Van der Waals equation

The van der Waals equation is a mathematical formula that describes the behavior of real gases. It is an equation of state that relates the pressure, volume, number of molecules, and temperature in a fluid. The equation modifies the ideal gas law in two ways: first, it considers particles to have a finite diameter (whereas an ideal gas consists of point particles); second, its particles interact with each other (unlike an ideal gas, whose particles move as though alone in the volume).

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 processes.

One explicit way to write the van der Waals equation is:

where ${\displaystyle p}$ is pressure, ${\displaystyle T}$ is temperature, and ${\displaystyle v=V/n=N_{\text{A}}V/N}$ is molar volume, the ratio of volume, ${\displaystyle V}$, to quantity of matter, ${\displaystyle n}$ (${\displaystyle N_{\text{A}}}$ is the Avogadro constant and ${\displaystyle N}$ the number of molecules). Also ${\displaystyle a}$ and ${\displaystyle b}$ are experimentally determinable, substance-specific constants, and ${\displaystyle R=kN_{\text{A}}}$ is the universal gas constant. 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 ${\displaystyle v=V/n}$, and multiplying both sides by ${\displaystyle n}$ it becomes the form that appears in Figure A.

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 ${\displaystyle V_{0}}$; this is what van der Waals used, and he found the total excluded volume was ⁠${\displaystyle B=4NV_{0}}$⁠, namely 4 times the volume of all the particles. The constant ⁠${\displaystyle b=BN_{\text{A}}/N}$⁠, has the dimension of molar volume, [v]. The constant ${\displaystyle a}$ 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, a, 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 ${\displaystyle F=-d\varphi /dr}$, where ${\displaystyle \varphi (r)}$ 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, ⁠${\displaystyle \sigma }$⁠, and a minimum energy, ${\displaystyle -\varepsilon }$ (with ⁠${\displaystyle \varepsilon \geq 0}$⁠). Two of the many such functions that have been suggested are shown in Fig. B.