Coefficient of the virial

D2. Fugacity Coefficient of the Virial Equation (Leiden Form)... 

Coefficients B2, B, etc. are called the second, third, etc. coefficient of the virial. We can calculate the first coefficients of the virial using the cubic state equations as we did for the Dietrerici equation, but it does not provide any additional information. For example, for the Van der Waals equation we obtain, respectively ... 

It is therefore possible to measnre the coefficients of the virial using pressure measurements at a constant volume for different values of the density p of the molecules. 

The interaction energies between molecules is a complex issue and is most frequently satisfied by approximations. For example, we have seen that (see section 6.6) the interaction between two molecules does not disturb their internal organizations and the mteractions forces only depend on the distance between the molecules (see section 6.10). Between two molecules, the corrective energetic term hnked to the interaction ab will appear, which is practically zero when both molecules A and B are far enough as in the example of the Letmard-Jones potential (see relations [6.100] or [6.103]). Calculating the partition function of a fluid, in particular the calculation of the second coefficient of the virial, is tricky many examples in the literature are approximate or full of poorly justified shortcuts. This is why we chose to... 

We can see that this expression is identical to the virial equation given by relation [7.20], limited to the second term by identification, the second coefficient of the virial is therefore ... 

The term Baa(T) represents the second coefficient of the virial reduced to one molecule. 

The second coefficient of the virial was calculated by Lennard-Jones using, in relation [7.76], the potential function deduced from the van der Waals forces model, i.e. ... 

We are now going to focus on the second coefficient of the virial, calculated from the microscopic properties of macroscopic state equations, to ultimately give a physical meaning to the parameters of these state equations. We are therefore going to assess the application of relations [7.97], [7.102] and [7.110] to relation [7.96] to calculate the second coefficient, then to [7.94] to compare several state equations. 

We are limited, when determining the second coefficient of the virial, to spheiieal molecules. Although this model is aceeptable for simple gases or polyatomic gases with an almost spherieal shape such as ammonia, other models should be developed for moleeules with elongated forms and for polymers. 

With regard to the imperfection of gases, we are limited to the forces between two molecules only (two body forces) which give the expressions of the second coefficient of the virial. Researchers have endeavored to calculate the third, fourth and fifth coefficient of the virial. Here the three body forces are involved for the third coefficient, four body forces for the fourth and compact packing of spheres for the fifth coefficient. As for the second coefficient, the authors initially stuck to the hard-sphere model without attraction force (see section 7.3.3.1 and Figure 7.6), and as in the case of the second coefficient, they obtained coefficients practically independent of temperature, which allowed Hirshfelder and Roseveare to propose a state equation in the form ... 

Figure l.W.Variation in the third coefficient of the virial with temperature according toMalijevsky et al. 

In the next section, we will see how to interpret the second coefficient of the virial, which is often sufficient, by developing the same method we used in section 7.4.3 for pure gases. 

This quadratic form in the variation of the second coefficient of the virial with the molar fraction of one of the gases is followed by a number of mixtures of binary gas compounds. Figure 8.1 shows the results for the hydrogen/helium, hydrogen/argon and helirrm/argon couples. 

Figure 8.1. Variations in the second coefficient of the virial with composition -I - HfHe, U - H/Ak in - He/Ar om [GIB 29])...

As is well-known, the virial expansion, Eq. (3), provides a simple and convenient description of the EoS for dilute gases, has been widely used for both practical and theoretical studies, and represents a flrst attempt to understand the relationship between molecular interactions and the macroscopic behavior of fluids. In particular, values for the second, third, fourth and flfth virial coefficients (all in reduced L-J units) are available [16,285,286,291] for 2D L-J fluids. These values have been fltted by Reddy et al. [195] as polynomials in (T ) for the second, third, and fourth coefficients, and as powers of T y for the fifth one. The coefficients are given in Table 13. The Reddy et al. [195] results then lead to a complete description of five coefficients of the virial EoS, which has been used to obtain approximate values of the critical point see Table 11. 

This model [GUG32] is extrapolated from the imperfect gas model, which can be used to calculate the second coefficient of the virial (see section A.3.4 in Appendix 3). The canonical partition function then takes the form of equation [A.3.41]. 

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