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 phase change, liquid ${\displaystyle \leftrightarrow }$ vapor. 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]

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

${\displaystyle p}$

${\displaystyle T}$

${\displaystyle v=VN_{\text{A}}/N}$

${\displaystyle N_{\text{A}}}$

${\displaystyle V}$

${\displaystyle N}$

${\displaystyle N/N_{\text{A}}}$

${\displaystyle R=N_{\text{A}}k}$

${\displaystyle k}$

${\displaystyle a}$

${\displaystyle b}$

The constant ${\displaystyle a}$ expresses the strength of the molecular interactions. It has dimension pressure times molar volume squared, which is also molar energy times molar volume. The constant ${\displaystyle b}$ denotes an excluded molar volume due to a finite molecule size, since the centers of two molecules cannot be closer than their diameter, ${\displaystyle \sigma }$. A theoretical calculation of these constants for spherical molecules with an interparticle potential characterized by, ${\displaystyle \sigma }$, and a minimum energy, ${\displaystyle -\varepsilon }$, as shown in the accompanying plot produces ${\displaystyle b=4N_{\text{A}}}$. 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 ${\displaystyle a=I\varepsilon N_{\text{A}}b}$ where ${\displaystyle I}$ is a number that depends on the shape of the potential function, ${\displaystyle \varphi (r)}$.^[5]

In his book (see references and ) Boltzmann wrote equations using ${\displaystyle V/M}$ (specific volume) in place of ${\displaystyle VN_{\text{A}}/N}$ (molar volume) used here, Gibbs did as well, so do most engineers. Also the property, ${\displaystyle V/N}$ 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 ${\displaystyle R}$, those using specific volume contain ${\displaystyle R/{\bar {m}}}$ (the substance specific ${\displaystyle {\bar {m}}}$ is the molar mass), 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, or when ${\displaystyle T>T_{\text{c}}}$ no matter how high the pressure), 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, $$Zdroj:https://en.wikipedia.org?pojem=Van_der_Waals_equation
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