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Mathematical and Empirical Validity
Ever since van der Waals presented his thesis, "[m]any derivations, pseudo-derivations, and plausibility arguments have been given" for it. However, of these only two are mathematically rigorous and will be summarized here. They both proceed from the same source, the canonical partition function, $${\cal Z}(V,T,N)$$, written for a macroscopic system $${\cal Z}=\frac{Q_N}{N!\,\Lambda^{3N}}\qquad\mbox{with}\qquad Q_N=\int_{V^{\bf N}}\,\exp\left[-\frac{\Phi({\bf r^N})}{kT}\right]\,d{\bf r^N}$$ where $$\Lambda$$ is the DeBroglie wavelength, $$Q_N(V,T,N)$$ is the configuration integral, and $$\Phi$$ is the intermolecular potential energy. They both make use of the pairwise additivity approximation, in which $$\Phi$$ is written as a sum of pair potentials $$\Phi({\bf r^N})=\sum_{i<j}\,u(r_{ij})$$ that have the dimension of energy, and depend on the distance between the two molecules. The connection with thermodynamics is made through the relation $$F=-kT\ln\cal{Z}$$, and then the equation of state follows from $$p=-\partial_VF|_T$$.

Derivation
Any equation of state relating $$p,v,T$$ with 3 constants, one universal, $$R$$, and two that describe the molecules composing the fluid, $$a, b$$, cannot accurately represent all real fluids. If this were otherwise, all such fluids would have the same saturation curve, as shown in Fig. 2. for the vdW equation, but that is not the actual case as shown in Figs. 3 and 4. Similar considerations for other properties led Pitzer to introduce a third constant to describe molecular character, and this produced better quantitative agreement for the macroscopic properties. Consequently the vdW equation with constants $$R, a, b$$ cannot be obtained from a mathematically rigorous derivation -- it must be an approximation, although as Goodstein has noted "[m]any derivations, pseudo-derivations, and plausibility arguments have been given..." for it.

Furthermore, theoretical considerations of statistical mechanics suggest that the partition function should give the greatest weight to the most likely states, so that a rigorous derivation should produce, in the coexistence region, the low energy,stable, heterogeneous states, and not the high energy, metastable/unstable, homogeneous states given by the vdW equation.

In this case the canonical partition function which is given by, is related to the system pressure by has the form $$w(r_{ij})= \varepsilon(\sigma/\ell)\varphi(r_{ij}/\ell)$$,

and regarding $$\Phi({\bf r^N})=\sum\sum q(r_{ij}) + \sum\sum w(r_{ij})$$ to be the sum of short-range repulsive potentials $$q(r/\sigma)$$ and long-range attractive ones $$w=-(\sigma/\ell)^3\varphi\{-r/\ell\}$$, is that in a double limit process ($$\ell, V\rightarrow\infty$$ such that $$N/(N_AV)=\rho, N_AF/N=f$$ are finite and $$\ell\ll V^{1/3}$$), in which the attractive part of the intermolecular potential becomes infinitely weak and infinitely long-range (as $$\ell\rightarrow\infty$$), then $$\lim_{\ell\rightarrow\infty}f(\rho,T,\ell)=\mbox{CE}[f^0(\rho,T)-a\rho]$$ where CE is the complex envelope. This is the greatest convex function which is $$\le$$ the function. For example in Fig.8 the CE is the sum of the solid green and solid black curves. Differentiating, making use of $$p=-\partial_vf|_T=\rho^2\partial_{\rho}f|_T$$ produces $$p(\rho, T)=p^0(\rho,T)-a\rho^2$$ This is the vdW equation, since $$p^0$$ is calculated for $$w=0$$; however, because of the CE the subcritical isotherms (see Fig.1) are cut off at $$v_f(T), v_g(T)$$ and the points connected by a horizontal line. These are the heterogeneous states that has lower free energy than the original curve; they are also the ones generated by the Maxwell condition.

Derivation
As Goodstein has noted, "Many derivations, pseudo-derivations, and plausibility arguments have been given for this equation." The derivation presented here uses the canonical ensemble of statistical mechanics applied to a moderately dense gas. This approach makes explicit the assumptions required in order to obtain the van der Waals equation.

The canonical partition function is defined as $$ {\cal Z}=\sum_j\,\exp\left(-\frac{E_j}{kT}\right)= \sum_{E_j}\rho_{E_j}\,\exp\left(-\frac{E_j}{kT}\right)$$ where the first sum is over all the states of the system while the second sum is over the discrete (quantized) energies, and $$\rho_{E_j}$$ is the number of states (degeneracy) with energy $$E_j$$. The function $${\cal Z}$$ specifies the thermodynamic Helmholtz free energy function by $$F(V,T)=-kT\ln[{\cal Z}(V,T)]$$ (here $$V$$ is the variable that specifies the $$E_j$$), and through it all the other macroscopic thermodynamic functions, $$p=-\partial_VF|_T\qquad S=\partial_TF|_V\qquad U=F+TS$$

On a macroscopic scale the discrete energy levels are closely spaced so little error is introduced by replacing them with a continuous variable and write $${\cal Z}=\int\,\rho(E)\exp\left(-\frac{E}{kT}\right)\,dE.$$ The integration is taken over all energies with little additional error incurred because the integrand is sharply peaked about its average value. Furthermore on this scale the system energy, apart from its internal molecular structure, can be expressed in terms of the momenta and positions of the $$N$$ molecules that comprise it $$E=\sum_{i=1}^N\frac{{\bf p}_i^2}{2m}+\Phi({\bf r^N})$$ Here $${\bf p}_i$$ is the vector momentum of the $$i$$th particle, and $$\Phi({\bf r^N})$$ is the potential energy of the $$N$$ particles relative to one another where $${\bf r^N}$$ is a shorthand way of designating the $$N$$ vector locations $$({\bf r}_1,{\bf r}_2\ldots {\bf r}_N)$$ of the particles. When the individual particles move in three dimensional space the system state is specified by $$6N$$ variables so $${\cal Z}$$ can be represented by a point in this phase space whose elemental volume, $$d{\bf p^N}d{\bf r^N}=\prod_{i=1}^Nd{\bf p}_id{\bf r}_i $$, has a dimension which is a power of action, [Et]3N=[mL2/t]3N. However, in order to make this calculation of $${\cal Z}$$ correspond to quantum statistical mechanical calculations, the elemental volume must be made dimensionless $$d{\bf p}^N\,d{\bf r}^N/(N!h^{3N})$$, where $$h$$ is Plank's constant. This results finally in the expression, $${\cal Z}=\frac{1}{N!\,h^{3N}}\int\int\,\exp\left(-\sum_{i=1}^N\frac{{\bf p}_i^2}{2mkT}\right)\exp\left[-\frac{\Phi({\bf r^N})}{kT}\right]\,d{\bf p^N}d{\bf r^N}$$

Of the 6$$N$$ integrals in this expression, $$3N$$ of them, corresponding to the momenta, can be evaluated. Since $$\exp\left(-\sum_{i=1}^N\frac{{\bf p}_i^2}{2mkT}\right)=\exp\left(-\sum_{i=1}^{3N}\frac{p_i^2}{2mkT}\right)=\prod_{i=1}^{3N}\exp\left(-\frac{p_i^2}{2mkT}\right)$$ they can all be written as a single definite integral with a well known value $$\left[(2mkT)^{1/2}\int_{-\infty}^\infty\,e^{-x^2}\,dx\right]^{3N}=(2\pi mkT)^{3N/2}$$ so that $$\cal{Z}$$ can be written simply as $${\cal Z}=\frac{Q_N}{N!\,\Lambda^{3N}}\qquad\mbox{where}\qquad Q_N=\int\,\exp\left[-\frac{\Phi({\bf r^N})}{kT}\right]\,d{\bf r^N}$$ Here $$\Lambda=[h^2/(2\pi mkT)]^{1/2}$$ is the thermal de Broglie wavelength and $$Q_N$$ is the configuration integral. The limits of integration of these integrals are specified by the volume occupied by the molecules. Since $$F=-kT\ln{\cal Z}=kT\{\ln[(N!\,\Lambda^{3N})]-\ln Q_N\}$$, and subsequently $$p=-\partial_VF|_T=kT\partial_V\ln Q_N|_T$$, it is the configuration integral alone that specifies the equation of state $$p=p(V,T)$$.

When the molecules do not interact, $$\Phi=0$$ so $$Q_N=V^N$$. Then taking its natural logarithm, differentiating with respect to $$V$$, and multiplying by $$kT$$, produces $$p=NkT/V$$, the ideal gas law.

At this point the system potential energy function, $$\Phi({\bf r^N})$$, must be specified in order to evaluate the $$3N$$ integrals that make up $$Q_N$$. This is difficult to do in general, but on making the approximation of pairwise additivity, $$\Phi$$ takes the form $$\Phi=\sum_{1\le i<j\le N}\,\varphi(r_{ij}).$$ When $$r_{ij}=|{\bf r}_j-{\bf r}_i|$$, the distance between the two molecules, this applies to symmetric molecules. More fundamentally this approximation neglects the effect on the force exerted by one molecule on another when another is brought into their vicinity. Although the error created by this and other similar neglects (more than one additional molecule) is unknown, it surely becomes smaller as the number density, $$N/V$$, becomes smaller. Using pairwise additivity $$Q_N$$ becomes $$Q_N=\int\,\exp\left[-\frac{\sum_{1\le i<j\le N}\varphi(r_{ij})}{kT}\right]\, d{\bf r^N}= \int\,\prod_{1\le i<j\le N}\exp\left[-\frac{\varphi(r_{ij})}{kT}\right]\,d{\bf r^N},$$ but the equality is only exactly true in the limit $$N/V\rightarrow 0$$.

Defining $$f_{ij}=\exp[-\varphi(r_{ij})/(kT)]-1$$ the integrand can be written as $$\begin{array}{lll} \prod_{i<j}(1+f_{ij}) & = & 1+ \\ &  & f_{12}+f_{13}+f_{23}+f_{14}+\cdots + \\ &  & f_{12}f_{23}+f_{13}f_{23}+f_{12}f_{24}+\cdots +\\ &  & \mbox{etc.} \end{array}$$ where the second line is a sum of terms that denote an interaction of two molecules, the third line is the interaction among three molecules, and so on. For a dilute gas, $$\rho$$ small enough, only interactions between two molecules are important, and in this case the partition function simplifies to $$Q_N=\int\,\left(1+\sum_{i<j}f_{ij}\,\right)\,d{\bf r}_{ij}d{\bf r}^{N-1}$$ Here the differential volume has been separated to emphasize that $$f_{ij}$$ is a function of $${\bf r}_{ij}$$ only. Now all $$3N$$ integrals can be evaluated for the first element of the integrand, and $$3(N-1)$$ integrals can be evaluated for the second giving $$Q_N=V^N+ V^{N-1}\sum_{ii$$), this becomes a single integral in which, because the molecules are spherical, the angular coordinates in $$d{\bf r}=r^2\sin\theta drd\theta d\phi$$ have also been integrated $$Q_N= V^N\left[1+\frac{N(N-1)}{2V}(4\pi) \int_0^\infty\,f(r)r^2\,dr\right]$$ With $$N\gg 1$$, then $$N(N-1)\approx N^2$$, and this is written finally as $$Q_N=V^N\left[1-\frac{N^2B(T)}{N_AV}\right]\quad\mbox{where}\quad B(T)=-2\pi N_A\int_0^\infty\,f(r)r^2\,dr$$ This form for $$Q_N$$ produces $$ p=kT\partial_V\left\{N\ln V+\ln\left[1-\frac{N^2B(T)}{N_AV}\right]\right\}$$

Now the infinite series $$\ln(1+x)=\sum_{i=1}^\infty(-1)^{i+1}x^i/i$$ converges for$$-1<x\le 1$$ so for small enough molar density, $$\rho=N/(N_AV)$$, this becomes more simply $$p=NkT\partial_V\left[\ln V-\frac{NB(T)}{N_AV}\right]$$ Here only the first term of the logarithmic series, linear in $$\rho=N/(N_AV)$$, has been written. All the remaining terms are $$O(\rho^2)$$ meaning they approach $$0$$ at the same rate as $$\rho^2$$. They are not included because terms of this order have already been dropped, namely those that represent the interaction of three or more molecules. Carrying out the differentiation gives the pressure as, $$p=NkT/V+N^2kTB(T)/(N_AV^2)=\rho RT[1+\rho B(T)],$$ This is just two terms of a virial equation of state, and $$B(T)$$ is called the second virial coefficient. Retaining the $$O(\rho^2)$$ terms that were dropped would have produced the entire virial equation of state, and this shows that the $$i$$th term of the expansion contains $$i$$ molecule force interactions. For the dilute gas described here $$\rho B(T)\ll 1$$, and the higher order terms are negligible. Recall that $$B(T)=-2\pi N_A\int_0^\infty\,f(r) r^2\,dr$$ where $$f(r)=\exp[-\varphi(r)/(kT)]-1$$. A characteristic $$\varphi(r)$$ is shown in dimensionless form in the accompanying plot. It is positive for $$r< \sigma$$, and negative for $$r>\sigma $$ with minimum $$\varphi(r_0)=-\epsilon$$ at some $$r_0>\sigma$$. Furthermore $$\varphi$$ increases so rapidly that whenever $$r<\sigma$$ then $$\exp[-\varphi(r)/(kT)]\approx 0$$. In addition for $$\epsilon/(kT)\ll 1$$, the normal case except when $$T$$ is near $$0$$, the exponential can be approximated for $$r>\sigma$$ by two terms of its power series expansion. In these circumstances $$f(r)$$ can be approximated as $$f(r)=\left\{\begin{array}{lll} -1 &                                                              & r<\sigma \\ -\varphi/(kT)=-\epsilon/(kT)\bar{\varphi}(r) & & r>\sigma \end{array}\right.$$ where $$\bar{\varphi}=\varphi/\epsilon$$ has the minimum value of $$-1$$. Then, on evaluating the integral between $$0$$ and $$\sigma$$, the second virial coefficient can be written as, $$B(T)=2\pi\sigma^3 N_A/3-2\pi\sigma^3 N_A\epsilon/(kT)\int_1^\infty\,|\bar{\varphi}(x)| x^2\,dx$$ Now $$2\pi\sigma^3/3N_A=4[4\pi/3(\sigma/2)^3]N_A=b$$ so $$B(T)=b-a/(RT)$$ where $$a=IN_A\epsilon b$$, and $$I$$ is a finite numerical factor depending on the dimensionless intermolecular potential function $$I=3\int_1^\infty\, |\bar{\varphi}(x)|x^2,dx$$ With these definitions the two term virial equation of state is $$p=\rho RT[1+\rho b-/\rho a/(RT)]=\rho RT[1+b/v-a/(vRT)]$$

The Taylor expansion of $$(1-\rho b)^{-1}$$ is given by $$(1-\rho b)^{-1}=1+\rho b+O(\rho^2b^2)$$, so when $$\rho b=b/v\ll 1$$ the terms $$O(\rho^2b^2)$$ are ignorable, as has been done throughout this derivation. In that case the expression for $$p$$ can be written equivalently as, $$p=\rho RT[(1+\rho b)^{-1}-\rho a/(RT)]=RT/(v-b)-a/v^2,$$ which is the vdW equation,

According to this derivation the vdW equation is an equivalent of the two term virial equation of statistical mechanics when $$\rho b\ll 1$$. Consequently it has been shown to be valid in a region where the gas is dilute, $$\rho B(T)\ll 1$$, or specifically $$\rho b=b/v\ll 1$$ [the requirement $$\rho a/(RT)=I\epsilon/(kT)(b/v)\ll 1$$ is true whenever $$b/v\ll 1$$]. However,as Goodstein has noted, the most interesting behavior of the vdW equation occurs in the vicinity of the critical point where $$\rho_cb=1/3$$, namely in a region where its validity is questionable. Thus he wrote that "Obviously the value of the van der Waals equation rests principally on its empirical behavior rather than its theoretical foundations."

Yet this very remarkable empirical behavior ,which has been described in earlier sections of this article, provides irreplaceable insights; as Boltzmann noted, "...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."