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Virial coefficient
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<h2 id="derivation">Derivation</h2> <p>The first step in obtaining a closed expression for virial coefficients is a <a href="/facts/Cluster_expansion/qzWAr0zq">cluster expansion</a><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> of the <a href="/facts/Partition_function_(statistical_mechanics)/q65sJrrn">grand canonical partition function</a> </p> Ξ = ∑ n λ n Q n = e ( p V ) / ( k B T ) {\displaystyle \Xi =\sum _{n}{\lambda ^{n}Q_{n}}=e^{\left(pV\right)/\left(k_{\text{B}}T\right)}} <p>Here p {\displaystyle p} is the pressure, V {\displaystyle V} is the volume of the vessel containing the particles, k B {\displaystyle k_{\text{B}}} is the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a>, T {\displaystyle T} is the absolute temperature, λ = exp [ μ / ( k B T ) ] {\displaystyle \lambda =\exp[\mu /(k_{\text{B}}T)]} is the <a href="/facts/Fugacity/C2gdkThT">fugacity</a>, with μ {\displaystyle \mu } the <a href="/facts/Chemical_Potential/IaMIgg2Q">chemical potential</a>. The quantity Q n {\displaystyle Q_{n}} is the <a href="/facts/Partition_function_(statistical_mechanics)/q65sJrrn">canonical partition</a> function of a subsystem of n {\displaystyle n} particles: </p> Q n = tr [ e − H ( 1 , 2 , … , n ) / ( k B T ) ] . {\displaystyle Q_{n}=\operatorname {tr} [e^{-H(1,2,\ldots ,n)/(k_{\text{B}}T)}].} <p>Here H ( 1 , 2 , … , n ) {\displaystyle H(1,2,\ldots ,n)} is the Hamiltonian (energy operator) of a subsystem of n {\displaystyle n} particles. The Hamiltonian is a sum of the <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energies</a> of the particles and the total n {\displaystyle n} -particle <a href="/facts/Potential_energy/I03KLbmn">potential energy</a> (interaction energy). The latter includes pair interactions and possibly 3-body and higher-body interactions. The <a href="/facts/Grand_partition_function/q65sJrrn">grand partition function</a> Ξ {\displaystyle \Xi } can be expanded in a sum of contributions from one-body, two-body, etc. clusters. The virial expansion is obtained from this expansion by observing that ln Ξ {\displaystyle \ln \Xi } equals p V / ( k B T ) {\displaystyle pV/(k_{B}T)} . In this manner one derives </p> B 2 = V ( 1 2 − Q 2 Q 1 2 ) {\displaystyle B_{2}=V\left({\frac {1}{2}}-{\frac {Q_{2}}{Q_{1}^{2}}}\right)} B 3 = V 2 [ 2 Q 2 Q 1 2 ( 2 Q 2 Q 1 2 − 1 ) − 1 3 ( 6 Q 3 Q 1 3 − 1 ) ] {\displaystyle B_{3}=V^{2}\left[{\frac {2Q_{2}}{Q_{1}^{2}}}{\Big (}{\frac {2Q_{2}}{Q_{1}^{2}}}-1{\Big )}-{\frac {1}{3}}{\Big (}{\frac {6Q_{3}}{Q_{1}^{3}}}-1{\Big )}\right]} . <p>These are quantum-statistical expressions containing kinetic energies. Note that the one-particle partition function Q 1 {\displaystyle Q_{1}} contains only a kinetic energy term. In the <a href="/facts/Classical_limit/Yu7aHtr6">classical limit</a> ℏ = 0 {\displaystyle \hbar =0} the kinetic energy operators <a href="/facts/Commutator/eYxzFmFY">commute</a> with the potential operators and the kinetic energies in numerator and denominator cancel mutually. The <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a> (tr) becomes an integral over the configuration space. It follows that classical virial coefficients depend on the interactions between the particles only and are given as integrals over the particle coordinates. </p><p>The derivation of higher than B 3 {\displaystyle B_{3}} virial coefficients becomes quickly a complex combinatorial problem. Making the classical approximation and neglecting non-additive interactions (if present), the combinatorics can be handled graphically as first shown by <a href="/facts/Joseph_E._Mayer/7syQj0gk">Joseph E. Mayer</a> and <a href="/facts/Maria_Goeppert-Mayer/Ung0ch6E">Maria Goeppert-Mayer</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>They introduced what is now known as the <a href="/facts/Mayer_function/luwrMbxn">Mayer function</a>: </p> f ( 1 , 2 ) = exp [ − u ( | r → 1 − r → 2 | ) k B T ] − 1 {\displaystyle f(1,2)=\exp \left[-{\frac {u(|{\vec {r}}_{1}-{\vec {r}}_{2}|)}{k_{B}T}}\right]-1} <p>and wrote the cluster expansion in terms of these functions. Here u ( | r → 1 − r → 2 | ) {\displaystyle u(|{\vec {r}}_{1}-{\vec {r}}_{2}|)} is the interaction potential between particle 1 and 2 (which are assumed to be identical particles). </p> <h2 id="definition-in-terms-of-graphs">Definition in terms of graphs</h2> <p>The virial coefficients B i {\displaystyle B_{i}} are related to the irreducible <a href="/facts/Mayer_cluster_integral/qzWAr0zq">Mayer cluster integrals</a> β i {\displaystyle \beta _{i}} through </p> B i + 1 = − i i + 1 β i {\displaystyle B_{i+1}=-{\frac {i}{i+1}}\beta _{i}} <p>The latter are concisely defined in terms of graphs. </p> β i = The sum of all connected, irreducible graphs with one white and i black vertices {\displaystyle \beta _{i}={\mbox{The sum of all connected, irreducible graphs with one white and}}\ i\ {\mbox{black vertices}}} <p>The rule for turning these graphs into integrals is as follows: </p> <ol><li>Take a graph and <a href="/facts/Vertex_labeling/ux5jKRD7">label</a> its white vertex by k = 0 {\displaystyle k=0} and the remaining black vertices with k = 1 , . . , i {\displaystyle k=1,..,i} .</li> <li>Associate a labelled coordinate <i>k</i> to each of the vertices, representing the continuous degrees of freedom associated with that particle. The coordinate 0 is reserved for the white vertex</li> <li>With each bond linking two vertices associate the <a href="/facts/Mayer_f-function/luwrMbxn">Mayer f-function</a> corresponding to the interparticle potential</li> <li>Integrate over all coordinates assigned to the black vertices</li> <li>Multiply the end result with the <a href="/facts/Symmetry_number/tmtq9goq">symmetry number</a> of the graph, defined as the inverse of the number of <a href="/facts/Permutation/JSw9ukmv">permutations</a> of the black labelled vertices that leave the graph topologically invariant.</li></ol> <p>The first two cluster integrals are </p> <table><tbody><tr><td> b 1 = {\displaystyle b_{1}=} </td><td></td><td> = ∫ d 1 f ( 0 , 1 ) {\displaystyle =\int d\mathbf {1} f(\mathbf {0} ,\mathbf {1} )} </td></tr><tr><td> b 2 = {\displaystyle b_{2}=} </td><td></td><td> = 1 2 ∫ d 1 ∫ d 2 f ( 0 , 1 ) f ( 0 , 2 ) f ( 1 , 2 ) {\displaystyle ={\frac {1}{2}}\int d\mathbf {1} \int d\mathbf {2} f(\mathbf {0} ,\mathbf {1} )f(\mathbf {0} ,\mathbf {2} )f(\mathbf {1} ,\mathbf {2} )} </td></tr></tbody></table> <p>The expression of the second virial coefficient is thus: </p> B 2 = − 2 π ∫ r 2 ( e − u ( r ) / ( k B T ) − 1 ) d r , {\displaystyle B_{2}=-2\pi \int r^{2}{{\Big (}e^{-u(r)/(k_{\text{B}}T)}-1{\Big )}}~\mathrm {d} r,} <p>where particle 2 was assumed to define the origin ( r → 2 = 0 → {\displaystyle {\vec {r}}_{2}={\vec {0}}} ). This classical expression for the second virial coefficient was first derived by <a href="/facts/Leonard_Ornstein/Kho3hv2s">Leonard Ornstein</a> in his 1908 <a href="/facts/Leiden_University/MqnmnTtF">Leiden University</a> Ph.D. thesis. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Boyle_temperature/9YY3VLfy">Boyle temperature</a> – temperature at which the second virial coefficient B 2 {\displaystyle B_{2}} vanishes</li> <li><a href="/facts/Excess_property/hkvadLNB">Excess property</a></li> <li><a href="/facts/Compressibility_factor/9GKBEmgN">Compressibility factor</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Dymond, J. H.; Smith, E. B. (1980). <i>The Virial Coefficients of Pure Gases and Mixtures: a Critical Compilation</i>. Oxford: Clarendon. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0198553617.</li> <li>Hansen, J. P.; McDonald, I. R. (1986). <i>The Theory of Simple Liquids</i> (2nd ed.). London: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 012323851X.</li> <li><a href="http://scitation.aip.org/content/aip/journal/jcp/50/10/10.1063/1.1670902">http://scitation.aip.org/content/aip/journal/jcp/50/10/10.1063/1.1670902</a></li> <li><a href="http://scitation.aip.org/content/aip/journal/jcp/50/11/10.1063/1.1670994">http://scitation.aip.org/content/aip/journal/jcp/50/11/10.1063/1.1670994</a></li> <li>Reid, C. R., Prausnitz, J. M., Poling B. E., Properties of gases and liquids, IV edition, Mc Graw-Hill, 1987</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hill, T. L. (1960). Introduction to Statistical Thermodynamics. Addison-Wesley. ISBN 9780201028409. <a href="9780201028409" target="_blank">9780201028409</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Mayer, J. E.; Goeppert-Mayer, M. (1940). Statistical Mechanics. New York: Wiley. <a href="https://archive.org/details/in.ernet.dli.2015.460487" target="_blank">https://archive.org/details/in.ernet.dli.2015.460487</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>