Van der Waals equation of state
The van der Waals equation of state, developed by Johannes Diderik van der Waals
Contents |
[edit] Equation of state
The van der Waals equation of state can be written as
\[\left(p + \frac{an^2}{V^2}\right)\left(V-nb\right) = nRT\]
where:
- \( p \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( T \) is the absolute temperature,
- \( R \) is the molar gas constant; \( R = N_A k_B \), with \( N_A \) being the Avogadro constant and \(k_B\) being the Boltzmann constant.
- \(a\) and \(b\) are constants that introduce the effects of attraction and volume respectively and depend on the substance in question.
[edit] Critical point
At the critical point one has \(\left.\frac{\partial p}{\partial v}\right|_{T=T_c}=0 \), and \(\left.\frac{\partial^2 p}{\partial v^2}\right|_{T=T_c}=0 \), leading to
\[T_c= \frac{8a}{27bR}\]
\[p_c=\frac{a}{27b^2}\]
\[\left.v_c\right.=3b\]
along with a critical point compressibility factor of
\[\frac{p_c v_c}{RT_c}= \frac{3}{8} = 0.375\]
which then leads to
\[a= \frac{27}{64}\frac{R^2T_c^2}{p_c}\]
\[b= \frac{RT_c}{8p_c}\]
[edit] Virial form
One can re-write the van der Waals equation given above as a virial equation of state as follows:
\[Z := \frac{pV}{nRT} = \frac{1}{1- \frac{bn}{V}} - \frac{an}{RTV} \]
Using the well known series expansion \((1-x)^{-1} = 1 + x + x^2 + x^3 + ...\)
one can write the first term of the right hand side as
\[\frac{1}{1- \frac{bn}{V}} = 1 + \frac{bn}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ... \]
Incorporating the second term of the right hand side in its due place leads to:
\[Z = 1 + \left( b -\frac{a}{RT} \right) \frac{n}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ...\].
From the above one can see that the second virial coefficient corresponds to
\[B_{2}(T)= b -\frac{a}{RT} \]
and the third virial coefficient is given by
\[B_{3}(T)= b^2 \]
[edit] Boyle temperature
The Boyle temperature of the van der Waals equation is given by
\[B_2\vert_{T=T_B}=0 = b -\frac{a}{RT_B} \]
leading to
\[T_B = \frac{a}{bR}\]
[edit] Dimensionless formulation
If one takes the following reduced quantities
\[\tilde{p} = \frac{p}{p_c};~ \tilde{V} = \frac{V}{V_c}; ~\tilde{t} = \frac{T}{T_c};\]
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 \(T/T_c\) = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:
[edit] Critical exponents
The critical exponents of the Van der Waals equation of state place it in the mean field universality class.
[edit] See also
[edit] References
- ↑ 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
- ↑ This expansion is valid as long as \(-1 < x < 1\), which is indeed the case for \(bn/V\)
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)
