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Over the years since van der Waals presented his thesis, "[m]any derivations, pseudo-derivations, and plausibility arguments have been given" for it.[1] However, of these only two are mathematically rigorous and are summarized here. They both proceed from the same source, the canonical partition function, , of statistical mechanics written for an particle macroscopic system
where is the DeBroglie wavelength, , is the particle configuration integral, the intermolecular potential energy is a function of the particle position vectors , and is the volume element of the dimensional space. The potential is written using the assumption of pairwise additivity,[2] which is valid for most molecule types,[3] in the form
The connection with thermodynamics is made through the Helmholtz free energy, , and then the equation of state follows from .[4][5][6]
This derivation is simplest when begun from the grand partition function, , wchich can be written ain terms of the canonical function,[7]
In this case the connection with thermodynamics is through , together with the most probable (also the average) number of particles . 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
expanding in its convergent power series and equating powers of produces relations that can be solved for these coefficients in terms of the configuration integrals, for example . From
the number density is expressed as the series The coefficients are given in terms of by a known formula, or determined simply by substituting into the series for and setting powers of to zero, thus . Finally, using this series in the series for produces the virial expansion ,[8]
This conditionally convergent series is also an asymptotic power series for the limit , and a finite number of terms is an asymptotic approximation to .[9] The dominant order approximation in this limit is , which is the ideal gas law. It can be written as an equality using order symbols,[10] for example , which states that the remaining terms approach zero in the limit, or , which states, more accurately, that they approach zero in proportion to . The two term approximation is , and the expression for is
where and is a dimensionless two particle potential function. For spherically symetric molecules this function can be represented most simply with two parameters, , a characteristic molecular diameter, and binding energy respectively as shown in the accompanying plot in which . Also for spherically symetric molecules 5 of the 6 integrals in the expression for can be done with the result
From its definition is positive for , and negative for with a minimum of at some . Furthermore increases so rapidly that whenever then . In addition in the limit the exponential can be approximated for by two terms of its power series expansion. In these circumstances can be approximated as
where has the minimum value of . On splitting the interval of integration into 2 parts, one less than and the other greater than , evaluating the first integral, and making the second integration variable dimensionless using produces,[11][12]
where and with a numerical factor whose value depends on the specific dimensionless intermolecular pair potential
Here where are the constants given in the introduction. The condition that be finite requires that be integrable over the range [1,). This result indicates that a dimensionless that is a function of a dimensionless molecular temperature is a universal function for all real gases with an intermolecular pair potential of the form ; it is one example of the principle of corresponding states on the molecular level.[13] Moreover this is true in general and has been developed extensively both theoretically and experimentally.[14][15]
Substituting the expression for into the two term approximation produces
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 is given by , so substituting for produces . Alternatively this is
the vdW equation.[16]
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 . Consequently the equation produces an accurate approximation in a region defined by , 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,[17] the average values, . 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."[18]
This derivation begins with the canonical partition function, however, the intermolecular potential function is split in two
Here the are short-range repulsive potentials and are long-range attractive ones . The new element here is the decay length, which also appears as a divisor of . Hence in the limit , the vdW limit, the decay length becomes infinitely long and simutaneouly the strength of the interaction becomes infinitely weak while remains unchanged. Here is evaluated from
Then in a double limit process ( such that are finite and ), in which the attractive part of the intermolecular potential becomes infinitely weak and infinitely long-range (as ), the result is
where CE is the complex envelope, which is the greatest convex function which is the function. For example in Fig.8 the CE is the sum of the solid green and solid black curves. Differentiating, making use of produces
This is the vdW equation, since is calculated for ; however, because of the CE the subcritical isotherms (see Fig.1) are cut off at 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.[19][20]