Quantum Second Virial Coefficients for Krypton-86 Gas

A quantum second virid coefficient for krypton gas is calculated using ab initio potential, Barker et al potential and Morse potential. Calculations have been made using Galitskii-Migdal-Feynman (GMF) formalism. Agreements with experiments is good though not perfect.


Introduction
Virial coefficients B i B appear as coefficients in the virial expansion of the pressure of a many-particle system -The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles-in powers of the density, providing systematic corrections to the ideal gas law.They are characteristic of the interaction potential between the particles and in general depend on the temperature.The second virial coefficient B 2 B depends only on the pair interaction between the particles.Many methods have been developed for calculating second virial coefficients for Kr gas: A differential method is described for measuring the second virial coefficient of a gas from approximately the normal boiling-point of the substance to room temperature, given the value of the virial coefficient at one temperature (Byrne et al.,1971).The experimental second virial coefficients of Ar, Kr, and Ar-Kr mixtures were determined for the temperature range 80-140K.These were 5-15% larger in magnitude than those calculated with the Lennard-Jones 12-6 potential (Fender & Halsey, 1962).In our work quantum second virial coefficient at temperatures above 120°K is calculated by using theoretical method at low density.

GMF T-matrix
The starting point in computing B q , then, is the determination of E δ  .This can be accomplished by solving the GMF integral equation for the T-matrix, using a matrix-inversion technique.The form the T-matrix is given by (Ghassib et al., 1976): Here: p and p′ are the relative incoming and outgoing momenta; P is the center-of-mass momentum.The the parameter s the total energy of the interacting pair in the center-of-mass frame, given by is the total energy of the pair and 2 Ρ is the energy carried by the center of mass.

The Second Virial Coefficient & the Beth-Uhlenbeck Formula
The Beth-Uhlenbeck formula gives a quantum expression of second virial coefficient B q (T) (Seguin et al., 1987): ) is the thermal de Broglie wavelength, E B denotes bound-state energies and ( ) is the 'effective' ℓ-partial phase shift corresponding to energy ( ) . The word 'effective' indicates that the quantities involved pertain to a many-body medium; they depend on the density and temperature.The first term on the right-hand side is the 'ideal Bose gas' term which diminishes with increasing temperature; it depends on statistics but not on the interaction.The second represents the contribution of bound states in the system; it is related to its discrete energy levels.However, this term can be neglected (Dardi & Dahler, 1991).The third term represents the contribution of scattering states; it is the major term in B q (Ghassib, 2014).

Results and Discussion
The results of our work and previous experimental (Sengers et al., 1971) are summarized in Table 1 and Figs.1-3.
Table 1.The quantum second virial coefficient, B q (cm 3 /mol), at different temperatures, T (K), at n = 1 × 10 20 atoms/ m -3 with previous experimental value Clearly, from table 1, the quantum second virial coefficient, B q (T), is an increasing function of temperature throughout most of the useful temperature range.(It does decrease slightly at high temperatures.).B q is predominantly negative; but it becomes less negative as T increases.This can be explained as follows: At low T, the mean energies of the atoms in the gas are of the same order of magnitude as the depth of the potential well; so that the interacting atoms spend most of their time in the attractive region of the potential.This results in a decrease in the gas pressure which leads to a negative B(T).At high T, the average energies of the atoms increase; so that the predominant contribution to B(T) arises from the repulsive portion of the potential.This causes an increase in the gas pressure and, hence, B(T) becomes less negative (Gans, 1994).

Conclusion
The second virial coefficient had roughly the correct behavior as it decreases monotonically when temperature is lowered.The second virial coefficients show good agreement with experimental values except at temperatures below 156 (Mores potential), 152 (Barker et al.) and 160 (ab initio potential) K where the experimental values become appreciably more negative than those calculated either with potentials in our work.

Figure 1 .
Figure 1.Deviation plot for second virial coefficients [cm 3 /mole] as a function of temperature T [K] where the potential is Mores potential