The van der Waals equation is an equation of state that relates the pressure, molar volume, and temperature in fluids. It describes both the liquid and gas states. It was the first successful thermodynamic model to treat fluids as being composed of molecules with finite size and with intermolecular interactions.
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. However, the equation accurately predicted the behavior of a fluid around its critical point, which had been discovered a few years earlier. Its qualitative and quantitative agreement with experiments ultimately cemented its acceptance in the scientific community. These accomplishments won Van der Waals the 1910 Nobel Prize in Physics. Today the equation is recognized as an important model of phase change in fluids.
The van der Waals equation represents fluids whose intermolecular potential can be approximated as hard repulsion with weak attraction at a distance.
Description The van der Waals equation can be written
where is pressure, is the gas constant, is absolute temperature, and is molar volume, the ratio of volume, , to the number of moles in the gas, . The constants and are experimentally determinable, substance-specific constants. This form is useful for plotting isotherms (constant temperature curves).
Van der Waals wrote it in an equivalent, explicit in temperature, form in his thesis (although he could not denote absolute temperature by its modern form in 1873)
This form is useful for plotting isobars (constant pressure curves). Writing , and multiplying both sides by , the equation has the form that appears in Figure A (in which pressure is capitalized and the number of moles is lower cased).
When Van der Waals created his equation, few scientists believed that fluids were composed of rapidly moving particles. Moreover, those who thought so did not know the atomic/molecular structure. The simplest conception of a particle, and the easiest to model mathematically, was a hard sphere of volume ; this is what Van der Waals used, and he found the total excluded volume was , namely 4 times the volume of all the particles. The constant , has the dimension of molar volume, [v]. The constant expresses the strength of the hypothesized inter-particle attraction. Van der Waals only had Newton's law of gravitation, in which two particles are attracted in proportion to the product of their masses, as a model. Thus he argued that, in his case, the attractive pressure was proportional to the density squared. The proportionality constant, , when written in the form used above, has the dimension [pv2] (pressure times molar volume squared).
The force magnitude between two spherically symmetric molecules is written as , where is the pair potential function, and the force direction is along the line connecting the two mass centers. The specific functional relation is most simply characterized by a single length, , and a minimum energy, (with ). Two of the many such functions that have been suggested are shown in Fig. B.
A modern theory based on statistical mechanics produces the same result for obtained by Van der Waals and his contemporaries. It also produces a constant value for when is small enough.
Once the constants and are known for a given substance, the van der Waals equation can be used to predict attributes like the boiling point at any given pressure, and the critical point. These predictions are accurate for only a few substances. Most simple fluids are only a valuable approximation.
Relationship to the ideal gas law
The ideal gas law follows from the van der Waals equation whenever the molar volume is sufficiently large (when , so ), or equivalently whenever the molar density, , is sufficiently small (when , so ).
When is large enough that both inequalities are satisfied, these two approximations reduce the van der Waals equation to , or . Using the definition given after Eq (1a), this becomes ; either of these forms expresses the ideal gas law. This is unsurprising since the van der Waals equation was constructed from the ideal gas equation to obtain an equation valid beyond the low-density limit of ideal gas behavior.
What is truly remarkable is the extent to which Van der Waals succeeded. Indeed, Paul Sophus Epstein in his classic thermodynamics textbook began his discussion of the van der Waals equation by writing, "Despite its simplicity, it comprehends both the gaseous and the liquid state and brings out, in a most remarkable way, all the phenomena pertaining to the continuity of these two states". Also, in Volume 5 of his Lectures on Theoretical Physics, Arnold Sommerfeld, in addition to noting that "BoltzmannBoltzmann, L. Enzykl. der Mathem. Wiss., V, (1), 550 described Van der Waals as the Newton of real gases", 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.
History
The first to propose a volume correction to Boyle's law was Daniel Bernoulli in his microscopic theory in Hydrodynamica, however this model was mostly ignored in 1738.
In 1857 Rudolf Clausius published The Nature of the Motion which We Call Heat. In it he derived the relation for the pressure in a gas, composed of particles in motion, with number density , mass , and mean square speed . He then noted that using the classical laws of Boyle and Jacques Charles, one could write with a constant of proportionality . Hence temperature was proportional to the average kinetic energy of the particles. This article inspired further work based on the twin ideas that substances are composed of indivisible particles, and that heat is a consequence of the particle motion; movement that evolves according to Newton's laws. The work, known as the kinetic theory of gases, was done principally by Clausius, James Clerk Maxwell, and Ludwig Boltzmann. At about the same time, Josiah Willard Gibbs advanced the work by converting it into statistical mechanics.
This environment influenced Johannes Diderik van der Waals. After initially pursuing a teaching credential, he was accepted for doctoral studies at the University of Leiden under Pieter Rijke. This led, in 1873, to a dissertation that provided a simple, particle-based equation that described the gas-liquid change of state, the origin of a critical temperature, and the concept of corresponding states. The equation is based on two premises: first, that fluids are composed of particles with non-zero volumes, and second, that at a large enough distance each particle exerts an attractive force on all other particles in its vicinity. Boltzmann called these forces van der Waals cohesive forces.
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, displayed an experimentally obtained set of isotherms of carbonic acid, , 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). Andrews called the isotherm at which the jump disappears the critical point. Given the similarity of the titles of this paper and Van der Waals' subsequent thesis, one might think that Van der Waals set out to develop a theoretical explanation of Andrews' experiments; however, this is not what happened. 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.
By 1877 sprays of both liquid oxygen and liquid nitrogen had been produced, and a new field of research, low-temperature physics, had been opened. The van der Waals equation played a part in all this, especially for the liquefaction of hydrogen and helium which was finally achieved in 1908. From measurements of and in two states with the same density, the van der Waals equation produces the values
Thus from two such measurements of pressure and temperature, one could determine and , and from these values calculate the expected critical pressure, temperature, and molar volume. David Goodstein summarized this contribution of the van der Waals equation as follows: 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 Physics in 1910, in recognition of the contribution of his formulation of this "equation of state for gases and liquids."
Use
Van der Waals assumed that the molecules represented in the equation of state had hard sphere repulsion and weak attraction at distance. The Sutherland potential is an example of this interaction. Many liquid metals are approximated by the Sutherland potential; and some are well represented by van der Waals’ equation of state.
The van der Waals equation has been, and remains, useful because: Its coefficient of thermal expansion has a simple analytic expression It explains the existence of the critical point, and establishes the theorem of corresponding states Its internal energy and entropy have simple analytic expressions Its specific heat at constant volume is a function of only Its specific heat at constant pressure, has a simple relationship with Its Joule–Thomson coefficient and associated inversion curve, which are instrumental in the commercial liquefaction of gases, have simple analytic expressions Together with the Maxwell construction it explains the existence of the liquid–vapor phase transition, including the observed metastable states In addition Its enthalpy and free energies all have simple analytic expressions Its isothermal compressibility has a simple analytic expression Its saturation curve has a simple analytic parametric solution It is an intermediate mathematical model that is useful as a pedagogical tool when teaching physics, chemistry, and engineering
and It plays an important role in the modern theory of phase transitions It is the completely accurate equation of state for substances whose intermolecular potential matches the Sutherland potential
Critical point and corresponding states
Figure 1 shows four isotherms of the van der Waals equation (abbreviated as vdW) on a (pressure, molar volume) plane. The essential character of these curves is that they come in three forms:
At some critical temperature (orange isotherm), the slope is negative everywhere except at a single inflection point: the critical point , where both the slope and curvature are zero, . At higher temperatures (red isotherm), the isotherm's slope is negative everywhere. (This corresponds to values of for which the vdW equation has one real root for ). At lower temperatures (green and blue isotherms), all isotherms have two points with zero slope. (This corresponds to values of , for which the vdW equation has three real roots for ).