2dr particles. This is a mean field approximation; the position of the particles is averaged. In reality the density close to the particle is different than far away as can be described by a pair correlation function. Furthermore, it is neglected that the fluid is enclosed between walls. Performing the integral we get:

\phi = -2 a' \frac{N}{V}\quad\hbox{with}\quad a' = \epsilon \frac{2\pi d^3}{3} =\epsilon b'. $$ Hence, we obtain:

\ln Z = N \ln {(V-Nb')} + \frac{N^2 a'}{V kT} - N \ln {(\Lambda^3)} -\ln {N!} $$ From statistical thermodynamics we know that:
 * $$ p = kT \frac{\partial \ln Z}{\partial V}

$$, so that we only have to differentiate the terms containing $$V$$. We get:

p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. $$