= Van der Waals equation
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The <Van der Waals equation> is a simple <real gas> <equation of state> that can also describe a <liquid-gas critical point>. With particle count $N$ fixed, write the <volume per particle> as $v=V/N$:
$$
p=\frac{k_BT}{v-b}-\frac{a}{v^2},\qquad v>b,\quad a>0,\quad b>0,
$$
or equivalently $(p+aN^2/V^2)(V-Nb)=Nk_BT$. Here $p$ is <pressure>, $T$ is absolute <temperature>, and $k_B$ is the <Boltzmann constant>. In <SI>, $v$ and $b$ have units $\mathrm{m^3}$ per particle, $a$ has units $\mathrm{Pa\,m^6}=\mathrm{J\,m^3}$, and $k_B$ has units $\mathrm{J\,K^{-1}}$; the particle count is dimensionless. The pole at $v=b$ excludes smaller <volume (thermodynamics)> values from this model.
The $b$ correction represents <excluded volume>; the attraction term lowers <pressure> at fixed <temperature> and particle <number density> $1/v$. This is a <mean-field theory>, replacing detailed intermolecular structure by two parameters. At $v\gg b$ with $a/v^2\ll k_BT/v$, it approaches the <ideal gas law> $pv=k_BT$. It supplies a useful qualitative route from a dilute <gas> to <phase coexistence>, without a claim of quantitative accuracy for every <fluid>.<Thermodynamics and Statistical Mechanics by Richard Fitzpatrick/Van der Waals Gas>[opening explanation and equation 6.156: excluded volume and attractive pressure correction; Fitzpatrick uses molar volume]{ref}
Do not mix this particle convention with the <molar volume> convention elsewhere in this article. With <amount of substance> $n_{\rm mol}=N/N_A$, <Avogadro constant> $N_A$, and <molar gas constant> $R=N_Ak_B$, one has $v_m=N_Av$, $b_m=N_Ab$, and $a_m=N_A^2a$, so $p=RT/(v_m-b_m)-a_m/v_m^2$. The molar parameters have units $\mathrm{m^3\,mol^{-1}}$ and $\mathrm{Pa\,m^6\,mol^{-2}}$, respectively.
<Johannes Diderik van der Waals> introduced this <equation of state> in 1873, connecting molecular attraction with departures from the dilute <ideal gas law>. Its historical significance is the unified treatment of <gases> and <liquids>, rather than a universally exact fit.<Johannes Diderik van der Waals, Facts by Nobel Prize Outreach>[Nobel Prize Outreach, Johannes Diderik van der Waals, Work: equation introduced in 1873 and molecular attraction]{ref}
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