Cluster integrals contributing to the fourth virial coefficient of hard convex bodies

Abstract

The fourth-order virial expansion represents an important tool in the description of the equilibrium behaviour of pure fluids and mixtures in the vicinit of their critical point/critical region. Dependences of cluster integrals D₄ ^HCB, D₅ ^HCB and D₆ ^HCB of hard convex bodies on the geometric characteristics (i.e. the volume, surface area and the mean curvature integral of the given body) form the basic information necessary for the evaluation of the fourth virial coefficient, D, of Kihara non-spherical molecules. We determined D₄ ^HCB, D₅ ^HCB and D₆ ^HCB for pure prolate and oblate hard spherocylinders with the non-sphericity parameter α E (1,3). A Monte Carlo integration technique was employed and the individual contributions D₄ ^HCB/V³, D₅ ^HCB/V³ and D₆ ^HCB/V³ were expressed as quadratic functions of α, with coefficients (integral quantities) obtained by a three-step fitting procedure. Values of the HCB fourth virial coefficient (obtained as an algebraic sum of m_iD_i ^HCB) for the individual types of molecules agree well with the pseudo-experimental data from the literature. The expressions for D_i ^PS and D_i ^OS as well as that for the total fourth virial coefficient for prolate and oblate spherocylinders differ considerably; none of the one-parameter equations of state (proposed for HCB systems) yields an expression predicting correctly the fourth virial coefficient of HCBs in the considered range of α. An attempt is made to express the fourth virial coefficient in terms of two non-sphericity parameters; different results for prolate and oblate hard spherocylinders were obtained.