Virial coefficients
appear as coefficients in the
virial expansion of the pressure of a
many-particle system in powers of the density, providing systematic corrections to the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
. They are characteristic of the interaction potential between the particles and in general depend on the temperature. The second virial coefficient
depends only on the pair interaction between the particles, the third (
) depends on 2- and non-additive 3-body interactions, and so on.
Derivation
The first step in obtaining a closed expression for virial coefficients is a
cluster expansion
In statistical mechanics, the cluster expansion (also called the high temperature expansion or hopping expansion) is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interac ...
of the
grand canonical partition function
:
Here
is the pressure,
is the volume of the vessel containing the particles,
is
Boltzmann's constant,
is the absolute temperature,
is the
fugacity
In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas whic ...
, with
the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
. The quantity
is the
canonical partition function of a subsystem of
particles:
:
Here
is the Hamiltonian (energy operator) of a subsystem of
particles. The Hamiltonian is a sum of the
kinetic energies of the particles and the total
-particle
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...
(interaction energy). The latter includes pair interactions and possibly 3-body and higher-body interactions. The
grand partition function
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggre ...
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
equals
. In this manner one derives
:
:
.
These are quantum-statistical expressions containing kinetic energies. Note that the one-particle partition function
contains only a kinetic energy term. In the
classical limit the kinetic energy operators
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with the potential operators and the kinetic energies in numerator and denominator cancel mutually. The
trace (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.
The derivation of higher than
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
Joseph E. Mayer and
Maria Goeppert-Mayer.
They introduced what is now known as the
Mayer function The Mayer f-function is an auxiliary function that often appears in the series expansion of thermodynamic quantities related to classical many-particle systems.Donald Allan McQuarrie, ''Statistical Mechanics'' (HarperCollins, 1976), page 228
It is ...
:
:
and wrote the cluster expansion in terms of these functions. Here
is the interaction potential between particle 1 and 2 (which are assumed to be identical particles).
Definition in terms of graphs
The virial coefficients
are related to the irreducible
Mayer cluster integrals
through
:
The latter are concisely defined in terms of graphs.
:
The rule for turning these graphs into integrals is as follows:
# Take a graph and
label its white vertex by
and the remaining black vertices with
.
# Associate a labelled coordinate ''k'' to each of the vertices, representing the continuous degrees of freedom associated with that particle. The coordinate 0 is reserved for the white vertex
# With each bond linking two vertices associate the
Mayer f-function corresponding to the interparticle potential
# Integrate over all coordinates assigned to the black vertices
# Multiply the end result with the
symmetry number of the graph, defined as the inverse of the number of
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s of the black labelled vertices that leave the graph topologically invariant.
The first two cluster integrals are
:
The expression of the second virial coefficient is thus:
:
where particle 2 was assumed to define the origin (
).
This classical expression for the second virial coefficient was first derived by
Leonard Ornstein
Leonard Salomon Ornstein (November 12, 1880 in Nijmegen, the Netherlands – May 20, 1941 in Utrecht, the Netherlands) was a Dutch physicist.
Biography
Ornstein studied theoretical physics with Hendrik Antoon Lorentz at University of Lei ...
in his 1908
Leiden University
Leiden University (abbreviated as ''LEI''; nl, Universiteit Leiden) is a public research university in Leiden, Netherlands. The university was founded as a Protestant university in 1575 by William, Prince of Orange, as a reward to the city o ...
Ph.D. thesis.
See also
*
Boyle temperature
The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_(T), becomes zero.
It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out
P = RT ...
- temperature at which the second virial coefficient
vanishes
*
Excess virial coefficient
*
Compressibility factor
In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas ...
References
Further reading
*
*
*http://scitation.aip.org/content/aip/journal/jcp/50/10/10.1063/1.1670902
*http://scitation.aip.org/content/aip/journal/jcp/50/11/10.1063/1.1670994
* Reid, C. R., Prausnitz, J. M., Poling B. E., Properties of gases and liquids, IV edition, Mc Graw-Hill, 1987
{{DEFAULTSORT:Virial Coefficient
Statistical mechanics