User:Airman72/sandbox

Mathematical and Empirical Validity

Since van der Waals presented his thesis, "[m]any derivations, pseudo-derivations, and plausibility arguments have been given" for it. 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. 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.

Review

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

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

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

... 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 verified by van Hove when he proved 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 $f,v$ are molar quantities). In addition its derivative exists and defines the pressure, which is a non increasing function of the volume per particle. 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,, that a system of $N$ hard rods of length $\delta$, constrained to move along a straight line of length $L>N\delta ,$, and exerting only direct contact forces on one another satisfy a vdW equation with $a=0$; Rayleigh also knew this. Later Tonks by evaluating the configuration integral, showed that the force exerted on a wall by this system is given by, $F_{W}=NkT/(L-N\delta )=kT/(l-\delta )\,{\mbox{with}}\,l=L/N.$ This can be put in a more recognizable, molar, form by dividing by the rod cross sectional area $A$, and defining $N_{A}AL/N=v>b=N_{A}A\delta$. This produces $p=RT/(v-b)$; clearly there is no condensation, $\partial _{v}p|_{T}<0$ for all $b\leq v<\infty$. 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, $x_{i},\,i=1,N$ satisfy the relations $\delta /2\leq x_{1}\leq x_{2}-\delta ,\cdots ,\leq x_{N}\leq L-\delta /2$. As a result the configuration integral is simply ${\cal {Z}}_{N}=(L-N\delta )^{N}$.

In 1959 this one-dimensional gas model was extended by Kac to include particle pair interactions through an attractive potential, $\varphi (x)=-\varepsilon \exp(-x/\ell ),\,x\geq \delta$. This specific form allowed evaluation of the grand partition function, $$Q_{G}(T,V,\mu )=\sum _{{\cal {N}}\geq 0}\,Q_{\cal {N}}(T,V)e^{\mu {\cal {N}}/kT},$$ in the thermodynamic limit, in terms of the eigenfunctions and eigenvalues of a homogeneous integral equation. 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.

Figure 9: Shows a subcritical isotherm of the vdW equation + the Maxwell construxtion. It is colored in green with a black section that is rendered in a different color because it is composed of heterogeneous states, liquid and vapor; the green sections of the curve contain only homogeneous states.

Four years later, in 1963, Kac together with Uhlenbeck and Hemmer modified the pair potential of Kac's previous work as $\varphi (x)=-\varepsilon (\delta /\ell )\exp(-x/\ell ),\,x\geq \delta$, so that $$a_{N}=-\int _{0}^{\infty }\,\varphi (x)\,dx=\varepsilon \delta$$ was independent of $\ell$. They found, that a second limiting process they called the van der Waals limit, $\ell \rightarrow \infty$ (in which the pair potential becomes both infinitely long range and infinity weak) and performed after the thermodynamic limit, produced the vdW equation, here rendered in molar form, $$p={\frac {RT}{v-b}}-{\frac {a}{v^{2}}}\quad {\mbox{in which}}\quad a=N_{A}^{2}Aa_{N}=N_{A}\varepsilon b$$ together with the Gibbs criterion, $\mu _{f}=\mu _{g}$ (equivalently the Maxwell construction). As a result all isotherms satisfy the condition $\partial _{v}p|_{T}\leq 0$ as shown in Fig. 9, and hence the first criticism of the vdW equation is not as serious as originally thought.

Then, in 1966, Lebowitz and Penrose generalized what they called the Kac potential to apply to an arbitrary number, $\nu$, of dimensions, $\varphi =-\varepsilon (\sigma /\ell )^{\nu }{\bar {\varphi }}(x/\ell )$. For $\nu =1$ and ${\bar {\varphi }}=\exp(-x/\ell )$ this reduces to the one-dimensional case considered by Kac, et. al. and for $\nu =3$ it is physical three dimensional space. The function ${\bar {\varphi }}(x/\ell )$ is a bounded, non-negative function whose integral $$a_{N}=-(1/2)\int _{V^{\nu }}\,\varphi (r/\ell )\,d{\bf {r}}^{\nu }=(\varepsilon \sigma ^{\nu }/2)\int _{V^{\nu }}\,{\bar {\varphi }}(x)\,d{\bf {x}}^{\nu }$$ is finite, independent of $\ell$. By obtaining upper and lower bounds on ${\cal {Z}}(V,T,N,\ell )$ and hence on $F$, taking the thermodynamic limit ($\lim N,V\rightarrow \infty {\mbox{ with }}N/V{\mbox{ finite}}$) to obtain upper and lower bounds on the function $F(V,T,N/V,\ell )/V$, 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, $$f(v,T)={\mbox{CE}}\{f^{0}(v,T)-a/v\}.$$ The abbreviation CE is for convex envelope which denotes the largest convex function that is less than or equal to the function. The function $f^{0}(v,T)$ is the limit function when $\varphi =0$; also here $a=N_{A}^{2}a_{N}$. This result is illustrated in the present context by the solid green curves and black line in Fig. 8. The corresponding limit for the pressure is a generalized form of the vdW equation $$p(v,T)=p^{0}(v,T)-a/v^{2}$$ together with the Gibbs criterion, $\mu _{f}=\mu _{g}$ (equivalently the Maxwell construction). Here $p^{0}=-\partial _{v}f^{0}|_{T}$ 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 $p^{0},$ the pressure that results from direct particle collisions (or more accurately the core repulsive forces), replaces $RT/(v-b)$. This is consistent with the second criticism that can be stated as $p^{0}\neq RT/(v-b){\mbox{ for all }}b\leq v<\infty$. Consequently the vdW equation cannot be rigorously derived from the configuration integral over the entire range of $v$.

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 $\lim v/b\rightarrow \infty$.

The virial equation of state

This derivation is simplest when begun from the grand partition function, $Q_{G}(V,T,\mu )$ (see above),

In this case the connection with thermodynamics is through $pV=kT\ln Q_{G}$, together with the number of particles $N=kT\partial _{\mu }\ln Q_{G}|_{T,V}$. Using the expression for the $Q_{N}$ written above in the series for $Q_{G}$, it becomes $$Q_{G}=e^{pV/kT}=1+\sum _{{\cal {N}}\geq 1}{\frac {{\cal {Z}}_{\cal {N}}}{{\cal {N}}!}}z^{\cal {N}}\quad {\mbox{where}}\quad z={\frac {\exp(\mu /kT)}{\Lambda ^{3}}}.\quad {\mbox{Writing}}\quad p/kT=\sum _{j\geq 1}\,b_{j}(T)z^{j},$$ expanding $\exp(pV/kT)$ in its convergent power series, using the series for $p/kT$ in each term, and equating powers of $z$ produces relations that can be solved for the $b_{j}$ in terms of the ${\cal {Z}}_{j}$. For example $b_{1}(T)={\cal {Z}}_{1}/V=1$, $b_{2}(T)=({\cal {Z}}_{2}-{\cal {Z}}_{1}^{2})/(2!V)$, and $b_{3}=({\cal {Z}}_{3}-3{\cal {Z}}{\cal {Z}}_{2}+2{\cal {Z}}_{1}^{2})/(3!V)$. From $N=kT\partial _{\mu }\ln Q_{G}|_{T,V}=kTV\partial _{\mu }(p/kT)=Vz\partial _{z}(p/kT)$, the number density, $\rho _{N}=N/V$, 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 equating powers of $\rho _{N}$, 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,, or virial equation of state $$p/\rho _{N}kT=Z(\rho _{N},T)=\sum _{i\geq 1}\,B_{k}(T)\rho _{N}^{k-1}\quad {\mbox{where}}\quad B_{1}(T)=1.$$

The second virial coefficient $B_{2}(T)$

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 $\tau =\varepsilon /kT\rightarrow 0$ ($\tau$ is a dimensionless coldness, and the quantity $\varepsilon /k$ is a characteristic molecular temperature) 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 $$f(r)=\left\{{\begin{array}{lll}-1&&r<\sigma \\-\tau {\bar {\varphi }}(r)+O[(\tau ^{2}]&&r>\sigma \end{array}}\right.$$ 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, $$B_{2}(T)=2\pi \sigma ^{3}/3+2\pi \sigma ^{3}\tau \int _{1}^{\infty }\,{\bar {\varphi }}(x)x^{2}\,dx+O(\tau ^{2})\sim b_{N}-a_{N}/kT,$$ 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 $B_{2}(\tau )/\sigma ^{3}$ that is a function of a dimensionless molecular temperature $\tau$ is a universal function for all real gases with an intermolecular pair potential of the form $\varphi =\varepsilon {\bar {\varphi }}(r/\sigma )$; this is an 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.

The van der Waals Approximation

Substituting the (approximate in $\tau$) expression for $B_{2}(T)$ into the two term virial approximation produces $$Z=1+[b_{N}-a_{N}/kT+O(\tau ^{2})]\rho _{N}+O(\rho _{N}^{2}b_{N}^{2})\sim 1+(1-a/bRT)\rho b+O(\rho ^{2}b^{2})+O(\tau ^{2}\rho b)=1+\rho b-\rho a/RT+O(\rho ^{2}b^{2})+O(\tau ^{2}\rho b).$$ 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 $(1-\rho b)^{-1}$, uniformly convergent for $\rho b\ll 1$, can be written as $(1-\rho b)^{-1}=1+\rho b+O(\rho ^{2}b^{2})$, so substituting for $1+\rho b$ produces $Z=p/\rho RT=(1-\rho b)^{-1}-a\rho /RT+O(\rho ^{2}b^{2})+O(\tau ^{2}\rho b)$. Alternatively this is $$p\sim RT\rho /(1+\rho b)-\rho ^{2}a=RT/(v-b)-a/v^{2},$$ the vdW equation.

Summary

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 $\tau ,\rho b\rightarrow 0$. Consequently the equation produces an accurate approximation in a region defined by $v\gg b,\,T\gg \varepsilon /k$ (on a molecular basis $\rho _{N}\sigma ^{3}\ll 1$), 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, and Rayleigh (see Rowlinson) 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. Moreover, 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) in order to manage the liquids, and gases of pure substances and mixtures, they encounter in practice.

This situation has been described by Boltzmann most aptly as follows:

...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.

References

• Hill, Terrell L. (1986). Statistical Thermodynamics. NY: Dover.
• Kac, Marc (1958). "On the Partition Function of a One-Dimensional Gas". Phys Fluids. 1: 8–12.
• Kac, M.; Uhlenbeck, G.E.; Hemmer, P.C. (1963). "On the van der Waals Theory of the Vapor-Liquid Equilibrium. 1. Discussion of a One-Dimensional Model". J. Math. Phys. 4: 216–228.
• Kontogeorgis, G.M.; Privat, R.; Jaubert, J-N.J. (2019). "Taking Another Look at the van der Waals Equation of State---Amost 150 Years Later". J. Chem. Eng. Data. 64: 4619–4637.
• Korteweg, D.T. (1891). "On Van Der Waals Isothermal Equation". Nature. 45 (1155): 152–154.
• Korteweg, D.T. (1891). "On Van Der Waals Isothermal Equation". Nature. 45 (1160): 277.
• Lebowitz, J.L. (1974). "Exact Derivation of the Van Der Waals Equation". Physica. 73: 48–60.
• Lebowitz, J.L.; Penrose, O. (1966). "Rigorous Treatment of the Van der Waals-Maxwell Theory of the Liquid-Vapor Transition". Jour Math Phys. 7: 98–113.
• Strutt, J.W., 3rd Baron Rayleigh (1891). "On the Virial of a System of Hard Colliding Bodies". Nature. 45: 80–82.{{cite journal}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
• Tonks, L. (1936). "The Complete Equation of State of One, Two, and Three-Dimensional Gases of Hard Elastic Spheres". Phys. Rev. 50: 955–963.
• Valderrama, J.O. (2003). "The State of the Cubic Equations of State". Ind. Chem. Eng. Res. 42: 1603–1618.
• Vera, J.H.; Prausnitz, J.M. (1972). "Generalized van der Waals Theory for Dense Fluids". Chem. Eng. Jour. 3: 1–13.
• van der Waals, Johannes D. (1967). "The Equation of State of Gases and Liquids". in Nobel Lectures, Physics 1901-1921. Amsterdam: Elsevier. pp. 254–265.
• van Hove, L. (1949). "Quelques Proprieties Generales De L'Integrale De Configuration D'Un Systeme De Particules Avec Interaction". Physica. 15: 951–961.