Editing Van der Waals equation of state

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>
<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, 
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==
The van der Waals equation of state can be written as

:<math>\left(p + \frac{an^2}{V^2}\right)\left(V-nb\right) = nRT</math>

where:
* <math> p </math> is the [[pressure]],
* <math> V </math> is the volume,
* <math> n </math> is the number of moles,
* <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>a</math> and <math>b</math> are constants that introduce the effects of attraction and volume respectively and depend on the substance in question.

==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

:<math>T_c= \frac{8a}{27bR}</math>


:<math>p_c=\frac{a}{27b^2}</math>


:<math>\left.v_c\right.=3b</math>


along with a critical point [[compressibility factor]] of 


:<math>\frac{p_c v_c}{RT_c}= \frac{3}{8} = 0.375</math>


which then leads to 


:<math>a= \frac{27}{64}\frac{R^2T_c^2}{p_c}</math>


:<math>b= \frac{RT_c}{8p_c}</math>

==Virial form==
One can re-write the van der Waals equation given above as a [[virial equation of state]] as follows:

:<math>Z :=  \frac{pV}{nRT} = \frac{1}{1- \frac{nb}{V}} - \frac{an}{VRT} </math>

Using the well known [http://mathworld.wolfram.com/SeriesExpansion.html series expansion] <math>(1-x)^{-1} = 1 + x + x^2 + x^3 + ...</math>
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>nb/V</math> </ref>:

:<math>\frac{1}{1- \frac{nb}{V}} = 1 +  \frac{nb}{V} +  \left( \frac{nb}{V} \right)^2 +  \left( \frac{nb}{V} \right)^3 + ... </math>

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

:<math>Z = 1 + \left( b -\frac{a}{RT} \right) \frac{n}{V} +  \left( \frac{nb}{V} \right)^2 + \left( \frac{nb}{V} \right)^3 +   ...</math>.

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

:<math>B_{2}(T)=  b -\frac{a}{RT} </math>

and the third virial coefficient is given by

:<math>B_{3}(T)=  b^2 </math>
==Boyle temperature==
The [[Boyle temperature]] of the van der Waals equation is given by

:<math>B_2\vert_{T=T_B}=0 =  b -\frac{a}{RT_B} </math>

leading to 

:<math>T_B = \frac{a}{bR}</math>
==Dimensionless formulation==
If one takes the following reduced quantities 

:<math>\tilde{p} = \frac{p}{p_c};~ \tilde{V} = \frac{V}{V_c}; ~\tilde{t} = \frac{T}{T_c};</math>

one arrives at

:<math>\tilde{p} = \frac{8\tilde{t}}{3\tilde{V} -1} -\frac{3}{\tilde{V}^2}</math>

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]]
==Critical exponents==
The [[critical exponents]] of the Van der Waals equation of state place it in the [[Universality classes#Mean-field | mean field universality class]].

==See also==
*[[Zeno line#Batchinsky law | Batchinsky law]]
==References==
<references/>
'''Related 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]
*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)
[[Category: equations of state]]