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The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]] | The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]], takes into account two features that are absent in the [[Equation of State: Ideal Gas | ideal gas]] equation of state; the parameter <math> b </math> introduces somehow the repulsive behavior between pairs of molecules at short distances, | ||
it represents the minimum molar volume of the system, whereas <math> a </math> 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. | it represents the minimum molar volume of the system, whereas <math> a </math> 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== | ==Equation of state== | ||
The van der Waals equation of state can be written as | The van der Waals equation of state can be written as | ||
:<math>\left | :<math> \left. p = \frac{ n R T}{V - n b } - a \left( \frac{ n}{V} \right)^2 \right. </math>. | ||
where: | where: | ||
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* <math> n </math> is the number of moles, | * <math> n </math> is the number of moles, | ||
* <math> T </math> is the absolute [[temperature]], | * <math> T </math> is the absolute [[temperature]], | ||
* <math> R </math> is the [[molar gas constant]]; <math> R = N_A k_B </math>, with <math> N_A </math> being the [[Avogadro constant]] and <math>k_B</math> being the [[Boltzmann constant]]. | * <math> R </math> is the [[molar gas constant]]; <math> R = N_A k_B </math>, with <math> N_A </math> being the [[Avogadro constant]] and <math>k_B</math> being the [[Boltzmann constant]]. | ||
==Critical point== | ==Critical point== | ||
At the [[Critical points |critical point]] one has <math>\left.\frac{\partial p}{\partial v}\right|_{T=T_c}=0 </math>, and <math>\left.\frac{\partial^2 p}{\partial v^2}\right|_{T=T_c}=0 </math>, leading to | At the [[Critical points |critical point]] one has <math>\left.\frac{\partial p}{\partial v}\right|_{T=T_c}=0 </math>, and <math>\left.\frac{\partial^2 p}{\partial v^2}\right|_{T=T_c}=0 </math>, leading to | ||
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:<math>\left. | :<math>\left.V_c\right.=3b</math>. | ||
and | |||
:<math>\frac{ | :<math>\frac{p_cV_c}{T_c}= \frac{3R}{8}</math> | ||
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:<math>a= \frac{27}{64}\frac{R^2T_c^2}{ | :<math>a= \frac{27}{64}\frac{R^2T_c^2}{P_c}</math> | ||
:<math>b= \frac{RT_c}{8P_c}</math> | |||
:<math> | |||
==Dimensionless formulation== | ==Dimensionless formulation== | ||
If one takes the following reduced quantities | If one takes the following reduced quantities | ||
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The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state: | The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state: | ||
[[Image:vdW_isotherms.png|center|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]] | [[Image:vdW_isotherms.png|center|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]] | ||
== | ==Maxwell's equal area construction== | ||
==Interesting reading== | |||
== | |||
= | |||
*[http://nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.pdf Johannes Diderik van der Waals "The Equation of State for Gases and Liquids", Nobel Lecture, December 12, 1910] | *[http://nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.pdf 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) | *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) | ||
==References== | |||
*J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873). | |||
English translation: | |||
*[http://store.doverpublications.com/0486495930.html J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930] | |||
[[Category: equations of state]] | [[Category: equations of state]] |