Virial equations

Virial equations are extensions of the general gas equation through a series expansion according to powers of . They represent approximate equations of state for real gases . If the series expansion is broken off after the first term, the general gas equation is obtained again. If one continues the series expansion, however, a potentially infinite number of equations of state with an increasing number of parameters arise. In implicit form, the general series expansion is as follows: ${\displaystyle 1/V_{\mathrm {m} }}$

${\displaystyle {\frac {p}{RT}}={\frac {1}{V_{\mathrm {m} }}}+{\frac {B_{2V}(T)}{V_{\mathrm {m} }^{2}}}+{\frac {B_{3V}(T)}{V_{\mathrm {m} }^{3}}}+\dotsb }$

The individual symbols stand for the following quantities :

• ${\displaystyle V_{\mathrm {m} }}$- molar volume
• ${\displaystyle T}$- temperature (in Kelvin )
• ${\displaystyle p}$- pressure
• ${\displaystyle R}$- universal gas constant
• ${\displaystyle B_{1V}=1}$(as with ideal gas )
• ${\displaystyle B_{2V}(T)}$ - second virial coefficient
• ${\displaystyle B_{3V}(T)}$ - third virial coefficient (etc.)

The name comes from the fact that the virial of the internal forces is developed in the gas equation (see virial theorem ). The following applies to the ideal gas: ${\displaystyle B_{2V}=B_{3V}=\dots =0}$

Virial coefficients

The virial coefficients result from the interactions between the molecules. They cannot be physically interpreted.

The second virial coefficient results from the pair potential between two molecules (i.e. the binding energy ): ${\displaystyle V(r)}$

${\displaystyle B_{2V}(T)=2\pi \cdot N_{\mathrm {A} }\cdot \int _{0}^{\infty }\left(1-\exp \left(-{\frac {V(r)}{kT}}\right)\right)r^{2}dr}$

With

• ${\displaystyle N_{\mathrm {A} }}$- Avogadro's constant
• ${\displaystyle k}$- Boltzmann's constant
• ${\displaystyle r}$ - distance between the molecules.

The third virial coefficient depends on the interactions within groups of three molecules; the same applies to all others.

The virial equation with two or three virial coefficients is only valid for moderate pressures.

If no experimental values ​​are available for the virial coefficients, these can be calculated using the empirical model according to Hayden-O'Connell. Here, the second virial coefficient is estimated from the critical temperature, critical pressure , dipole moment and radius of gyration.

Connection to the Van der Waals equation

Another, more widely used equation of state for real gases is the Van der Waals equation . A connection can be made between this and the virial equation with two virial coefficients via a simplification.

${\displaystyle pV_{\mathrm {m} }=RT+B'\rho +C'\rho ^{2}+\dotsb }$

with the mass density . ${\displaystyle \rho =1/V_{\mathrm {m} }}$

If the series development is aborted, this correction factor is calculated according to the correspondence principle from the critical state of the respective substance. ${\displaystyle B'}$

At the same time, this form of the virial equation is also a simplified form of the Redlich-Kwong equation .

${\displaystyle pV_{\mathrm {m} }=RT+B'\rho }$

With

• ${\displaystyle B'=b-{\frac {a}{RT}}}$
• the parameters and the Van der Waals equation. ${\displaystyle a}$ ${\displaystyle b}$

Is equal to the Boyle temperature , the following applies . ${\displaystyle T}$ ${\displaystyle B'=0}$

Virial development: consideration of statistical mechanics

For real gases, i.e. H. interacting particles, the sum of states of a statistical ensemble cannot be evaluated exactly. However, this can be calculated approximately for gases of low density. The virial expansion is a development of the thermal equation of state in the particle density (particles per volume): ${\displaystyle \rho =N/V}$

${\displaystyle {\frac {p}{kT}}=\sum _{i=1}^{\infty }\rho ^{i}B_{i}(T)=\rho \ +\rho ^{2}\,B_{2}(T)+\rho ^{3}\,B_{3}(T)+{\mathcal {O}}(\rho ^{4})}$

where the -th is the virial coefficient (where , since becomes an ideal gas for the real gas). The virial coefficients depend on the interaction potential between the particles and, in general, on the temperature. ${\displaystyle B_{i}}$ ${\displaystyle i}$ ${\displaystyle B_{1}=1}$ ${\displaystyle \rho \to 0}$

For and one obtains with the above representation of the virial equation if one defines the virial coefficients as . ${\displaystyle N=N_{\mathrm {A} }}$ ${\displaystyle V=V_{\mathrm {m} }}$ ${\displaystyle N_{\mathrm {A} }k=R}$ ${\displaystyle B_{i\,V}}$ ${\displaystyle B_{i\,V}=N_{\mathrm {A} }^{i-1}B_{i}}$

Derivation in the Grand Canonical Ensemble

The virial development can be derived in the grand canonical ensemble . The grand-canonical partition function is defined as ${\displaystyle \Xi =\Xi (T,V,\mu )}$

${\displaystyle \Xi =\sum _{N=0}^{\infty }Z_{N}z^{N}=1+Z_{1}z+Z_{2}z^{2}+Z_{3}z^{3}+\dotsb }$

It is the canonical partition of particles in the volume at temperature , the fugacity , the inverse temperature and the chemical potential . The grand potential is related to the logarithm of the partition: . Thus one obtains an equation of state as a power series in the fugacity ( is expanded by ): ${\displaystyle Z_{N}:=Z(T,V,N)}$ ${\displaystyle N}$ ${\displaystyle V}$ ${\displaystyle T}$ ${\displaystyle z:=e^{\beta \mu }}$ ${\displaystyle \beta :=(kT)^{-1}}$ ${\displaystyle \mu }$ ${\displaystyle \Omega =-pV}$ ${\displaystyle \Omega =-kT\,\ln \Xi }$ ${\displaystyle \ln \Xi }$ ${\displaystyle z=0}$

${\displaystyle {\begin{aligned}{\frac {pV}{kT}}&=-{\frac {\Omega }{kT}}=\ln \Xi =\ln \!\left(1+Z_{1}z+Z_{2}z^{2}+Z_{3}z^{3}+Z_{4}z^{4}+\dotsb \right)\\&=Z_{1}z+\left(Z_{2}-{\frac {1}{2}}Z_{1}^{2}\right)z^{2}+\left(Z_{3}-Z_{1}Z_{2}+{\frac {1}{3}}Z_{1}^{3}\right)z^{3}+\left(Z_{4}-Z_{1}Z_{3}-{\frac {1}{2}}Z_{2}^{2}+Z_{1}^{2}Z_{2}-{\frac {1}{4}}Z_{1}^{4}\right)z^{4}+\dotsb \end{aligned}}}$

In order to convert this equation of state into a thermal equation of state - this relates the state variables pressure , volume , temperature and number of particles - the fugacity must be expressed by the number of particles . ${\displaystyle p}$ ${\displaystyle V}$ ${\displaystyle T}$ ${\displaystyle N}$ ${\displaystyle z}$ ${\displaystyle N}$

By deriving the grand canonical potential according to the chemical potential , the negative mean number of particles is obtained : ${\displaystyle \mu }$ ${\displaystyle N}$

${\displaystyle {\begin{aligned}N&=-\left.{\frac {\partial \Omega }{\partial \mu }}\right|_{T,V}=-{\frac {\partial z}{\partial \mu }}\left.{\frac {\partial \Omega }{\partial z}}\right|_{T,V}=-\beta z\left.{\frac {\partial (-\beta ^{-1}\ln \Xi )}{\partial z}}\right|_{T,V}=z{\frac {1}{\Xi }}\left.{\frac {\partial \Xi }{\partial z}}\right|_{T,V}\end{aligned}}}$

So follows

${\displaystyle N\Xi =z\left.{\frac {\partial \Xi }{\partial z}}\right|_{T,V}\quad \Longleftrightarrow \quad N\left(1+Z_{1}z+Z_{2}z^{2}+Z_{3}z^{3}+\dotsb \right)=Z_{1}z+2Z_{2}z^{2}+3Z_{3}z^{3}+\dotsb }$

which can be solved with the power series approach using coefficient comparison for powers in . The first summands are ${\displaystyle z=\sum _{n=1}^{\infty }\alpha _{n}N^{n}}$ ${\displaystyle N}$

${\displaystyle {\begin{aligned}z=&{\frac {N}{Z_{1}}}+{\frac {N^{2}}{Z_{1}^{2}}}\left({\frac {Z_{1}^{2}-2Z_{2}}{Z_{1}}}\right)+{\frac {N^{3}}{Z_{1}^{3}}}\left({\frac {Z_{1}^{4}-5Z_{1}^{2}Z_{2}+8Z_{2}^{2}-3Z_{1}Z_{3}}{Z_{1}^{2}}}\right)+\\&+{\frac {N^{4}}{Z_{1}^{4}}}\left({\frac {Z_{1}^{6}-9Z_{1}^{4}Z_{2}-11Z_{1}^{3}Z_{3}+32Z_{1}^{2}Z_{2}^{2}+30Z_{1}Z_{2}Z_{3}-40Z_{2}^{3}-4Z_{1}^{2}Z_{4}}{Z_{1}^{3}}}\right)+\dotsb \end{aligned}}}$

Inserted into the above equation of state after ordering the powers of, this results in the thermal equation of state ${\displaystyle N}$

${\displaystyle {\begin{aligned}{\frac {pV}{kT}}=&N+N^{2}\left({\frac {1}{2}}-{\frac {Z_{2}}{Z_{1}^{2}}}\right)+N^{3}\left({\frac {1}{3}}-{\frac {2Z_{2}}{Z_{1}^{2}}}+{\frac {4Z_{2}^{2}}{Z_{1}^{4}}}-{\frac {2Z_{3}}{Z_{1}^{3}}}\right)+\\&+N^{4}\left({\frac {1}{4}}-{\frac {3Z_{2}}{Z_{1}^{2}}}+{\frac {27Z_{2}^{2}}{2Z_{1}^{4}}}-{\frac {20Z_{2}^{3}}{Z_{1}^{6}}}-{\frac {6Z_{3}}{Z_{1}^{3}}}+{\frac {18Z_{2}Z_{3}}{Z_{1}^{5}}}-{\frac {3Z_{4}}{Z_{1}^{4}}}\right)+\dotsb \end{aligned}}}$

On the right-hand side, the ensemble-averaged size (sum of all spatial coordinates times force) can be identified according to the equipartition theorem :

${\displaystyle {\frac {pV}{kT}}=N-{\frac {1}{3}}\sum _{i=1}^{3N}\left\langle q_{i}{\frac {\partial (\Phi /kT)}{\partial q_{i}}}\right\rangle =N+{\frac {1}{3kT}}\sum _{i=1}^{3N}\left\langle q_{i}F_{i}\right\rangle =N+{\frac {1}{3kT}}\sum _{i=1}^{N}\left\langle {\boldsymbol {q}}_{i}\cdot {\boldsymbol {F}}_{i}\right\rangle }$

If one introduces the particle number density , one obtains the virial expansion, i. H. a development of the thermal equation of state according to powers of the particle density: ${\displaystyle \rho =N/V}$

${\displaystyle {\frac {p}{kT}}=\rho \underbrace {1} _{B_{1}}+\rho ^{2}\underbrace {V\left({\frac {1}{2}}-{\frac {Z_{2}}{Z_{1}^{2}}}\right)} _{B_{2}(T)}+\rho ^{3}\underbrace {V^{2}\left({\frac {1}{3}}-{\frac {2Z_{2}}{Z_{1}^{2}}}+{\frac {4Z_{2}^{2}}{Z_{1}^{4}}}-{\frac {2Z_{3}}{Z_{1}^{3}}}\right)} _{B_{3}(T)}+\dotsb }$

whereby the virial coefficients could be identified. In the limit of vanishing particle density ( ) one obtains the ideal gas law in leading order ${\displaystyle B_{i}(T)}$ ${\displaystyle \rho \to 0}$

${\displaystyle {\frac {p}{kT}}=\rho \equiv {\frac {N}{V}}}$

Classic gas

The canonical partition function for particles is in the classical limit case (here the kinetic and the potential energy term are exchanged) with the Hamilton function given by ${\displaystyle Z_{N}}$ ${\displaystyle N}$ ${\displaystyle H=\sum _{i=1}^{N}{\frac {{\boldsymbol {p}}_{i}^{2}}{2m}}+U_{N}}$

${\displaystyle Z_{N}={\frac {1}{N!h^{3N}}}\underbrace {\int \!d^{3}{\boldsymbol {p}}_{1}\dotso d^{3}{\boldsymbol {p}}_{N}\,\exp \!\left(-\beta \sum _{i=1}^{N}{\frac {{\boldsymbol {p}}_{i}^{2}}{2m}}\right)} _{=(h/\Lambda )^{3N}}\underbrace {\int \!d^{3}{\boldsymbol {r}}_{1}\dotso d^{3}{\boldsymbol {r}}_{N}\,\exp \!\left(-\beta U_{N}\right)} _{=:Q_{N}}={\frac {Q_{N}}{N!\Lambda ^{3N}}}}$

where the pulse integrations could be carried out ( is the thermal wavelength ) and the configuration integral was introduced. The total potential energy of particles is composed of an external potential ( ) and internal potentials between the gas molecules, resulting from two-particle and multi-particle interactions ( ) : ${\displaystyle \Lambda }$ ${\displaystyle Q_{N}}$ ${\displaystyle U_{N}}$ ${\displaystyle N}$ ${\displaystyle u_{1}}$ ${\displaystyle u_{2},u_{3},\dotsc }$

${\displaystyle U_{N}=U_{N}({\boldsymbol {r}}_{1},\dotsc ,{\boldsymbol {r}}_{N})=\sum _{i=1}^{N}u_{1}({\boldsymbol {r}}_{i})+\sum _{\underset {i<j}{i,j=1}}^{N}u_{2}({\boldsymbol {r}}_{i},{\boldsymbol {r}}_{j})+\sum _{\underset {i<j<k}{i,j,k=1}}^{N}u_{3}({\boldsymbol {r}}_{i},{\boldsymbol {r}}_{j},{\boldsymbol {r}}_{k})+\dotsb +u_{N}({\boldsymbol {r}}_{1},\dotsc ,{\boldsymbol {r}}_{N})}$

Multi-particle interactions can e.g. B. by exchange interaction , induction and dispersion (such as the Axilrod-Teller triple dipole effect).

In particular, the following applies to the sums of state with the lowest particle numbers : ${\displaystyle Z_{N}}$ ${\displaystyle N}$

${\displaystyle Z_{0}=1\ ,\quad Z_{1}={\frac {Q_{1}}{\Lambda ^{3}}}\ ,\quad Z_{2}={\frac {Q_{2}}{2\Lambda ^{6}}}\ ,\quad Z_{3}={\frac {Q_{3}}{6\Lambda ^{9}}}}$

where the configuration integrals are as follows (the integrations over the spatial coordinates extend to the available volume , thus ): ${\displaystyle V}$ ${\displaystyle \int \!d^{3}{\boldsymbol {r}}=V}$

${\displaystyle {\begin{aligned}Q_{1}&=\int \!d^{3}{\boldsymbol {r}}_{1}\,e^{-\beta u_{1}({\boldsymbol {r}}_{1})}\\Q_{2}&=\int \!d^{3}{\boldsymbol {r}}_{1}\int \!d^{3}{\boldsymbol {r}}_{2}\,e^{-\beta \left[u_{1}({\boldsymbol {r}}_{1})+u_{1}({\boldsymbol {r}}_{2})+u_{2}({\boldsymbol {r}}_{1},{\boldsymbol {r}}_{2})\right]}\\Q_{3}&=\int \!d^{3}{\boldsymbol {r}}_{1}\int \!d^{3}{\boldsymbol {r}}_{2}\int \!d^{3}{\boldsymbol {r}}_{3}\,e^{-\beta \left[u_{1}({\boldsymbol {r}}_{1})+u_{1}({\boldsymbol {r}}_{2})+u_{1}({\boldsymbol {r}}_{3})+u_{2}({\boldsymbol {r}}_{1},{\boldsymbol {r}}_{2})+u_{2}({\boldsymbol {r}}_{1},{\boldsymbol {r}}_{3})+u_{2}({\boldsymbol {r}}_{2},{\boldsymbol {r}}_{3})+u_{3}({\boldsymbol {r}}_{1},{\boldsymbol {r}}_{2},{\boldsymbol {r}}_{3})\right]}\end{aligned}}}$

The virial development is thus written as

${\displaystyle {\frac {p}{kT}}=\rho \underbrace {1} _{B_{1}}+\rho ^{2}\underbrace {V\left({\frac {1}{2}}-{\frac {1}{2}}{\frac {Q_{2}}{Q_{1}^{2}}}\right)} _{B_{2}(T)}+\rho ^{3}\underbrace {V^{2}\left({\frac {1}{3}}-{\frac {Q_{2}}{Q_{1}^{2}}}+{\frac {Q_{2}^{2}}{Q_{1}^{4}}}-{\frac {1}{3}}{\frac {Q_{3}}{Q_{1}^{3}}}\right)} _{B_{3}(T)}+\dotsb }$

and the first virial coefficients are:

${\displaystyle {\begin{aligned}B_{1}&=1\\B_{2}&={\frac {V}{2}}\left(1-{\frac {2Z_{2}}{Z_{1}^{2}}}\right)={\frac {V}{2}}\left(1-{\frac {Q_{2}}{Q_{1}^{2}}}\right)\\B_{3}&=V^{2}\left({\frac {1}{3}}\left[1-{\frac {6Z_{3}}{Z_{1}^{3}}}\right]-{\frac {2Z_{2}}{Z_{1}^{2}}}\left[1-{\frac {2Z_{2}}{Z_{1}^{2}}}\right]\right)=V^{2}\left({\frac {1}{3}}\left[1-{\frac {Q_{3}}{Q_{1}^{3}}}\right]-{\frac {Q_{2}}{Q_{1}^{2}}}\left[1-{\frac {Q_{2}}{Q_{1}^{2}}}\right]\right)\end{aligned}}}$

The first virial coefficient is identical to 1, the second depends on the interaction with an external potential and pair interactions , the third on the interaction with an external potential, pair interactions and non-additive 3-particle interactions , etc. ${\displaystyle u_{1}}$ ${\displaystyle u_{2}}$ ${\displaystyle u_{3}}$

Special case of interaction: only distance-dependent pair interaction

The second virial coefficient is simplified for the special case that there is no external potential ( ) and the two- particle interactions only depend on the distance between the particles with and to ${\displaystyle u_{1}=0}$ ${\displaystyle |{\tilde {\boldsymbol {r}}}|=|{\boldsymbol {r}}_{1}-{\boldsymbol {r}}_{2}|}$ ${\displaystyle u_{2}({\boldsymbol {r}}_{1},{\boldsymbol {r}}_{2})=u_{2}(|{\tilde {\boldsymbol {r}}}|)}$ ${\displaystyle Q_{1}=\int d^{3}{\boldsymbol {r}}=V}$ ${\displaystyle Q_{2}=V\int \!d^{3}{\tilde {\boldsymbol {r}}}\,e^{-\beta \,u_{2}(|{\tilde {\boldsymbol {r}}}|)}}$

${\displaystyle {\begin{aligned}B_{2}&={\frac {1}{2}}\left[V-{\frac {V^{2}}{V^{2}}}\int \!d^{3}{\tilde {\boldsymbol {r}}}\,e^{-\beta \,u_{2}(|{\tilde {\boldsymbol {r}}}|)}\right]={\frac {1}{2}}\int \!d^{3}{\tilde {\boldsymbol {r}}}\,\underbrace {\left[1-e^{-\beta \,u_{2}(|{\tilde {\boldsymbol {r}}}|)}\right]} _{=:-f(|{\tilde {\boldsymbol {r}}}|)}\\&=-{\frac {1}{2}}\int \!d^{3}{\tilde {\boldsymbol {r}}}\,f(|{\tilde {\boldsymbol {r}}}|)=-{\frac {4\pi }{2}}\int _{0}^{\infty }d{\tilde {r}}\,{\tilde {r}}^{2}\,f({\tilde {r}})\end{aligned}}}$

The Mayer function was introduced in the second step and spherical coordinates were used in the third due to the symmetry. ${\displaystyle f({\boldsymbol {r}}):=e^{-\beta \,u_{2}({\boldsymbol {r}})}-1}$

If is in addition , then applies and the third virial coefficient becomes ${\displaystyle u_{3}=0}$ ${\displaystyle Q_{3}=V\int \!d^{3}{\tilde {\boldsymbol {r}}}_{1}d^{3}{\tilde {\boldsymbol {r}}}_{2}\,e^{-\beta \,\left[u_{2}(|{\tilde {\boldsymbol {r}}}_{1}|)+u_{2}(|{\tilde {\boldsymbol {r}}}_{2}|)+u_{2}(|{\tilde {\boldsymbol {r}}}_{1}-{\tilde {\boldsymbol {r}}}_{2}|)\right]}}$

${\displaystyle {\begin{aligned}B_{3}&=\int \!d^{3}{\tilde {\boldsymbol {r}}}_{1}\int \!d^{3}{\tilde {\boldsymbol {r}}}_{2}\,\left\{{\frac {1}{3}}\left(1-e^{-\beta \,\left[u_{2}(|{\tilde {\boldsymbol {r}}}_{1}|)+u_{2}(|{\tilde {\boldsymbol {r}}}_{2}|)+u_{2}(|{\tilde {\boldsymbol {r}}}_{1}-{\tilde {\boldsymbol {r}}}_{2}|)\right]}\right)-e^{-\beta \,u_{2}(|{\tilde {\boldsymbol {r}}}_{1}|)}\left(1-e^{-\beta \,u_{2}(|{\tilde {\boldsymbol {r}}}_{2}|)}\right)\right\}\\&=-{\frac {1}{3}}\int \!d^{3}{\tilde {\boldsymbol {r}}}_{1}\int \!d^{3}{\tilde {\boldsymbol {r}}}_{2}\ f(|{\tilde {\boldsymbol {r}}}_{1}|)\,f(|{\tilde {\boldsymbol {r}}}_{2}|)\,f(|{\tilde {\boldsymbol {r}}}_{1}-{\tilde {\boldsymbol {r}}}_{2}|)\\&=-{\frac {1}{3V}}\int d^{3}{\boldsymbol {r}}_{1}\int d^{3}{\boldsymbol {r}}_{2}\int d^{3}{\boldsymbol {r}}_{3}\,f_{1,2}f_{2,3}f_{3,1}\end{aligned}}}$

The last step was used. ${\displaystyle f_{i,j}=f(|{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}|)}$

Neglecting non-additive multi-particle interactions ( ) the fourth virial coefficient is ${\displaystyle u_{n\geq 3}=0}$

${\displaystyle {\begin{aligned}B_{4}&=-{\frac {1}{8}}\int d^{3}{\boldsymbol {r}}_{2}\int d^{3}{\boldsymbol {r}}_{3}\int d^{3}{\boldsymbol {r}}_{4}\,f_{1,2}f_{2,3}f_{3,4}f_{4,1}{\Bigl (}3+3f_{1,3}+3f_{2,4}+f_{1,3}f_{2,4}{\Bigr )}\\&=-{\frac {1}{8V}}\int d^{3}{\boldsymbol {r}}_{1}\int d^{3}{\boldsymbol {r}}_{2}\int d^{3}{\boldsymbol {r}}_{3}\int d^{3}{\boldsymbol {r}}_{4}\,f_{1,2}f_{2,3}f_{3,4}f_{4,1}{\Bigl (}3+6f_{1,3}+f_{1,3}f_{2,4}{\Bigr )}\end{aligned}}}$

and the fifth

${\displaystyle {\begin{aligned}B_{5}=&-{\frac {1}{30V}}\int d^{3}{\boldsymbol {r}}_{1}\int d^{3}{\boldsymbol {r}}_{2}\int d^{3}{\boldsymbol {r}}_{3}\int d^{3}{\boldsymbol {r}}_{4}\int d^{3}{\boldsymbol {r}}_{5}\,f_{1,2}f_{2,3}f_{3,4}f_{4,5}f_{5,1}\cdot \\&\cdot {\Bigl (}12+60f_{2,4}+10f_{1,4}f_{2,5}/f_{5,1}+60f_{1,3}f_{3,5}+30f_{1,4}f_{2,5}+10f_{1,4}f_{2,5}f_{2,4}/f_{5,1}+\\&\ \ +15f_{1,3}f_{3,5}f_{2,4}+30f_{1,4}f_{2,5}f_{2,4}+10f_{1,3}f_{3,5}f_{1,4}f_{2,5}+f_{1,3}f_{2,4}f_{3,5}f_{1,4}f_{2,5}{\Bigr )}\end{aligned}}}$

This example shows that the calculation of higher virial coefficients requires the evaluation of complex integrals. The integrands can, however, be determined systematically with the help of graph theory . If one neglects non-additive multi-particle interactions ( ), the virial coefficients can be determined graphically according to Mayer (cluster expansion): ${\displaystyle u_{n\geq 3}=0}$

${\displaystyle {\begin{aligned}B_{i+1}&=-{\frac {i}{i+1}}\beta _{i}=-{\frac {i}{i+1}}{\frac {1}{i!}}\left\{\beta _{i}\right\}_{\text{labelled}}=-{\frac {1}{i+1}}{\frac {1}{(i-1)!}}\left\{\beta _{i}\right\}_{\text{labelled}}\\&=-{\frac {i}{V}}\alpha _{i+1}=-{\frac {i}{V}}{\frac {1}{(i+1)!}}\left\{\alpha _{i+1}\right\}_{\text{labelled}}=-{\frac {1}{i+1}}{\frac {1}{(i-1)!}}{\frac {1}{V}}\left\{\alpha _{i+1}\right\}_{\text{labelled}}\end{aligned}}}$

for . The irreducible cluster integrals are graphs with nodes and the Mayer function as a link, with integration being performed via nodes. They are irreducible cluster integrals over graphs with nodes and the Mayer function as a link, with integration over all nodes. If a connected graph cannot be divided into two disconnected graphs by cutting one edge, it is called irreducible. ${\displaystyle i\geq 1}$ ${\displaystyle \beta _{i}}$ ${\displaystyle (i+1)}$ ${\displaystyle i}$ ${\displaystyle \alpha _{i}}$ ${\displaystyle i}$ ${\displaystyle i}$

Calculation of the second virial coefficient for example potential

Many realistic gases show a strong repulsion for small distances between molecules (Pauli repulsion when the atomic shells overlap) and a weak attraction for large distances (such as like ). A simple two-particle interaction potential can be modeled as follows: ${\displaystyle r^{-6}}$

${\displaystyle u_{2}(r)={\begin{cases}\infty &{\text{für}}\quad r<\sigma \\v(r)&{\text{für}}\quad r\geq \sigma \end{cases}}}$

The potential is infinite for distances smaller than the hard sphere radius and slightly negative for distances greater than this: and . The Mayer function can thus be approximated as ${\displaystyle \sigma =2r_{0}}$ ${\displaystyle v(r)<0}$ ${\displaystyle |v(r)|\ll kT}$

${\displaystyle f(r)\approx {\begin{cases}-1&{\text{für}}\quad r<\sigma \\-{\frac {v(r)}{kT}}&{\text{für}}\quad r\geq \sigma \end{cases}}}$

and the second is virial coefficient

${\displaystyle B_{2}=-{\frac {4\pi }{2}}\int _{0}^{\infty }dr\,r^{2}\,f(r)=2\pi \underbrace {\int _{0}^{\sigma }dr\,r^{2}} _{\sigma ^{3}/3}-{\frac {2\pi }{kT}}\int _{\sigma }^{\infty }dr\,r^{2}\,\left[-v(r)\right]=:b-{\frac {a}{kT}}}$

Here, the intrinsic volume linked ${\displaystyle b}$ ${\displaystyle V_{0}}$

${\displaystyle b:={\frac {2\pi }{3}}\sigma ^{3}=4{\frac {4\pi }{3}}r_{0}^{3}=4V_{0}>0}$

and is a measure of the mean external (attractive) potential ${\displaystyle a}$

${\displaystyle a:=2\pi \int _{\sigma }^{\infty }dr\,r^{2}\,\underbrace {\left[-v(r)\right]} _{>0}>0}$

Thus in this approximation it is negative for small temperatures and positive for large ones with the limit value for . The temperature at which the second virial coefficient disappears is called the Boyle temperature . ${\displaystyle B_{2}=b-a/(kT)}$ ${\displaystyle B_{2}=b}$ ${\displaystyle T\to \infty }$ ${\displaystyle T=a/(bk)}$

The virial development is thus:

${\displaystyle {\frac {p}{kT}}=\rho +\rho ^{2}\left(b-{\frac {a}{kT}}\right)+{\mathcal {O}}(\rho ^{3})}$

For low density ( small) we only consider the development only up to the second order. Reshaping and utilizing for (the mean volume available for each particle is much larger than its own volume ) yields: ${\displaystyle \rho }$ ${\displaystyle (1-\rho b)^{-1}\approx 1+\rho b}$ ${\displaystyle \rho b\ll 1}$ ${\displaystyle \rho ^{-1}}$ ${\displaystyle b}$

${\displaystyle p+a\rho ^{2}=kT\rho \left(1+\rho b\right)\approx kT{\frac {\rho }{1-\rho b}}=kT{\frac {N}{V-Nb}}}$

${\displaystyle \Longrightarrow \quad \left(p+a{\frac {N^{2}}{V^{2}}}\right)\left(V-Nb\right)\approx NkT}$

The last equation is the so-called Van der Waals equation .

Equation of state of hard balls

For a system of hard spheres with a radius , the interaction potential or the Mayer function (with ) is ${\displaystyle R}$ ${\displaystyle 2R=\sigma }$

${\displaystyle u_{2}(r)={\begin{cases}\infty &{\text{für}}\quad r<2R\\0&\mathrm {f{\ddot {u}}r} \quad r\geq 2R\end{cases}}\quad \Rightarrow \quad f(r)={\begin{cases}-1&{\text{für}}\quad r<2R\\0&{\text{für}}\quad r\geq 2R\end{cases}}}$

Although this system is never realized in nature, it is often used in statistical mechanics, since the structure of real liquids is mainly determined by repulsive forces. This model is thus the simplest model with liquid-like properties at high densities and is used as a reference system for perturbation calculations. It already shows rich structural and thermodynamic properties, such as B. a liquid-solid phase transition in the range 0.494 to 0.545 of the packing density. The greatest possible packing density is reached at 0.7405 (see closest packing of spheres ).

The first four virial coefficients can be calculated analytically for the hard sphere model ( hereinafter referred to as the sphere volume ). The second virial coefficient results analogously to the Van der Waals gas without an attractive part . The third can be calculated as follows: ${\displaystyle V_{0}}$ ${\displaystyle V_{0}={\tfrac {4}{3}}\pi R^{3}}$ ${\displaystyle B_{2}=4V_{0}}$

${\displaystyle {\begin{aligned}B_{3}&=-{\frac {1}{3}}\int \!\mathrm {d} ^{3}{\boldsymbol {r}}\int \!\mathrm {d} ^{3}{\boldsymbol {r}}'\,f(|{\boldsymbol {r}}|)f(|{\boldsymbol {r}}'|)f(|{\boldsymbol {r}}-{\boldsymbol {r}}'|)\quad {\text{mit}}\quad {\boldsymbol {r}}_{2}:={\boldsymbol {r}}'-{\boldsymbol {r}}\,,\ {\boldsymbol {r}}_{1}:={\frac {{\boldsymbol {r}}+{\boldsymbol {r}}'}{2}}\\&=-{\frac {1}{3}}\int \!\mathrm {d} ^{3}{\boldsymbol {r}}_{2}\,f(|{\boldsymbol {r}}_{2}|)\int \!d^{3}{\boldsymbol {r}}_{1}\,f\left(\left|{\boldsymbol {r}}_{1}-{\frac {1}{2}}{\boldsymbol {r}}_{2}\right|\right)f\left(\left|{\boldsymbol {r}}_{1}+{\frac {1}{2}}{\boldsymbol {r}}_{2}\right|\right)\end{aligned}}}$

The integral over can be evaluated geometrically: It corresponds to the overlap volume of two spheres with a radius whose distance is. This volume is twice the volume of a spherical cap with height . The height is half of the "overlap distance" between the ball centers. Then you can integrate into spherical coordinates via (the minus sign disappears because the sphere is around the origin): ${\displaystyle r_{1}}$ ${\displaystyle \sigma }$ ${\displaystyle r_{2}}$ ${\displaystyle \pi h^{2}(3\sigma -h)/3}$ ${\displaystyle h=\sigma -r_{2}/2}$ ${\displaystyle 2\sigma -r_{2}}$ ${\displaystyle r_{2}}$ ${\displaystyle f=-1}$

${\displaystyle B_{3}=-{\frac {1}{3}}\int _{0}^{\sigma }\!\mathrm {d} r_{2}\,4\pi r_{2}^{2}\,(-1){\frac {2}{3}}\pi \left(\sigma -{\frac {1}{2}}r_{2}\right)^{2}\left[3\sigma -\left(\sigma -{\frac {1}{2}}r_{2}\right)\right]={\frac {5}{18}}\pi ^{2}\sigma ^{6}=10V_{0}^{2}}$

The analytical calculation of the fourth virial coefficient requires a lengthy calculation. The result is:

${\displaystyle B_{4}={\frac {2707\pi +(438{\sqrt {2}}-4131\arccos {\frac {1}{3}})}{70\pi }}\,V_{0}^{3}\approx 18{,}364768\,V_{0}^{3}}$

The higher virial coefficients were calculated numerically. Using the packing density, the first ten terms of the virial expansion are : ${\displaystyle \eta =\rho V_{0}}$

${\displaystyle {\begin{aligned}{\frac {p}{kT}}=\rho &{\big [}1+4\,\eta +10\,\eta ^{2}+18{,}364768\,\eta ^{3}+28{,}22445\,\eta ^{4}+39{,}81545\,\eta ^{5}+\\&+53{,}3418\,\eta ^{6}+68{,}534\,\eta ^{7}+85{,}805\,\eta ^{8}+105{,}8\,\eta ^{9}+\dotsb {\big ]}\end{aligned}}}$

The virial coefficients do not depend on the temperature here. The hard sphere system, like the ideal gas, is an 'athermal' system. The canonical partition functions and structural properties are independent of the temperature, they only depend on the packing density. The free energy comes purely from entropy, not from contributions of potential energy.

If the first five terms are approximated by the neighboring integer (4, 10, 18, 28, 40), then these can be calculated using a simple rule for . If you use this to calculate all coefficients, you get an approximate equation of state for hard balls: ${\displaystyle B_{i+1}=(i^{2}+3i)V_{0}^{i}}$ ${\displaystyle i\geq 1}$

${\displaystyle {\frac {p}{kT\rho }}\approx 1+\sum _{i=1}^{\infty }(i^{2}+3i)\eta ^{i}={\frac {1+\eta +\eta ^{2}-\eta ^{3}}{(1-\eta )^{3}}}}$

In the last step, the first two derivatives of the geometric series were used. Despite the approximation for higher virial coefficients, this equation of state according to Carnahan and Starling agrees very well with simulation results for the liquid phase (largest deviations approx. 1%). The phase transition and the equation of state of the solid phase are not included in the Carnahan-Starling approximation.

literature

• Torsten Fließbach: Statistical Physics. Spectrum Academic Publishing House, 2010, ISBN 978-3-8274-2527-0 .
• Franz Schwabl : Statistical Mechanics. 3rd, updated edition. Springer, Berlin a. a. 2006, ISBN 3-540-31095-9 .
• Hermann Schulz: Statistical Physics. Based on quantum theory. An introduction. Harri Deutsch, Frankfurt am Main 2005, ISBN 3-8171-1745-0 .