Virial expansion
Virialgleichungen are extensions of the general gas equation by a series expansion in powers of. Make approximate equations of state for real gases dar. When canceling the series expansion after the first term is again obtained the general gas equation. Leads to the further expansion, however, creates a potentially infinite number of state equations with an increasing number of parameters. In implicit form the general series expansion is as follows:
The individual symbols stand for the following sizes:
- Vm - molar volume
- T - temperature
- P - pressure
- R - universal gas constant
- - Second virial coefficient
- - Third virial coefficient (etc.)
Virial coefficient
The virial coefficients result from the interactions between the molecules. You are not physically interpretable.
The second virial coefficient is obtained approximately from the pair potential V (r) between the molecules:
- NA - Avogadro constant
- K - Boltzmann constant
- V - pair potential
- R - distance between the molecules
The third virial coefficient depends on the interactions within a group of three molecules, for all further applies correspondingly. The virial equation with two or three virial coefficient is valid only in moderate pressures. If there are no experimental values for the virial coefficients are available, they can be calculated using the empirical model to Hayden - O'Connell. Here, the second virial coefficient of critical temperature, critical pressure, dipole moment and radius of gyration is estimated.
Connection to the van der Waals equation
Another, more common equation of state for real gases, the van der Waals equation dar. About a simplification can be produced between this and the virial equation with two virial coefficients of a connection.
If the series expansion terminated, this correction factor based on the principle of correspondence from the critical state of the substance. At the same time, this form of the virial equation is a simplified form of the Redlich - Kwong equation.
For the parameters A and B, see Van der Waals equation. If T is equal to the Boyle temperature, so have B ' = 0
Virial expansion: Consideration of statistical mechanics
For real gases, that is, interacting particles, the partition function of a statistical ensemble can not evaluate exactly. For gases of low density, however, can this be calculated approximately. The virial expansion is a development of the thermal equation of state in the particle density ( particles per volume):
Here is the -th virial coefficient. The virial coefficients depend on between the particles and in general on the temperature of the interaction potential.
For, and, with above presentation of the virial equation, if one defines the virial coefficients as.
Derivation in the Grand Canonical Ensemble
In the grand canonical ensemble, the virial expansion can be derived. The grand canonical partition function is defined as
Here is the canonical partition function for particles in volume at temperature the fugacity, the inverse temperature and the chemical potential. The grand potential is related to the logarithm of the partition. Thus we obtain an equation of state as a power series in the fugacity ( is designed to ):
To be in a thermal equation of state, this equation of state - this sets the variables pressure, volume, temperature and number of particles in relationship - to convert the fugacity must be expressed by the number of particles.
By deriving the grand canonical potential with respect to the chemical potential yields the negative mean numbers of particles:
This follows
That can be solved with the power series approach by comparing coefficients of powers in. The first summand are
This is substituted into the above equation of state order of magnitude of the thermal equation of state
On the right side, the ensemble - averaged size can be according to the equipartition theorem (sum of all spatial coordinates times force) identify:
Performs to the particle number density a, we obtain the virial expansion, ie a development of the thermal equation of state in powers of the particle density:
The virial coefficients could be identified. In the limit of vanishing particle density ( ), we obtain in leading order the ideal gas law
Classical gas
The canonical partition function for particles in the classical limit ( here reversed the kinetic to the potential energy term ) with the Hamiltonian give by
Wherein the pulse integrations were performed ( the thermal wavelength) and the integral configuration has been 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 Mehrteilchenwechselwirkungen () together:
Mehrteilchenwechselwirkungen can, for example, by exchange interaction, by induction and dispersion ( about the Axilrod -Teller triple -dipole effect) caused.
In particular, for the partition functions with the lowest number of particles:
The configuration integrals are as follows ( the integrations over the spatial coordinates extend the available volume, thus ):
Thus, the virial writes as
And the first virial coefficients are:
The first virial coefficient is the same as 1, the second depends on the interaction with an external potential and pair interactions, the third of the interaction with an external potential, pair interactions and non-additive three - particle interactions, and so forth
Special case of interaction: only distance-dependent pair interaction
The second virial coefficient is simplified for the special case that no external potential is present () and the Zweiteilchenwechselwirkungen depend only on the distance of the particles and
The Mayer function was introduced in the second step and used in the third due to the symmetry of spherical coordinates.
If, in addition, shall apply and the third virial coefficient becomes
In the last step was used.
Neglecting non- additive Mehrteilchenwechselwirkungen () is the fourth virial coefficient
And the fifth
From this example you can see that the calculation of higher virial coefficient requires the evaluation of complex integrals. The integrand, however, can be determined systematically with the help of the graph theory. Neglecting non-additive Mehrteilchenwechselwirkungen (), then the virial leave any graph-theoretical Mayer (cluster development ) determine:
For. Here are the irreducible cluster integrals over graphs with nodes and the Mayer function as a bond, is integrated over nodes. They are irreducible cluster integrals over graphs with nodes and the Mayer function as a bond, is integrated over all nodes. Can not be divided by cutting an edge into two unconnected graph, a connected graph, it is called irreducible.
Calculation of the second virial coefficient for example potential
Many realistic gases show small molecule distances a strong repulsion ( Pauli repulsion in overlap of the atomic shells ) and for large distances a weak attraction (such as ). A simple Zweiteilchenwechselwirkungs potential can be modeled as follows:
The potential is infinite for distances smaller than the hard-sphere radius and for distances greater than this slightly negative: and. Thus, the Mayer function can approximate as
And the second virial coefficient is
This is associated with the internal volume
And is a measure of the average outer ( attractive ) Potential
Thus, in this approximation is for small temperatures and negative for large positive with the limit for. The temperature at which the second virial coefficient disappears is Boyle temperature.
The virial expansion is:
For low density ( small ), we consider only the development only up to second order. Forming and for exploitation of (average for each of the available volume of the particles is much larger than the intrinsic volume ) yields:
Last equation is the so-called van der Waal's equation.
For a system of hard spheres with radius the interaction potential and the Mayer function is ( with )
Although this system is never realized in nature, it is often used in statistical mechanics, since the structure of real fluids is mainly determined by repulsive forces. Thus, this model is the simplest model with liquid-like properties at high density, and is used as a reference system for error calculations. It already shows extensive structural and thermodynamic properties, such as a liquid-solid phase transition in the range from 0.494 to 0.545 of the packing density. The maximum packing density is reached at 0.7405 (see densest packing of spheres ).
The first four virial coefficients can be calculated analytically for the hard-sphere model ( hereafter denote the spherical volume ). The second virial coefficient is obtained analogous to the van der Waals gas with no attractive part to. The third can be calculated as follows:
The integral can be evaluated over geometrically: It corresponds to the overlap volume of two spheres with radius, is the distance between them. This volume is twice the volume of a spherical cap with height. The amount is half of the " Überlappstrecke " between the ball centers. Then can be over in spherical coordinates integrate (the minus sign disappears, there is within the sphere around the origin):
The analytical calculation of the fourth virial coefficient requires a lengthy statement. The result is:
The higher virial coefficients were calculated numerically. The first ten terms of the virial expansion loud using the packing density:
The virial coefficients do not depend here on the temperature. The hard-sphere system is like the ideal gas, athermal ' system. The canonical partition function and structural properties are independent of temperature, they only depend on the packing density. The free energy comes in from the entropy, not contributions of potential energy.
Approaching the first five terms by the neighboring integer (4, 10, 18, 28, 40), so these can be calculated by a simple rule for. If one uses this to calculate all the coefficients, we obtain an approximate equation of state for hard spheres:
In the last step with the first two derivatives of the geometric series, were used. This equation of state by Carnahan and Starling is true in spite of the approximation for higher virial coefficients very well with simulation results for the liquid phase match ( largest deviations about 1%). The phase transition and the equation of state of the solid phase is not included in the Carnahan -Starling approximation.