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 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 b } introduces somehow the repulsive behavior between pairs of molecules at short distances, it represents the minimum molar volume of the system, whereas 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 } 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 (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(p + \frac{an^2}{V^2}\right)\left(V-nb\right) = nRT}

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,
• 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 } 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} 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 b} are constants that introduce the effects of attraction and volume respectively and depend on the substance in question.

Critical point

At the critical point one has 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.\frac{\partial p}{\partial v}\right|_{T=T_c}=0 } , 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 \left.\frac{\partial^2 p}{\partial v^2}\right|_{T=T_c}=0 } , leading to

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}}

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} .

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 \frac{p_cV_c}{T_c}= \frac{3R}{8}}

which then leads to

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}}

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 b= \frac{RT_c}{8P_c}}

Virial form

One can re-write the van der Waals equation given above as a virial equation of state as follows:

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 Z := \frac{pV}{nRT} = \frac{1}{1- \frac{bn}{V}} - \frac{an}{RTV} }

Using the well known series expansion 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 (1-x)^{-1} = 1 + x + x^2 + x^3 + ...} one can write the first term of the right hand side as :

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 \frac{1}{1- \frac{bn}{V}} = 1 + \frac{bn}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ... }

Incorporating the second term of the right hand side in its due place leads to:

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 Z = 1 + \left( b -\frac{a}{RT} \right) \frac{n}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ...} .

From the above one can see that the second virial coefficient corresponds to

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 B_{2}(T)= b -\frac{a}{RT} }

and the third virial coefficient is given by

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 B_{3}(T)= b^2 }

Boyle temperature

The Boyle temperature of the van der Waals equation is given by

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 B_2\vert_{T=T_B}=0 = b -\frac{a}{RT_B} }

leading to

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_B = \frac{a}{bR}}

Dimensionless formulation

If one takes the following reduced 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

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{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

See also

• Batchinsky law

References

Related reading

• 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)