View source for Van der Waals equation

{{short description|Gas equation of state based on plausible reasons why real gases do not follow the ideal gas law.}}
{{Multiple issues|
{{More citations needed|date=June 2015}} in other words-vander waal research approach indicates that volume of gas molecule can not be neglected and gas molecule say b so gas molecules are moving in the remaining volumen(v-b) instead of v i.e, v is replaced by (v-b)
{{format footnotes|date=July 2015}}
{{citation style|date=June 2015}}
{{Expert needed|Fluid dynamics|talk=Final comments and addition of expert needed tag|reason=the derivations are unsourced, and the text needs review by a physical chemistry expert to determine attribution of content|date=June 2015}}
}}
<!--Topics that this article should cover:  ==Liquid-gas phase transition==  ==Fits to experimental data== -->

The '''van der Waals equation''' (or '''van der Waals equation of state'''; named after [[Johannes Diderik van der Waals]]) is an [[equation of state]] that generalizes the [[ideal gas law]] based on plausible reasons that real gases do not act ideally. The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions.<ref name=PhysicalChemistry>{{cite book | first1 = Robert J. | last1 = Silbey | first2 = Robert A. | last2 = Alberty | first3 = Moungi G. | last3 = Bawendi | title = Physical Chemistry | edition = 4th | year = 2004 | publisher = Wiley | isbn = 978-0471215042}}</ref> The [[ideal gas law]] states that volume (''V'') occupied by n moles of any gas has a pressure (''P'') at temperature (''T'') in [[kelvin]]s<!-- lower-case and pluralized – see Kelvin#Usage conventions --> given by the following relationship, where ''R'' is the [[gas constant]]:
: ''PV'' = ''nRT'',

To account for the volume that a real gas molecule takes up, the van der Waals equation replaces ''V'' in the ideal gas law with <math>(V_m-b)</math>, where ''V<sub>m</sub>'' is the [[molar volume]] of the gas and ''b'' is the volume that is occupied by one mole of the molecules. This leads to:<ref name=PhysicalChemistry />

:<math>P(V_m - b)=R T</math>

The second modification made to the ideal gas law accounts for the fact that gas molecules do in fact attract each other and that real gases are therefore more compressible than ideal gases. Van der Waals provided for intermolecular attraction by adding to the observed pressure ''P'' in the equation of state a term <math>a/V_m^2</math>, where ''a'' is a constant whose value depends on the gas. The '''van der Waals equation''' is therefore written as:<ref name=PhysicalChemistry />

:<math>(P+a\frac1{V_m^2})(V_m-b)=R T</math>
and can also be written as the equation below

:<math>(P+a \frac{n^2}{V^2})(V-n b)=n R T</math>

where ''V<sub>m</sub>'' is the [[molar volume]] of the gas, ''R'' is the universal gas constant, ''T'' is temperature, ''P'' is pressure, and ''V'' is volume. When the molar volume ''V<sub>m</sub>'' is large, ''b'' becomes negligible in comparison with ''V<sub>m</sub>'', ''a/V<sub>m</sub><sup>2</sup>'' becomes negligible with respect to ''P'', and the van der Waals equation reduces to the ideal gas law, ''PV<sub>m</sub>=RT''.<ref name=PhysicalChemistry />

It is available via its traditional derivation (a mechanical equation of state), or via a derivation based in [[statistical thermodynamics]], the latter of which provides the [[partition function (statistical mechanics)|partition function]] of the system and allows thermodynamic functions to be specified. It successfully approximates the behavior of real fluids above their [[critical temperature]]s and is qualitatively reasonable for their liquid and low-pressure gaseous states at low temperatures. However, near the [[phase transition|transition]]s between gas and liquid, in the range of ''p'', ''V'', and ''T'' where the liquid [[phase (matter)|phase]] and the [[gas]] phase are in equilibrium, the ''van der Waals equation'' fails to accurately model observed experimental behaviour, in particular that ''p'' is a constant function  of ''V'' at given temperatures. As such, the van der Waals model is not useful only for calculations intended to predict real behavior in regions near the critical point. Empirical corrections to address these predictive deficiencies have been inserted into the van der Waals model, e.g., by [[James Clerk Maxwell]] in his [[#Maxwell equal area rule|equal area rule]], and related but distinct theoretical models, e.g., based on the [[principle of corresponding states]], have been developed to achieve better fits to real fluid behaviour in equations of more comparable complexity.

==Overview and history==
{{expand section|a proper lay explanation of the equation and the history and context of its discovery|small=no|date=June 2015}}

The van der Waals equation is a [[thermodynamic]] [[equation of state]] based on the [[theory]] that [[fluid]]s are composed of particles with non-zero volumes, and subject to a (not necessarily pairwise) inter-particle attractive [[force]].{{citation needed|date=June 2015}} It was based on work in theoretical [[physical chemistry]] performed in the late 19th century by [[Johannes Diderik van der Waals]], who did related work on the attractive [[van der Waals force|force]] that also bears his name.{{citation needed|date=June 2015}} The equation is known to be based on a traditional set of derivations deriving from van der Waals' and related efforts,{{citation needed|date=June 2015}} as well as a set of derivation based in [[statistical thermodynamics]],{{citation needed|date=June 2015}} see below.

Van der Waals' early interests were primarily in the field of [[thermodynamics]], where a first influence was [[Rudolf Clausius]]'s published work on heat in 1857; other significant influences were the writings by [[James Clerk Maxwell]], [[Ludwig Boltzmann]], and [[Willard Gibbs]].<ref name="vdW1910nobel">{{cite book | chapter-url=http://nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.pdf | title=Nobel Lectures, Physics 1901–1921 | publisher=Elsevier Publishing Company | year=1967 | location=Amsterdam | pages=254–265 | chapter=J. D. Van der Waals, The equation of state for gases and liquids}}</ref> After initial pursuit of teaching credentials, van der Waals' undergraduate coursework in mathematics and physics at the [[University of Leiden]] in the [[Netherlands]] led (with significant hurdles) to his acceptance for doctoral studies at Leiden under [[Pieter Rijke]]. While his dissertation helps to explain the experimental observation in 1869 by Irish professor of chemistry [[Thomas Andrews (scientist)|Thomas Andrews]] ([[Queen's University Belfast]]) of the existence of a [[critical point (thermodynamics)|critical point]] in fluids,<ref name=Andrews1869>{{cite journal | author=Andrews, T. | year = 1869 | title = The Bakerian Lecture: On the Gaseous State of Matter | journal=Philosophical Transactions of the Royal Society of London | volume = 159 | pages = 575–590 | doi = 10.1098/rstl.1869.0021 }}{{primary source inline|date=June 2015}}</ref> science historian [[Martin J. Klein]] states that it is not clear whether van der Waals was aware of Andrews' results when he began his doctorate work.<ref>{{cite journal |last1=Klein |first1=M. J. |year=1974 |title=The Historical Origins of the van der Waals Equation |journal=Physica |volume=73 |issue=1 |page=31 |doi=10.1016/0031-8914(74)90224-9|bibcode=1974Phy....73...28K }}</ref> Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi-quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, ''Over de Continuïteit van den Gas-en Vloeistof[-]toestand'' (Dutch; in English, ''On the Continuity of the Gas- and Liquid-State''); it was in this dissertation that the first derivations of what we now refer to as the ''van der Waals equation'' appeared.<ref name="VDWdissertation">{{cite book | url=http://www.scs.uiuc.edu/~mainzv/exhibit/vanderwaals.htm | title=Over de Continuiteit van den Gas- en Vloeistoftoestand | publisher=[[University of Leiden]] |author1=van der Waals |author2=J. D. | year=1873}}</ref> James Clerk Maxwell reviewed and lauded its published content in the British science journal ''[[Nature (journal)|Nature]]'',<ref name=Maxwell1874>{{cite journal | author=Maxwell, J.C. | year = 1874 | title = Over de Continuïteit van den Gas-en Vloeistof[-]toestand, Academisch Proefschrift | journal=Nature | volume = 10 | pages = 477–480 | doi = 10.1038/010477a0 | issue=259 |bibcode = 1874Natur..10..477C }}</ref><ref name="MaxwellPapersIIReview">{{cite book | url=https://books.google.com/?id=LrI-AQAAMAAJ | title=The scientific papers of James Clerk Maxwell, vol II | publisher=Cambridge University Press | author=Maxwell, J.C. | editor=Niven, W. D. | authorlink=James Clerk Maxwell | year=1890 | pages=407–415 | chapter=LXIX. Van der Waals on the Continuity of the Gaseous and Liquid States}}</ref> and van der Waals began independent work that would result in his receipt of the [[Nobel Prize]] in 1910, which emphasized the contribution of his formulation of this "equation of state for gases and liquids."<ref name="vdW1910nobel" />

==Equation==
{{refimprove science|section|date=June 2015}}
<!--The figure legend for the top image fails to provide for the reader a description of what is being seen, and how to interpret it. Instead, it provides interpretive and concluding comments that belong in the text, after a thorough description of the model. Second image and Legend offered in parallel, as easier to explain and understand. Note title, and explanation of image, in the legend.-->

[[Image:Waals2.svg|thumb|right|300px|'''van der Waals isotherms.'''{{citation needed|date=June 2015}} The model correctly predicts a mostly incompressible liquid phase, but the 
oscillations in the [[phase transition]] zone do not fit experimental data.]]

[[Image:Van der Waals Isotherms.PNG|thumb|right|300px|'''van der Waals isotherms and oscillations, temperatures, ''T'', below and above ''T''<sub>C</sub>.'''{{citation needed|date=June 2015}} Shown in this ''PV'' diagram are a series of 5 isotherms for temperatures ranging from below (blue) to above (red) ''T''<sub>C</sub> (the critical temperature), where the upward arise and downward dip in the isotherm predicted by the van der Waals equation (the "oscillation", especially evident for the two isotherms for values of ''T'' < ''T''<sub>C</sub>) is evident in the region of the gas-liquid phase transition.]]
The equation relates four state variables: the [[pressure]] of the fluid ''p'', the total volume of the fluid's container ''V'', the [[number of particles]] ''N'', and the [[temperature|absolute temperature]] of the system ''T''.

The [[Intensive and extensive properties|intensive]], microscopic form of the equation is:

:<math>\left(p + \frac{a'}{v^2}\right)\left(v-b'\right) = k_\text{B} T</math><ref name=vdW1910nobel/>

where

:<math>v=V/N</math>

is the volume of the container occupied by each particle (not the [[velocity]] of a particle), and ''k''<sub>B</sub> is the [[Boltzmann constant]]. It introduces two new parameters: ''a''′, a measure of the average attraction between particles, and ''b''′, the volume excluded from ''v'' by one particle.

The equation can be also written in extensive, molar form:

:<math>\left(p + \frac{n^2 a}{V^2}\right)\left(V-nb\right) = nRT</math><ref>{{cite book |last=Chang |first=Raymond |date=2014 |title=Physical Chemistry for the Chemical Sciences |publisher=University Science Books |page=14 |isbn=978-1891389696}}</ref>

where

:<math>a = N_A^2a' </math>

is a measure of the average attraction between particles,

:<math>b = N_\text{A} b' </math>

is the volume excluded by a mole of particles,

:<math>n=N / N_\text{A}</math>

is the number of moles,

:<math>R = N_\text{A} k_\text{B}</math>

is the universal [[gas constant]], ''k''<sub>B</sub> is the Boltzmann constant, and ''N''<sub>A</sub> is the [[Avogadro constant]].

A careful distinction must be drawn between the volume ''available to'' a particle and the volume ''of'' a particle.{{according to whom|date=June 2015}} In the intensive equation, ''v'' equals the total space available to each particle, while the parameter ''b''′ is proportional to the proper volume of a single particle – the volume bounded by the [[atomic radius]]. This is subtracted from ''v'' because of the space taken up by one particle.{{citation needed|date=June 2015}} In van der Waals' original derivation, given below, ''b&apos;'' is four times the proper volume of the particle. Observe further that the pressure ''p'' goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when ''V'' = ''nb''.<ref name=Purdue>{{cite web|title=Deviations from Ideal Gas Law Behavior|url=http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch4/deviation5.html#analysis|website=Purdue University}}</ref>

===Gas mixture===
If a mixture of <math>n</math> gases is being considered, and each gas has its own <math>a</math> (attraction between molecules) and <math>b</math> (volume occupied by molecules) values, then <math>a</math> and <math>b</math> for the mixture can be calculated as
:<math>m</math> = total number of moles of gas present,
:for each <math>i</math>, <math>m_i</math> = number of moles of gas <math>i</math> present, and <math>x_i = \frac{m_i}{m}</math>
:<math>a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j})</math><ref name=vw>[http://www.nigelhewitt.co.uk/diving/maths/vdw.html Who was Van der Waals anyway and what has he to do with my Nitrox fill?]</ref><ref name=slp>[https://slideplayer.com/slide/9376976/28/images/34/Mixing+Rules+for+Simple+Equations+of+State.jpg Mixing Rules for Simple Equations of State]</ref>
:<math>b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j})</math><ref name=vw/><ref name=slp/>

and the rule of adding [[partial pressure]]s becomes invalid if the numerical result of the equation <math>\left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT</math> is significantly different from the [[ideal gas equation]] <math>pV = nRT</math> .

===Reduced form===
The van der Waals equation can also be expressed in terms of [[reduced properties]]:

:<math>\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r</math>

This yields a critical [[compressibility factor]] of 3/8. Reasons for modification of ideal gas equation: The equation state for ideal gas is PV=RT. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made.

==Validity==
{{refimprove section|date=June 2015}}
The ''van der Waals equation'' is mathematically simple, but it nevertheless predicts the experimentally observed [[phase transition|transition]] between vapor and liquid, and predicts [[Critical point (thermodynamics)#Liquid-vapor critical point|critical behaviour]].<ref name=Hill12>Terrell L. Hill, 2012 [1960], "An Introduction to Statistical Thermodynamics" [Dover Books on Physics], Chicago: R.R. Donnelly (Courier/Dover), {{ISBN|0486130908}}. Note, this Donnelly edition is a reprint of the 1986 Dover edition, which itself reprinted the 1962 corrected version of the original 1960 text [Addison Wesley Series in Chemistry, Francis T. Bonner & George C. Pimentel, Eds.], Reading, MS: Addison-Wesley.</ref>{{rp|289}} It also adequately predicts and explains the [[Joule–Thomson effect]], which is not possible in ideal gas.

Above the [[critical temperature]], ''T<sub>C</sub>'', the van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., ''T'' < ''T''<sub>''C''</sub>, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (''p, V, T'') where a liquid [[phase (matter)|phase]] and a [[gas]] phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that <var>p</var> is typically observed to be constant as a function of ''V'' for a given temperature in the two-phase region. This apparent discrepancy is resolved in the context of vapour–liquid equilibrium: at a particular temperature, there exist two points on the van der Waals isotherm that have the same chemical potential, and thus a system in thermodynamic equilibrium will appear to traverse a straight line on the ''p''–''V'' diagram as the ratio of vapour to liquid changes. However, in such a system, there are really only two points present (the liquid and the vapour) rather than a series of states connected by a line, so connecting the locus of points is incorrect: it is not an equation of multiple states, but an equation of (a single) state. It is indeed possible to compress a gas beyond the point at which it would typically condense, given the right conditions, and it is also possible to expand a liquid beyond the point at which it would usually boil. Such states are called "metastable" states. Such behaviour is qualitatively (though perhaps not quantitatively) predicted by the van der Waals equation of state.<ref name=Sandler>S. I. Sandler, "Chemical and Engineering Thermodynamics", Third Edition. New York: Wiley, 1999, p. 273.</ref>

However, the values of physical quantities as predicted with the van der Waals equation of state "are in very poor agreement with experiment", so the model's utility is limited to qualitative rather than quantitative purposes.<ref name=Hill12/>{{rp|289}} Empirically-based corrections can easily be inserted into the van der Waals model (see [[#Maxwell equal area rule|Maxwell's correction]], below), but in so doing, the modified expression is no longer as simple an analytical model; in this regard, other models, such as those based on the [[principle of corresponding states]], achieve a better fit with roughly the same work.{{citation needed|date=June 2015}}
Even with its acknowledged shortcomings, the pervasive use of the ''van der Waals equation'' in standard university [[physical chemistry]] textbooks makes clear its importance as a pedagogic tool to aid understanding fundamental physical chemistry ideas involved in developing theories of vapour–liquid behavior and equations of state.<ref>Peter Atkins & Julio de Paula, 2006, "Physical chemistry", 8th Edn., pp. 17–22, 104''f,'' 632–641, New York: Macmillan, {{ISBN|0716787598}}.</ref><ref>R. Stephen Berry, Stuart A. Rice & John Ross, 2000, "Physical Chemistry", pp. 298–306 and ''passim'', Oxford: Oxford University Press, {{ISBN|0195105893}}</ref><ref>Ken A. Dill & Sarina Bromberg, 2003, "Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology", pp. 457–462, New York: Garland Science, {{ISBN|0815320515}}.</ref> In addition, other (more accurate) equations of state such as the Redlich–Kwong and Peng–Robinson [[equation of state]] are essentially modifications of the van der Waals equation of state.

==Derivation==
{{unreferenced section|date=June 2015}}
Textbooks in physical chemistry generally give two [[Formal proof|derivations]] of the title equation.{{who|date=June 2015}} One is the conventional derivation that goes back to van der Waals, a mechanical equation of state that cannot be used to specify all thermodynamic functions; the other is a statistical mechanics derivation that makes explicit the intermolecular potential neglected in the first derivation.{{citation needed|date=June 2015}} A particular advantage of the statistical mechanical derivation is that it yields the partition function for the system, and allows all thermodynamic functions to be specified (including the mechanical equation of state).{{citation needed|date=June 2015}}

===Conventional derivation===
Consider one mole of gas composed of non-interacting point particles that satisfy the ideal gas law:(see any standard Physical Chemistry text, op. cit.)

:<math> p = \frac{RT}{V_\mathrm{m}}.</math>

Next, assume that all particles are hard spheres of the same finite radius ''r'' (the [[van der Waals radius]]). The effect of the finite volume of the particles is to decrease the available void space in which the particles are free to move. We must replace ''V'' by ''V''&nbsp;&minus;&nbsp;''b'', where ''b'' is called the ''excluded volume'' or "co-volume". The corrected equation becomes

:<math> p = \frac{RT}{V_\mathrm{m}-b}.</math>

The excluded volume <math>b</math> is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times that volume. To see this, we must realize that a particle is surrounded by a sphere of radius 2''r'' (two times the original radius) that is forbidden for the centers of the other particles. If the distance between two particle centers were to be smaller than 2''r'', it would mean that the two particles penetrate each other, which, by definition, hard spheres are unable to do.

The excluded volume for the two particles (of average diameter ''d'' or radius ''r'') is

:<math>b'_2 = 4\pi d^3/3 = 8\times (4\pi r^3/3)</math>,
which, divided by two (the number of colliding particles), gives the excluded volume per particle:

:<math>b' = b'_2/2 \quad \rightarrow \quad b'=4\times (4\pi r^3/3)</math>,

So ''b&prime;'' is four times the proper volume of the particle. It was a point of concern to van der Waals that the factor four yields an upper bound; empirical values for ''b&prime;'' are usually lower. Of course, molecules are not infinitely hard, as van der Waals thought, and are often fairly soft.

Next, we introduce a (not necessarily pairwise) attractive force between the particles. van der Waals assumed that, notwithstanding the existence of this force, the density of the fluid is homogeneous; furthermore, he assumed that the range of the attractive force is so small that the great majority of the particles do not feel that the container is of finite size.{{citation needed|date=June 2015}} Given the homogeneity of the fluid, the bulk of the particles do not experience a net force pulling them to the right or to the left. This is different for the particles in surface layers directly adjacent to the walls. They feel a net force from the bulk particles pulling them into the container, because this force is not compensated by particles on the side where the wall is (another assumption here is that there is no interaction between walls and particles, which is not true, as can be seen from the phenomenon of droplet formation; most types of liquid show adhesion). This net force decreases the force exerted onto the wall by the particles in the surface layer. The net force on a surface particle, pulling it into the container, is proportional to the number density

:<math>C=N_\mathrm{A}/V_\mathrm{m}</math>.

The number of particles in the surface layers is, again by assuming homogeneity, also proportional to the density. In total, the force on the walls is decreased by a factor proportional to the square of the density, and the pressure (force per unit surface) is decreased by 
:<math>a'C^2= a' \left(\frac{N_\mathrm{A}}{V_\mathrm{m}}\right)^2 = \frac{a}{V_\mathrm{m}^2} </math>,
so that

:<math> p = \frac{RT}{V_\mathrm{m}-b}-\frac{a}{V_\mathrm{m}^2} \Rightarrow
\left(p + \frac{a}{V_\mathrm{m}^2}\right)(V_\mathrm{m}-b) = RT.
</math>
Upon writing ''n'' for the number of moles and ''nV''<sub>m</sub> = ''V'', the equation obtains the second form given above,
:<math> \left(p + \frac{n^2 a}{V^2}\right)(V-nb) = nRT.
</math>

It is of some historical interest to point out that van der Waals, in his Nobel prize lecture, gave credit to [[Laplace]] for the argument that pressure is reduced proportional to the square of the density.{{citation needed|date=June 2015}}
<!--WHAT IS THIS AND WHY IS IT HERE? 
This causes the average [[Helmholtz energy]] per particle to be reduced by an amount proportional to the fluid [[density]]. However, the pressure obeys the thermodynamic relation

:<math> p = - \left(\frac{\partial A^*}{\partial v}\right)_T </math>

where ''A*'' is the [[Helmholtz energy]] per particle. The attraction therefore reduces the pressure by an amount proportional to ''1/v²''. Denoting the constant of proportionality by ''a'', we obtain
:<math> p = \frac{kT}{v-b}-\frac{a}{v^2}</math>

which is the van der Waals equation.
-->

===Statistical thermodynamics derivation===
The canonical [[partition function (statistical mechanics)|partition function]] ''Z'' of an ideal gas consisting of ''N = nN''<sub>A</sub> identical ('''non-interacting''') particles, is:<ref>{{cite book|last1=Hill|first1=Terrell L.|title=An Introduction to Statistical Thermodynamics|date=1960|publisher=Courier Corporation|page=77}}</ref><ref>{{cite book | title = Modern Thermodynamics | first = John | last = Denker | year = 2014 | publisher = CreateSpace Independent Publishing Platform | isbn = 978-1502530356 | chapter-url = https://www.av8n.com/physics/thermo/z-particles.html#sec-particle-box | chapter = Chapter 26.9, Derivation: Particle in a Box }}</ref>
:<math>
Z = \frac{z^N}{N!}\quad \hbox{with}\quad z = \frac{V}{\Lambda^3}
</math>
where <math>\Lambda</math> is the [[thermal de Broglie wavelength]],
:<math>
\Lambda = \sqrt{\frac{h^2}{2\pi m k T}}
</math>
with the usual definitions: ''h'' is [[Planck's constant]], ''m'' the mass of a particle, ''k'' [[Boltzmann's constant]] and ''T'' the absolute temperature. In an ideal gas ''z'' is the partition function of a single particle in a container of volume ''V''. In order to derive the van der Waals equation we assume now that each particle moves independently in an average potential field offered by the other particles. The averaging over the particles is easy because we will assume that the particle density of the van der Waals fluid is homogeneous.
The interaction between a pair of particles, which are hard spheres, is taken to be
:<math>
u(r) =
\begin{cases}
\infty                  &\hbox{when}\quad r < d, \\
-\epsilon \left(\frac{d}{r}\right)^6 & \hbox{when}\quad r \ge d,
\end{cases}
</math>
''r'' is the distance between the centers of the spheres and ''d'' is the distance where the hard spheres touch each other (twice the van der Waals radius). The depth of the van der Waals well is <math>\epsilon</math>.

Because the particles are not coupled under the mean field Hamiltonian, the mean field approximation of the total partition function still factorizes,

:<math>Z= z^N/N!</math>,
but the intermolecular potential necessitates two modifications to ''z''. First, because of the finite size of the particles, not all of ''V'' is available, but only ''V − Nb''', where (just as in the conventional derivation above)

:<math> b' =  2\pi d^3/3 </math>.
Second, we insert a Boltzmann factor
exp['' - ϕ/2kT''] to take care of the average intermolecular potential. We divide here the potential by two because this interaction energy is shared between two particles. Thus
:<math>
z= \frac{(V-Nb') \, e^{-\phi/(2kT)}}{\Lambda^3}.
</math>

The total attraction felt by a single particle  is
:<math>
\phi = \int_d^{\infty} u(r) \frac{N}{V} 4\pi r^2 dr ,
</math>

where we assumed that in a shell of thickness d''r'' there are ''N/V''&nbsp;4''π'' ''r''<sup>2</sup>''dr'' particles. This is a mean field approximation; the position of the particles is averaged. In reality the density close to the particle is different than far away as can be described by a pair correlation function. Furthermore, it is neglected that the fluid is enclosed 
between walls.  Performing the integral we get
:<math>
\phi = -2 a' \frac{N}{V}\quad\hbox{with}\quad a' = \epsilon \frac{2\pi d^3}{3} =\epsilon b'.
</math>
Hence, we obtain,
:<math>
\ln Z = N \ln {(V-Nb')} + \frac{N^2 a'}{V kT} - N \ln {(\Lambda^3)} -\ln {N!} 
</math>
From statistical thermodynamics we know that
:<math> p = kT \frac{\partial \ln Z}{\partial V}
</math>,
so that we only have to differentiate the terms containing V. We get
:<math>
p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow
\left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT.
</math>

==Maxwell equal area rule==
{{refimprove science|section|date=June 2015}}
{{main|Maxwell construction}}

[[Image:MaxwellEqArea.svg|thumb|400px|'''van der Waals isotherm oscillation and Maxwell's equal areas.'''<ref name=Sandler284>{{cite book|last1=Sandler|first1=Stanley I.|title=Chemical, Biochemical, and Engineering Thermodynamics|date=2006|publisher=John Wiley & Sons|location=New York|isbn=978-0-471-66174-0|page=284|edition=4th}}</ref> Maxwell's rule eliminates the oscillating behavior of the [[isotherm (contour line)|isotherm]] in the phase transition zone by defining it as a certain [[Contour line#Barometric pressure|isobar]] in that zone. The above isotherm is for a reduced temperature of ''T<sub>R</sub>''=0.9. The Maxwell correction is at a vapor pressure of ''p<sub>V</sub>''≈0.64700 between the reduced volume of the pure liquid ''V<sub>L</sub>''≈0.60340 and the pure gas ''V<sub>G</sub>''≈2.3488 at the vapor pressure.]]

Below the critical temperature, the van der Waals equation seems to predict qualitatively incorrect relationships. Unlike for ideal gases, the p-V isotherms oscillate with a relative minimum (''d'') and a relative maximum (''e''). Any pressure between ''p<sub>d</sub>'' and ''p<sub>e</sub>'' appears to have 3 stable volumes, contradicting the experimental observation that two state variables completely determine a one-component system's state.<ref name=Sandler284/> Moreover, the isothermal [[compressibility]] is negative between ''d'' and ''e'' (equivalently <math>\scriptstyle \left({{\partial P}/{\partial V}}\right)_{T,N} > 0 </math>), which cannot describe a system at equilibrium.<ref name=Sandler287>{{cite book|last1=Sandler|first1=Stanley I.|title=Chemical, Biochemical, and Engineering Thermodynamics|date=2006|publisher=John Wiley & Sons|location=New York|isbn=978-0-471-66174-0|page=287|edition=4th}}</ref>

To address these problems, [[James Clerk Maxwell]] replaced the isotherm between points ''a'' and ''c'' with a horizontal line positioned so that the areas of the two shaded regions would be equal (replacing the ''a''-''d''-''b''-''e''-''c'' curve with a straight line from ''a'' to ''c''); this portion of the isotherm corresponds to the liquid-vapor equilibrium. The regions of the isotherm from ''a''&ndash;''d'' and from ''c''&ndash;''e'' are interpreted as metastable states of super-heated liquid and super-cooled vapor, respectively.<ref name="Clerk-Maxwell1875">{{cite journal|last1=Clerk-Maxwell|first1=J.|title=On the Dynamical Evidence of the Molecular Constitution of Bodies|journal=Nature|volume=11|issue=279|year=1875|pages=357–359|issn=0028-0836|doi=10.1038/011357a0|bibcode = 1875Natur..11..357C }}</ref>
<ref name="MaxwellPapersII">{{cite book | url=https://books.google.com/?id=LrI-AQAAMAAJ | title=The scientific papers of James Clerk Maxwell, vol II | publisher=Cambridge University Press | author=Maxwell, J.C. | editor=Niven, W. D. | authorlink=James Clerk Maxwell | year=1890 | pages=418–438 | chapter=LXXI. On the Dynamical Evidence of the Molecular Constitution of Bodies}}</ref> The equal area rule can be expressed as:
:<math>p_V(V_G-V_L)=\int_{V_L}^{V_G}p\,dV</math>

where ''p<sub>V</sub>'' is the vapor pressure (flat portion of the curve), ''V<sub>L</sub>'' is the volume of the pure liquid phase at point ''a'' on the diagram, and ''V<sub>G</sub>'' is the volume of the pure gas phase at point ''c'' on the diagram.{{citation needed|date=June 2015}} A two-phase mixture at ''p<sub>V</sub>'' will occupy a total volume between ''V<sub>L</sub>'' and ''V<sub>G</sub>'', as determined by Maxwell's [[lever rule]].

Maxwell justified the rule based on the fact that the area on a ''pV'' diagram corresponds to mechanical work, saying that work done on the system in going from ''c'' to ''b'' should equal work released on going from ''a'' to ''b''. This is because the change in free energy ''A''(''T'',''V'') equals the work done during a reversible process, and, as a [[state variable]], the free energy must be path-independent. In particular, the value of ''A'' at point ''b'' should be the same regardless of whether the path taken is from left or right across the horizontal isobar, or follows the original van der Waals isotherm.{{citation needed|date=June 2015}}

This derivation is not entirely rigorous, since it requires a reversible path through a region of thermodynamic instability, while ''b'' is unstable.{{clarify|date=June 2015}}{{citation needed|date=June 2015}} Nevertheless, modern derivations from [[chemical potential]] reach the same conclusion, and it remains a necessary modification to the van der Waals and to any other analytic equation of state.<ref name=Sandler284/>

=== From chemical potential ===
The Maxwell equal area rule can also be derived from an assumption of equal [[chemical potential]] ''μ'' of coexisting liquid and vapour phases.<ref>{{cite journal | title=The area method for pure fluids and an analysis of the two-phase region |author1=Elhassan, A. E. |author2=Craven, R.J.B. |author3=de Reuck, K. M. | journal=Fluid Phase Equilibria |volume=130 | year=1997 | issue=1-2 | pages=167–187 | doi=10.1016/S0378-3812(96)03222-0}}</ref>{{primary source inline|date=June 2015}} On the isotherm shown in the above plot, points ''a'' and ''c'' are the only pair of points which fulfill the equilibrium condition of having equal pressure, temperature and chemical potential. It follows that systems with volumes intermediate between these two points will consist of a mixture of the pure liquid and gas with specific volumes equal to the pure liquid and gas phases at points ''a'' and ''c''.

The van der Waals equation may be solved for ''V<sub>G</sub>'' and ''V<sub>L</sub>'' as functions of the temperature and the vapor pressure ''p<sub>V</sub>''.{{citation needed|date=June 2015}} Since:{{according to whom|date=June 2015}}
:<math>p=-\left(\frac{\partial A}{\partial V}\right)_{T,N}</math>

where ''A'' is the Helmholtz free energy, it follows that the equal area rule can be expressed as:

:<math>p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}</math>is

Since the gas and liquid volumes are functions of ''p<sub>V</sub>'' and ''T'' only, this equation is then solved numerically to obtain ''p<sub>V</sub>'' as a function of temperature (and number of particles ''N''), which may then be used to determine the gas and liquid volumes.{{citation needed|date=June 2015}}

[[File:Vdw5annotatedwith dropping.png|thumb|800px|Locus of coexistence for two phases of van der Waals fluid]]
A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown
in the accompanying figure. One sees that the two locii meet at the critical point (1,1,1) smoothly. An isotherm of the van der Waals fluid taken at ''T <sub> r</sub>'' = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal.

== Other parameters, forms and applications ==
{{unreferenced section|date=June 2015}}

===Other thermodynamic parameters===
We reiterate that the extensive volume ''V''&nbsp; is related to the volume per particle ''v=V/N''&nbsp; where ''N = nN''<sub>A</sub>&nbsp; is the number of particles in the system.
The equation of state does not give us all the thermodynamic parameters of the system. We can take the equation for the [[Helmholtz energy]] ''A'' <ref name=rut>Michael Gershenson, Date Unknown, "Lecture 16. The van der Waals Gas (Ch. 5)," unpublished course slide set from prior teaching of ''Physics 351, Thermal Physics (Statistics and Thermodynamics), [https://www.physics.rutgers.edu/ugrad/351/]'' Rutgers University, Department of Physics and Astronomy, see [http://www.physics.rutgers.edu/ugrad/351/oldslides/Lecture16.pdf], accessed 25 June 2015. Possibly sourced from Daniel V. Schroeder, 2013, "An Introduction to Thermal Physics," New York: Pearson Education, {{ISBN|1292026219}}.</ref>
:<math>
A = -kT \ln Q.\,
</math>
From the equation derived above for ln''Q'', we find
:<math>A(T,V,N)=-NkT\left[1+\ln\left(\frac{(V-Nb')T^{3/2}}{N\Phi}\right)\right]
-\frac{a' N^2}{V}.</math>

Where Φ is an undetermined constant, which may be taken from the [[Sackur–Tetrode equation]] for an ideal gas to be:

:<math>\Phi=T^{3/2}\Lambda^3=\left(\frac{h}{\sqrt{2\pi m k}}\right)^3</math>

This equation expresses ''A''&nbsp; in terms of its [[Thermodynamic potential|natural variables]] ''V''&nbsp; and ''T''&nbsp;, and therefore gives us all thermodynamic information about the system. The mechanical equation of state was already derived above

:<math>p = -\left(\frac{\partial A}{\partial V}\right)_T 
= \frac{NkT}{V-Nb'}-\frac{a' N^2}{V^2}.</math>

The entropy equation of state yields the [[entropy]] (''S''&nbsp;)

:<math>S = -\left(\frac{\partial A}{\partial T}\right)_V
=Nk\left[ \ln\left(\frac{(V-Nb')T^{3/2}}{N\Phi}\right)+\frac{5}{2} \right]</math>

from which we can calculate the [[internal energy]]

:<math>U = A+TS = \frac{3}{2}\,NkT-\frac{a'N^2}{V}.</math>

Similar equations can be written for the other [[thermodynamic potential]] and the chemical potential, but expressing any potential as a function of pressure ''p''&nbsp; will require the solution of a third-order polynomial, which yields a complicated expression. Therefore, expressing the [[enthalpy]] and the [[Gibbs energy]] as functions of their natural variables will be complicated.
<!-- As I don't agree with U=A+TS and don't have patience to read thoroughly all related formulas, I offer another, more simple derivation of formula for internal energy:
: <math>dU = dQ-dA= TdS-PdV.</math>
: <math>\left(\frac{dU}{dV}\right)_{T=const}=T\frac{dS}{dV}-P=T\frac{dP}{dT}-P=T\frac{R}{\frac{V}{n}-b}-P=P+\frac{an^2}{V^2}-P=\frac{an^2}{V^2}</math>
:<math>\left(\frac{dU}{dT}\right)_{V=const}=\left(T\frac{dS}{dT}\right)_{V=const}=\left(\frac{dQ}{dT}\right)_{V=const}=\nu C_V</math>-->

===Reduced form===
Although the material constant ''a'' and ''b'' in the usual form of the van der Waals equation differs for every single fluid considered, the equation can be recast into an invariant form applicable to ''all'' fluids.

Defining the following reduced variables (''f<sub>R</sub>'', ''f<sub>C</sub>'' are the reduced and [[critical variable]] versions of ''f'', respectively),

:<math>p_R=\frac{p}{p_C},\qquad
v_R=\frac{v}{v_C},\quad\hbox{and}\quad
T_R=\frac{T}{T_C}</math>,
where
:<math>
p_C=\frac{a'}{27b'^2}, \qquad \displaystyle{v_C=3b'},\quad\hbox{and}\quad kT_C=\frac{8a'}{27b'}
</math>

as shown by Salzman.<ref>{{cite web 
 |url= http://www.chem.arizona.edu/~salzmanr/480a/480ants/vdwcrit/vdwcrit.html
 |title= Critical Constants of the van der Waals Gas
 |last1= Salzman
 |first1= W. R.
 |date= 2004
 |website= Chemical Thermodynamics
 |publisher= Arizona State University
 |accessdate= 2015-07-07 
}}</ref>

The first form of the van der Waals equation of state given above can be recast in the following reduced form:

:<math>\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R</math>)

This equation is ''invariant'' for all fluids; that is, the same reduced form equation of state applies, no matter what ''a'' and ''b'' may be for the particular fluid.

This invariance may also be understood in terms of the principle of corresponding states. If two fluids have the same reduced pressure, reduced volume, and reduced temperature, we say that their states are corresponding. The states of two fluids may be corresponding even if their measured pressure, volume, and temperature are very different. If the two fluids' states are corresponding, they exist in the same regime of the reduced form equation of state. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly.

===Cubic equation===
The van der Waals equation is a [[Cubic function|cubic]] equation of state; in the reduced formulation the cubic equation is:

:<math>{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0</math>

At the critical temperature, where <math>T_R=p_R=1</math> we get as expected

:<math>{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1</math>

For ''T<sub>R</sub>'' < 1, there are 3 values for ''v<sub>R</sub>''.
For ''T<sub>R</sub>'' > 1, there is 1 real value for ''v<sub>R</sub>''.

The solution of this equation for the case where there are three separate roots may be found at [[Maxwell construction]]

=== Application to compressible fluids ===
The equation is also usable as a [[PVT (physics)|PVT]] equation for [[compressible fluid]]s (e.g. [[polymers]]). In this case [[specific volume]] changes are small and it can be written in a simplified form:

:<math>(p+A)(V-B)=CT, \,</math>

where ''p'' is the [[pressure]], ''V'' is [[specific volume]], ''T'' is the [[temperature]] and ''A, B, C'' are parameters.

== See also ==
<!--For consistecy of format, moving phrase "To solve implicit equations like van der Waals" from the "Iteration" bullet up to here, until someone can check these wikilinks, mark the ones in the article as being so, remove ones unneeded, and for ones kept but not in the article, annotate them consistently.-->
* [[Gas laws]]
* [[Ideal gas]]
* [[Inversion temperature]]
* [[Iteration]]
* [[Maxwell construction]]
* [[Real gas]]
* [[Theorem of corresponding states]]
* [[van der Waals constants (data page)]]

==References==
{{Reflist}}

==Further reading==
* [http://www.its.caltech.edu/~mcc/index.html Michael Cross], 2004, ''[http://www.pmaweb.caltech.edu/~mcc/Ph127/b/Lecture3.pdf Lecture 3: First Order Phase Transitions,]'' in ''[http://www.pmaweb.caltech.edu/~mcc/Ph127/index.html Physics 127: Statistical Mechanics (Second Term)],'' Pasadena, CA: Division of Physics, Mathematics, and Astronomy, California Institute of Technology.
* David Chandler, 1987, ''Introduction to Modern Statistical Mechanics'', pp.&nbsp;287–295, Oxford: Oxford University Press, {{ISBN|0195042778}}.
* Charles Kittel & Herbert Kroemer, 1980, ''Thermal Physics'', Rev. Edn., pp.&nbsp;287–295, New York: Macmillan, {{ISBN|0716710889}}.
* A. Dalgarno & W.D. Davison, 1966, "The calculation of van der Waals interactions," in ''Advances in Atomic and Molecular Physics,'' '''1''', 195.

== External links ==
* [http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Thermal/vdWaalEquatOfState.html Some values of a and b in the 2nd equation]

{{Statistical mechanics topics}}

[[Category:Equations of state]]
[[Category:Gas laws]]
[[Category:Engineering thermodynamics]]
[[Category:Johannes Diderik van der Waals|Equation]]