Van der Waals equation

The equation is named after Dutch physicist Johannes Diderik van der Waals, who first derived it in 1873 as part of his doctoral thesis.

Van der Waals based the equation on the idea that fluids are composed of discrete particles, which few scientists believed existed.

[3] This application, expanded to treat multi-component mixtures, has extended the predictive ability of the equation to fluids of industrial and commercial importance.

In this arena it has spawned many similar equations in a continuing attempt by engineers to improve their ability to understand and manage these fluids;[4] it remains relevant to the present.

Van der Waals and his contemporaries used an alternative, but equivalent, analysis based on the mean free path between molecular collisions that gave this result.

[20] Also, in Volume 5 of his Lectures on Theoretical Physics, Sommerfeld, in addition to noting that "Boltzmann[21] described van der Waals as the Newton of real gases",[22] 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.

[23] The van der Waals equation has been, and remains, useful because:[24] In addition, its saturation curve has an analytic solution, which can depict the liquid metals (mercury and cesium) quantitatively, and describes most real fluids qualitatively.

[33] 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,[34] displayed an experimentally obtained set of isotherms of carbonic acid, H2CO3, 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).

Van der Waals began work by trying to determine a molecular attraction that appeared in Laplace's theory of capillarity, and only after establishing his equation he tested it using Andrews' results.

Goodstein summarized this contribution of the van der Waals equation as follows:[39]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".

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.

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.

The dashed green lines represent metastable states (superheated liquid and subcooled vapor) that are created in the process of phase transition, have a short lifetime, and 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 metastable states in a section titled "Undercooling, Delayed evaporation".

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 that can be dangerous.

The vertical axis is logarithmic in order to show the behavior at pressures closer to zero, where differences among the various substances (indicated by varying values of

[95] 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 the absence of experimental data, or computer modeling results to estimate their value the empirical combining rules, geometric and algebraic means can be used, respectively:[97]

These expressions come into use when mixing gases in proportion, such as when producing tanks of air for diving[99] and managing the behavior of fluid mixtures in engineering applications.

However, more sophisticated mixing rules are often necessary, in order to obtain satisfactory agreement with reality over the wide variety of mixtures encountered in practice.

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".

[107] However, no mathematically rigorous derivation of the equation over its entire range of molar volume that begins from a statistical mechanical principle exists.

[108][109][110] Goodstein writes, "Obviously the value of the van der Waals equation rests principally on its empirical behavior rather than its theoretical foundation.

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[12] ... the weak point in the study of the equation of state.

In 1949 the first criticism was confirmed by van Hove when he showed that in the thermodynamic limit, hard spheres with finite-range attractive forces have a finite Helmholtz free energy per particle.

This specific form allowed evaluation of the grand partition function, in the thermodynamic limit, in terms of the eigenfunctions and eigenvalues of a homogeneous integral equation.

[123] 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.

It is empirically well established that many systems whose molecules have attractive potentials that are neither long-range nor weak conform nearly quantatively to the Van der Waals model.

The Sutherland potential (orange) represents two hard spheres that attract according to an inverse power law, and the Lennard-Jones potential (black) represents the induced-dipole–induced-dipole interaction of two non-polar molecules. Both are simple realistic molecular models. [ 14 ]
Constant pressure curves of the van der Waals equation. On the green isobar ( ⁠ ⁠ ), the saturated liquid ( ⁠ ⁠ ) and saturated vapor ( ⁠ ⁠ ) points at temperature are given by green circles.
Figure A: The surface generated by the van der Waals equation.
Van der Waals equation on a wall in Leiden
Figure 1: Four isotherms of the van der Waals equation along with the spinodal curve (black dash-dot curve) and the coexistence (saturation) curve (red dash-dot curve), plotted using reduced (dimensionless) variables. The critical point lies at the inflection point on the orange isotherm.
Figure 2a: The coexistence (saturation) curve is the locus of the saturated liquid and vapor states, which smoothly meet at the critical point. The isotherm is also displayed, showing the intersection.
Figure 2b: The black dash-dot curve is the spinodal curve (stability limit) and the blue dash-dot curve is the coexistence (saturation) curve, plotted in the plane.
Figure 3: The family of saturation curves, showing the vdW curve as a member (blue curve). The blue dots are calculated from Lekner's solution. The orange dots are calculated from data in the ASME Steam Tables Compact Edition, 2006.
Figure 4: A plot of the correlation including data from various substances.
Figure 5: Curves of constant enthalpy in this plane have negative slope above this (green) inversion curve, positive slope below it, and zero slope on it; they are S-shaped. A gas entering a throttle at a state corresponding to a point on this curve to the right of its maximum will cool if the final state is below the curve. Right: a close-up of the region between zero and the critical point (1,1), showing the overlap between the inversion curve (green) and saturation curve (dashed purple).
Figure 6: Isotherms plotted against the compressibility factor . The horizontal axis is . The isotherms, spinodal curve, and coexistence curve here are the same as in Fig. 1. Additionally plotted are the isotherm , which has zero slope at the origin, and the isotherm .
Figure 7: Generalized compressibility chart for a van der Waals gas.
Figure 8: The tangent line (black) to the curve (green) at the two points and . The line's slope, given by , is corresponding to . This is the same data as isotherm in Fig. 1. The intercept on the line is , but its numerical value is arbitrary due to a constant of integration.
Figure 9: Isotherms of the vdW equation + the Maxwell construction. The subcritical isotherm ( ) is a composite curve: the black section is composed of heterogeneous states (liquid and vapor), while the green sections contain homogeneous states.