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| The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]] <ref>J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873)</ref> | | The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]], can be written as |
| <ref>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]</ref>, 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,
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| 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.
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| ==Equation of state==
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| The van der Waals equation of state can be written as
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| :<math>\left(p + \frac{an^2}{V^2}\right)\left(V-nb\right) = nRT</math> | | :<math> \left. p = \frac{ n R T}{V - n b } - a \left( \frac{ n}{V} \right)^2 \right. </math>. |
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| where: | | where: |
| * <math> p </math> is the [[pressure]], | | * <math> p </math> is the pressure |
| * <math> V </math> is the volume,
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| * <math> n </math> is the number of moles,
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| * <math> T </math> is the absolute [[temperature]],
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| * <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]].
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| *<math>a</math> and <math>b</math> are constants that introduce the effects of attraction and volume respectively and depend on the substance in question.
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| ==Critical point==
| | * <math> V </math> is the volume |
| 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>T_c= \frac{8a}{27bR}</math>
| | * <math> n </math> is the number of moles |
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| | * <math> T </math> is the absolute temperature |
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| :<math>p_c=\frac{a}{27b^2}</math>
| | * <math> R </math> is the Gas constant; <math> R = N_A k_B </math>, with <math> N_A </math> being [[Avogadro constant]] |
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| | The van der Waals equation of state 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. |
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| :<math>\left.v_c\right.=3b</math> | | :<math>a= \frac{27}{64}\frac{R^2T_c^2}{P_c}</math> |
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| along with a critical point [[compressibility factor]] of
| | :<math>b= \frac{RT_c}{8P_c}</math> |
| | ==Critical point== |
| | The [[Critical points |critical point]] for the van der Waals equation of state can be found at |
| | :<math>T_c= \frac{8a}{27bR}</math>, |
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| | | :<math>p_c=\frac{a}{27b^2}</math> |
| :<math>\frac{p_c v_c}{RT_c}= \frac{3}{8} = 0.375</math> | | and at |
| | | :<math>\left.v_c\right.=3b</math>. |
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| which then leads to
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| :<math>a= \frac{27}{64}\frac{R^2T_c^2}{p_c}</math>
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| :<math>b= \frac{RT_c}{8p_c}</math> | |
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| ==Virial form==
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| One can re-write the van der Waals equation given above as a [[virial equation of state]] as follows:
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| :<math>Z := \frac{pV}{nRT} = \frac{1}{1- \frac{bn}{V}} - \frac{an}{RTV} </math>
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| Using the well known [http://mathworld.wolfram.com/SeriesExpansion.html series expansion] <math>(1-x)^{-1} = 1 + x + x^2 + x^3 + ...</math>
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| one can write the first term of the right hand side as <ref>This expansion is valid as long as <math>-1 < x < 1</math>, which is indeed the case for <math>bn/V</math> </ref>:
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| :<math>\frac{1}{1- \frac{bn}{V}} = 1 + \frac{bn}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ... </math>
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| Incorporating the second term of the right hand side in its due place leads to:
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| :<math>Z = 1 + \left( b -\frac{a}{RT} \right) \frac{n}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ...</math>.
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| From the above one can see that the [[second virial coefficient]] corresponds to
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| :<math>B_{2}(T)= b -\frac{a}{RT} </math>
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| and the third virial coefficient is given by
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| :<math>B_{3}(T)= b^2 </math>
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| ==Boyle temperature==
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| The [[Boyle temperature]] of the van der Waals equation is given by
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| :<math>B_2\vert_{T=T_B}=0 = b -\frac{a}{RT_B} </math>
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| leading to
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| :<math>T_B = \frac{a}{bR}</math>
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| ==Dimensionless formulation== | | ==Dimensionless formulation== |
| If one takes the following reduced quantities | | If one takes the following quantities |
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| :<math>\tilde{p} = \frac{p}{p_c};~ \tilde{V} = \frac{V}{V_c}; ~\tilde{t} = \frac{T}{T_c};</math> | | :<math>\tilde{p} = \frac{p}{p_c};~ \tilde{v} = \frac{v}{v_c}; ~\tilde{t} = \frac{T}{T_c};</math> |
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| one arrives at | | one arrives at |
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| :<math>\tilde{p} = \frac{8\tilde{t}}{3\tilde{V} -1} -\frac{3}{\tilde{V}^2}</math> | | :<math>\tilde{p} = \frac{8\tilde{t}}{3\tilde{v} -1} -\frac{3}{\tilde{v}^2}</math> |
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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]] |
| ==Critical exponents==
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| The [[critical exponents]] of the Van der Waals equation of state place it in the [[Universality classes#Mean-field | mean field universality class]].
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| ==See also== | | ==Interesting reading== |
| *[[Zeno line#Batchinsky law | Batchinsky law]] | | *[http://store.doverpublications.com/0486495930.html J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930] |
| | *[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) |
| ==References== | | ==References== |
| <references/>
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| '''Related reading'''
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| *[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]
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| *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)
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| [[Category: equations of state]] | | [[Category: equations of state]] |