Van der Waals equation

The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them (both of which depend on the specific substance). As a result the equation is able to model the liquid–vapor phase change; it is the first equation that did this, and consequently it had a substantial impact on physics at that time. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is^[1][2][3]

$${\displaystyle p={\frac {RT}{v-b}}-{\frac {a}{v^{2}}},}$$

where ${\displaystyle p}$ is pressure, ${\displaystyle T}$ is temperature, and ${\displaystyle v=VN_{\text{A}}/N}$ is molar volume, ${\displaystyle N_{\text{A}}}$ is the Avogadro constant, ${\displaystyle V}$ is the volume, and ${\displaystyle N}$ is the number of molecules (the ratio ${\displaystyle N/N_{\text{A}}}$ is called the amount of substance). In addition, ${\displaystyle R=N_{\text{A}}k}$ is the universal gas constant, ${\displaystyle k}$ is the Boltzmann constant, and ${\displaystyle a}$ and ${\displaystyle b}$ are experimentally determinable, substance-specific constants.

The constant ${\displaystyle a}$ expresses the strength of the molecular interactions. It has dimension of pressure times molar volume squared [pv²], which is also molar energy times molar volume. The constant ${\displaystyle b}$ denotes an excluded molar volume; it is some multiple of the molecular volume, because the centers of two hard spheres can never be closer than their diameter. It has dimension molar volume [v].

A theoretical calculation of these constants at low density for spherical molecules with an interparticle potential characterized by a length ${\displaystyle \sigma }$ and a minimum energy ${\displaystyle -\varepsilon }$ (with ${\displaystyle \varepsilon \geq 0}$), as shown in the accompanying plot produces ${\displaystyle b=4N_{\text{A}}[(4\pi /3)(\sigma /2)^{3}]}$. Multiplying this by the number of moles, ${\displaystyle N/N_{\text{A}}}$, gives the excluded volume as 4 times the volume of all the molecules.^[4] This theory also produces ${\displaystyle a=IN_{\text{A}}\varepsilon b,}$ where ${\displaystyle I}$ is a number that depends on the shape of the potential function ${\displaystyle \varphi (r)/\varepsilon }$.^[5]

In his book Boltzmann wrote equations using ${\displaystyle V/M}$ (specific volume) in place of ${\displaystyle VN_{\text{A}}/N}$ (molar volume) used here;^[6] Gibbs did as well, so do most engineers. Also the property ${\displaystyle V/N=1/\rho _{N},}$ the reciprocal of number density, is used by physicists, but there is no essential difference between equations written with any of these properties. Equations of state written using molar volume contain ${\displaystyle R}$, while those using specific volume contain ${\displaystyle R/{\bar {m}}}$ (where ${\displaystyle {\bar {m}}=N_{\text{A}}m_{p}}$ is the molar mass of a substance whose particle mass is ${\displaystyle m_{p}}$), and those written with number density contain ${\displaystyle k}$.

Once ${\displaystyle a}$ and ${\displaystyle b}$ are experimentally determined for a given substance, the van der Waals equation can be used to predict the boiling point at any given pressure, the critical point (defined by pressure and temperature values, ${\displaystyle p_{\text{c}}}$, ${\displaystyle T_{\text{c}}}$ such that the substance cannot be liquefied either when ${\displaystyle p>p_{\text{c}}}$ no matter how low the temperature is, or when ${\displaystyle T>T_{\text{c}}}$ no matter how high the pressure is), and other attributes. These predictions are accurate for only a few substances. For most simple fluids they are only a valuable approximation. The equation also explains why superheated liquids can exist above their boiling point and subcooled vapors can exist below their condensation point.

The graph on the right is a plot of ${\displaystyle T}$ vs ${\displaystyle v}$ calculated from the equation at four constant pressure values. On the red isobar, ${\displaystyle p=2p_{\text{c}}}$, the slope is positive over the entire range, ${\displaystyle b\leq v<\infty }$ (although the plot only shows a finite quadrant). This describes a fluid as a gas for all ${\displaystyle T}$, and is characteristic of all isobars ${\displaystyle p>p_{\text{c}}.}$ The green isobar, ${\displaystyle p=0.2\,p_{\text{c}}}$, has a physically unreal negative slope, hence shown dotted gray, between its local minimum, ${\displaystyle T_{\text{min}},v_{\text{min}}}$, and local maximum, ${\displaystyle T_{\text{max}},v_{\text{max}}}$. This describes the fluid as two disconnected branches; a gas for ${\displaystyle v\geq v_{\text{min}}}$, and a denser liquid for ${\displaystyle v\leq v_{\text{max}}\leq v_{\text{min}}}$.^[7]

The thermodynamic requirements of mechanical, thermal, and material equilibrium together with the equation specify two points on the curve, ${\displaystyle (T_{s},v_{f})}$, and ${\displaystyle (T_{s},v_{g})}$, shown as green circles that designate the coexisting boiling liquid and condensing gas respectively. Heating the fluid in this state increases the fraction of gas in the mixture; its ${\displaystyle v}$, an average of ${\displaystyle v_{f}}$ and ${\displaystyle v_{g}}$ weighted by this fraction, increases while ${\displaystyle T_{s}}$ remains the same. This is shown as the dotted gray line, because it does not represent a solution of the equation; however, it does describe the observed behavior. The points above ${\displaystyle T_{s}}$, superheated liquid, and those below it, subcooled vapor, are metastable; a sufficiently strong disturbance causes them to transform to the stable alternative (like a ball trapped in a local minimum of a sloping curve that has a lower minimum; the ball has a higher energy than the minimum possible, but can only get there by a push that gets it over the local hill). Consequently they are shown dashed. Finally the points in the region of negative slope are unstable. All this describes a fluid as a stable gas for ${\displaystyle T>T_{\text{s}}}$, a stable liquid for ${\displaystyle T<T_{\text{s}}}$, and a mixture of liquid and gas at ${\displaystyle T=T_{\text{s}}}$, that also supports metastable states of subcooled gas and superheated liquid. It is characteristic of all isobars ${\displaystyle 0<p<p_{\text{c}}}$, where ${\displaystyle T_{\text{s}}}$ is a function of ${\displaystyle p}$.^[8] The orange isobar is the critical one on which the minimum and maximum are equal. The black isobar is the limit of positive pressures, although drawn solid none of its points represent stable solutions, they are either metastable (positive or zero slope) or unstable (negative slope. All this is a good explanation of the observed behavior of fluids.