Van der Waals equation of state

The van der Waals equation of state, developed by Johannes Diderik van der Waals, takes into account two features that are absent in the ideal gas equation of state; the parameter $b$ introduces somehow the repulsive behavior between pairs of molecules at short distances, it represents the minimum molar volume of the system, whereas $a$ measures the attractive interactions between the molecules. The van der Waals equation of state leads to a liquid-vapor equilibrium at low temperatures, with the corresponding critical point.

Equation of state

The van der Waals equation of state can be written as

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.p={\frac {nRT}{V-nb}}-a\left({\frac {n}{V}}\right)^{2}\right.} .

where:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } is the pressure,
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } is the volume,
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } is the number of moles,
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } is the absolute temperature,
• $R$ is the molar gas constant; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = N_A k_B } , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_A } being the Avogadro constant and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} being the Boltzmann constant.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a= \frac{27}{64}\frac{R^2T_c^2}{P_c}}

$b={\frac {RT_{c}}{8P_{c}}}$

Critical point

The critical point for the van der Waals equation of state can be found at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c= \frac{8a}{27bR}} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_c=\frac{a}{27b^2}}

and at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.v_c\right.=3b} .

Dimensionless formulation

If one takes the following quantities

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{p} = \frac{p}{p_c};~ \tilde{v} = \frac{v}{v_c}; ~\tilde{t} = \frac{T}{T_c};}

one arrives at

${\tilde {p}}={\frac {8{\tilde {t}}}{3{\tilde {v}}-1}}-{\frac {3}{{\tilde {v}}^{2}}}$

The following image is a plot of the isotherms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T/T_c} = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the Van der Waals equation of state:

Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the Van der Waals equation of state

Interesting reading

• J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873).

English translation:

• J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930
• Johannes Diderik van der Waals "The Equation of State for Gases and Liquids", Nobel Lecture, December 12, 1910
• Luis Gonzalez MacDowell and Peter Virnau "El integrante lazo de Van der Waals", Anales de la Real Sociedad Española de Química 101 #1 pp. 19-30 (2005)