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Mathematical and Empirical Validity

Over the years 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 are summarized here. They both proceed from the same source, the canonical partition function, $Q(V,T,N)$, of statistical mechanics written for an $N$ particle macroscopic system $$Q_{N}={\frac {{\cal {Z}}_{N}}{N!\,\Lambda ^{3N}}}\qquad {\mbox{with}}\qquad {\cal {Z}}_{N}=\int _{V^{\bf {N}}}\,\exp \left[-{\frac {\Phi ({\bf {r^{N}}})}{kT}}\right]\,d{\bf {r^{N}}}$$ where $\Lambda =h(2\pi mkT)^{-1/2}$ is the DeBroglie wavelength, $Q_{N}(V,T)\equiv Q(v,T,N)$, ${\cal {Z}}_{N}(V,T)\equiv {\cal {Z}}(V,T,N)$ is the $N$ particle configuration integral, $\Phi$ the intermolecular potential energy is a function of the $N$ particle position vectors ${\bf {r^{N}}}={\bf {r}}_{1},{\bf {r}}_{2}\ldots {\bf {r}}_{N}$, and $d{\bf {r^{N}}}=d{\bf {r}}_{1}d{\bf {r}}_{2}\ldots d{\bf {r}}_{N}$ is the volume element of the $V^{\bf {N}}$ dimensional space. The potential is written using the assumption of pairwise additivity, which is valid for most molecule types, in the form $$\Phi ({\bf {r^{N}}})=\sum _{i<j}\,\varphi ({\bf {r}}_{ij})\qquad {\mbox{where}}\qquad {\bf {r}}_{ij}={\bf {r}}_{i}-{\bf {r}}_{j}.$$ The connection with thermodynamics is made through the Helmholtz free energy, $F=-kT\ln Q_{N}$, and then the equation of state follows from $p=-\partial _{V}F|_{T}=kT\partial _{V}\ln {\cal {{Z}_{N}}}$.

Derivation via the virial equation of state

This derivation is simplest when begun from the grand partition function, $Q_{G}(V,T,\mu )$, wchich can be written ain terms of the canonical function, $$Q_{G}(T,V,\mu )=\sum _{{\cal {N}}\geq 0}\,Q_{\cal {N}}(T,V)e^{\mu {\cal {N}}/(kT)}.$$ In this case the connection with thermodynamics is through $pV=kT\ln Q_{G}$, together with the most probable (also the average) number of particles $N=kT\partial _{\mu }\ln Q_{G}|_{T,V}$. Using this connection with thermodynamics, the expression for the grand function in terms of the sum of canonicals, and the expression for the canonical function in terms of the configuration integral produces $$Q_{G}=e^{pV/kT}=1+\sum _{{\cal {N}}\geq 1}{\frac {{\cal {Z}}_{\cal {N}}}{{\cal {N}}!}}z^{\cal {N}}\quad {\mbox{where}}\quad z={\frac {\exp(\mu /kT)}{\Lambda ^{3}}}.\quad {\mbox{Writing}}\quad pV/kT=\sum _{j\geq 1}\,b_{j}(T)Vz^{j},$$ expanding $\exp(pV/kT)$ in its convergent power series and equating powers of $z$ produces relations that can be solved for these coefficients in terms of the configuration integrals, for example $b_{1}(T)={\cal {Z}}_{1}/V=1,\quad b_{2}(T)=({\cal {Z}}_{2}-{\cal {Z}}_{1}^{2})/(2V)$. From $N=kT\partial _{\mu }\ln Q_{G}|_{T,V}=kTV\partial _{\mu }(p/kT)=Vz\partial _{z}(p/kT)$ the number density is expressed as the series $$\rho _{N}=\sum _{j\geq 1}\,jb_{j}(T)z^{j}\quad {\mbox{which can be inverted to give}}\quad z=\sum _{i\geq 1}\,a_{i}\rho _{N}^{i}.$$ The coefficients $a_{i}$ are given in terms of $b_{j}$ by a known formula, or determined simply by substituting $z$ into the series for$\rho _{N}$ and setting powers of $\rho _{N}$ to zero, thus $a_{1}=1/b_{1}=1,a_{2}=-2b_{2},a_{3}=-3b_{3}+8b_{2}^{2},\cdots$. Finally, using this series in the series for $p/kT$ produces the virial expansion , $$p/\rho _{N}kT=Z(\rho _{N},T)=\sum _{i\geq 1}\,B_{k}(T)\rho _{N}^{k-1}\quad {\mbox{where}}\quad B_{1}(T)=1.$$

This conditionally convergent series is also an asymptotic power series for the limit $\rho _{N}\rightarrow 0$, and a finite number of terms is an asymptotic approximation to $Z(\rho _{N},T)$. The dominant order approximation in this limit is $Z\sim 1$, which is the ideal gas law. It can be written as an equality using order symbols, for example $Z=1+o(1)$, which states that the remaining terms approach zero in the limit, or $Z=1+O(\rho _{N})$, which states, more accurately, that they approach zero in proportion to $\rho _{N}$. The two term approximation is $Z=1+B_{2}(T)\rho _{N}+O(\rho _{N}^{2})$, and the expression for $B_{2}(T)$ is $$B_{2}=-b_{2}=-({\cal {Z}}_{2}-{\cal {Z}}_{1}^{2})/(2V)=-(2V)^{-1}\int _{V_{1}}\int _{V_{2}}\,f({\bf {r}}_{12})\,d{\bf {r}}_{1}d{\bf {r}}_{2},$$

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

From its definition $\varphi (r)$ is positive for $r<\sigma$, and negative for $r>\sigma$ with a minimum of $\varphi (r_{0})=-\varepsilon$ at some $r_{0}>\sigma$. Furthermore $\varphi$ increases so rapidly that whenever $r<\sigma$ then $\exp[-\varphi (r)/(kT)]\approx 0$. In addition in the limit $\delta =\varepsilon /kT\rightarrow 0$ 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 $$f(r)=\left\{{\begin{array}{lll}-1&&r<\sigma \\-\delta {\bar {\varphi }}(r)+O[(\delta ^{2}]&&r>\sigma \end{array}}\right.$$ where ${\bar {\varphi }}=\varphi /\varepsilon$ has the minimum value of $-1$. On splitting the interval of integration into 2 parts, one less than and the other greater than $\sigma$, evaluating the first integral, and making the second integration variable dimensionless using $\sigma$ produces, $$B_{2}(T)=2\pi \sigma ^{3}/3+2\pi \sigma ^{3}\delta \int _{1}^{\infty }\,{\bar {\varphi }}(x)x^{2}\,dx+O(\delta ^{2})\sim b_{N}-a_{N}/kT,$$ where $b_{N}=2\pi \sigma ^{3}/3=4[4\pi /3(\sigma /2)^{3}]$ and $a_{N}=\varepsilon b_{N}I$ with $I$ a numerical factor whose value depends on the specific dimensionless intermolecular pair potential $$I=-3\int _{1}^{\infty }\,{\bar {\varphi }}(x)x^{2}\,dx.$$ Here $b_{N}=b/N_{\text{A}},a_{N}=a/N_{\text{A}}^{2}$ where $a,b$ are the constants given in the introduction. The condition that $I$ be finite requires that ${\bar {\varphi }}$ be integrable over the range [1,$\infty$). This result indicates that a dimensionless $B_{2}(\delta )/\sigma ^{3}$ that is a function of a dimensionless molecular temperature ${\bar {T}}=1/\delta$ is a universal function for all real gases with an intermolecular pair potential of the form $\varphi =\varepsilon {\bar {\varphi }}(r/\sigma )$; it is one example of the principle of corresponding states on the molecular level. Moreover this is true in general and has been developed extensively both theoretically and experimentally.

Substituting the expression for $B_{2}(T)$ into the two term approximation produces $$Z=1+[b_{N}-a_{N}/kT+O(\delta ^{2})]\rho _{N}+O(\rho _{N}^{2}b_{N}^{2})\sim 1+(1-a/bRT)\rho b+O(\rho ^{2}b^{2})=1+\rho b-\rho a/RT+O(\rho ^{2}b^{2}).$$ 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 $(1-\rho b)^{-1}$ is given by $(1-\rho b)^{-1}=1+\rho b+O(\rho ^{2}b^{2})$, so substituting for $1+\rho b$ produces $Z=p/\rho RT=(1-\rho b)^{-1}-a\rho /RT+O(\rho ^{2}b^{2})$. Alternatively this is $$p\sim RT\rho /(1+\rho b)-\rho ^{2}a=RT/(v-b)-a/v^{2},$$ the vdW equation.

According to this derivation the vdW equation is an equivalent of the two term approximation of the virial equation of statistical mechanics in the limit $\varepsilon /kT\rightarrow 0$. Consequently the equation produces an accurate approximation in a region defined by $\rho _{N}\ll 1/\sigma ^{3},\,T\gg \varepsilon /k$, corresponding to a dilute gas. However the most useful predictions of the vdW equation are the existence of a critical point and at lower temperatures (and pressures) a discontinuous change in density (phase change). Molecular and thermodynamic measurements have produced, for a collection of simple molecules, the average values, $\rho _{Nc}\sigma ^{3}=0.472,\,\varepsilon /kT_{c}=0.775$. Although each of these is small neither is very small, which is a possible explanation for the qualitative, rather than quantitative, predictions of the vdW equation. On the other hand these limitations may be simply artifacts of the derivation, in the next derivation they are both eliminated, replaced by a third. In any event the very remarkable empirical behavior of the vdW equation, which has been described in earlier sections of this page, 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."

Derivation via the Helmholtz free energy function

This derivation begins with the canonical partition function, however, the intermolecular potential function is split in two $$\Phi ({\bf {r^{N}}})=\sum _{j>i}q(r_{ij})+\sum _{j>i}w(r_{ij}).$$ Here the $q_{ij}$ are short-range repulsive potentials $q(r/\sigma )$ and $w_{ij}$ are long-range attractive ones $w=-(\sigma /\ell )^{3}\varphi \{r/\ell \}=-\varepsilon (\sigma /\ell )^{3}{\bar {\varphi }}(r/\ell )$. The new element here is the decay length, $\ell$ which also appears as a divisor of $\varphi$. Hence in the limit $\ell \rightarrow \infty$, the vdW limit, the decay length becomes infinitely long and simutaneouly the strength of the interaction becomes infinitely weak while $\varepsilon$ remains unchanged. Here $F$ is evaluated from $$F=-kT\ln$$

Then in a double limit process ($\ell ,V\rightarrow \infty$ such that $N/(N_{A}V)=\rho ,N_{A}F/N=f$ 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 $\ell \rightarrow \infty$), the result is $$\lim _{\ell \rightarrow \infty }f(\rho ,T,\ell )={\mbox{CE}}[f^{0}(\rho ,T)-a\rho ]$$ where CE is the complex envelope, which is the greatest convex function which is $\leq$ the function. For example in Fig.8 the CE is the sum of the solid green and solid black curves. Differentiating, making use of $p=-\partial _{v}f|_{T}=\rho ^{2}\partial _{\rho }f|_{T}$ 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.