Virial equations of state of d-dimensional semi classical Lennard-Jones fluid mixture

The virial coefficient of d-dimensional Lennard-Jones (LJ) (12-6) fluid mixture in the semi classical limit are studied. Explicit expressions for classical and first quantum correction of the second virial coefficient of the LJ (12-6) fluid mixture are derived. The results are discussed. The excess second virial coefficient depends on the dimensionality ‘d’.


INTRODUCTION
In recent years, d-dimensional fluids have been a subject of considerable inherent.However, this is confined to the one-component fluid 1,2 .No attempt have been made to study the equilibrium properties of d-dimensional fluid mixture.
In the present paper, we investigate the classical and quantum corrections to the virial coefficient of the d-dimensional fluid mixture.The usual way of studying the properties of the semi classical system is to expand them in the ascending power of Planck's constant  3 .Extensive work has been done for three-dimensional fluid 3 .However, no work is available for d-dimensional fluid mixture.
The purpose of present investigation is to derive unified expressions for the virial coefficient of the d-dimensional fluid mixture, whose molecules interact via the Lennard-Jones (LJ) (12-6) potential.

Virial expansion of equation of state of fluid mixture
In this low density limit, the equation of state is given by 3. where m  = m  is the mass of a molecule of species.

Second virial coefficient for d-dimensional Lennard-Jones (12-6) fluid mixture
We consider a fluid mixture whose molecules interact via the LJ (12-6) potential.
where  ab represents the depth of the potential well and  ab the diameter of the molecule.For the unlike species,  12 and  12 are given by where 12 is an adjustable parameter, which is fairly close to unity.using the relation 1 ... (7)   where the function f(r) depends on the distance r=|r| and S d is the surface area of the unit sphere, given by 1,4 S d = d d/2 (1+d/2) ...(8) Here (1) =1! is the Gamma function.The Eq. ( 3) and ( 4

Quantum correction to the second virial coefficient
We introduce the reduced quantum parameters

RESULTS AND DISCUSSION
We have calculated the reduced second virial Table 1 coefficient for d-dimensional He 4 -He 3 mixture for 1d5 as reported in Table 1., which are reported in Table 2.We see that the excess second virial coefficient depends on the dimensionality 'd'.Summary We have given as explicit unified expressions for the second virial coefficient for the d-dimensional LJ (12-6) model in the semi classical limit.From the study, we find that the second virial coefficient depends on the dimensionality.
The quantum effect increases with decrease of temperature and at lower temperature one should take the higher order terms to estimate the quantum effects for He 4 -He 3 mixture for which the quantum parameters are large.

Table 2 :
The excess second virial coefficient B * 43 (d)for He 4 -He 3 in A 0 units