Internal energy hard spheres

If we assume that molecules can be considered as billiard balls (hard spheres) without internal degrees of freedom, then the probability of reaction between, say, A and B depends on how often a molecule of A meets a molecule of B, and also if during this collision sufficient energy is available to cross the energy barrier that separates the reactants, A and B, from the product, AB. Hence, we need to calculate the collision frequency for molecules A and B. 

FIG. 4 The calculated internal energy of a 1-1 salt (line) is compared with the corresponding simulation results (open circles) obtained by Van Megen and Snook (Ref. 20). The Debye-Hiickel (DH, dashed line) and corrected Debye-Hiickel (CDH, full line) theory were used together with a GvdW(I) treatment of the uncharged hard-sphere mixture. The ion diameter was 4.25 A, the temperature was 298 K and the dielectric constant e was 78.36. 

In order to utilise our colloids as near hard spheres in terms of the thermodynamics we need to account for the presence of the medium and the species it contains. If the ions and molecules intervening between a pair of colloidal particles are small relative to the colloidal species we can treat the medium as a continuum. The role of the molecules and ions can be allowed for by the use of pair potentials between particles. These can be determined so as to include the role of the solution species as an energy of interaction with distance. The limit of the medium forms the boundary of the system and so determines its volume. We can consider the thermodynamic properties of the colloidal system as those in excess of the solvent. The pressure exerted by the colloidal species is now that in excess of the solvent, and is the osmotic pressure II of the colloid. These ideas form the basis of pseudo one-component thermodynamics. This allows us to calculate an elastic rheological property. Let us consider some important thermodynamic quantities for the system. We may apply the first law of thermodynamics to the system. The work done in an osmotic pressure and volume experiment on the colloidal system is related to the excess heat adsorbed d Q and the internal energy change d E ... 

Given the expression for K(T), one can construct an EOS by modeling the excess free energy density by = HS + u + ID + DI + DD + where is summed over contributions from hard-sphere (HS), ion-ion (II), ion-dipole (ID), dipole-ion (DI), and dipole-dipole interactions (DD), respectively. 4>ex also contains the contribution due to the internal partition function of the ion pair, = — p lnK(T). Pairing theories differ in the terms retained in the expression for ex. 

For the more complicated molecular models such as, for example, those that assume central forces, we replace the above set of parameters by a new set involved in defining the force field. If we add to this the problem of complex molecules (i.c., those with internal structure), then there is the additional set of parameters needed to define the interactions between the internal molecular motions and the external force fields. From the point of view of the hard sphere model this would involve the definition of still more accommodation coefficients to describe the efficiency of transfer of internal energy between colliding molecules. 

It is interesting to see how the statistical treatment of equilibrium sys-tem may be applied to the calculation of kinetic data, at least to the same accuracy as was obtained when the assumption of a Maxwellian distribution was employed. Let us assume that we have a mixed gas of hard sphere molecules A and B, capable of forming a weakly bound complex AB. Let us further assume that the molecules A and B possess no internal energy. We can then write the molecular partition functions for A, B, and AB ... 

Fig. 18. Average fractional energy transfer of diretly scattered oxygen atoms as a function of deflection angle, x. for i i) = 47 kJ mol and 6i = 60° (circles). The dashed line is the hard-sphere model prediction based on the effective surface mass, ms, shown. The solid line is the revised prediction after the hard-sphere model is corrected for the internal excitation of the interacting surface fragment. The correction is derived from a kinematic analysis of scattering in the c.m. reference frame.

Estimate the internal energy, Helmholtz energy, and entropy of a hard-sphere fluid in units of Nk Tassuming a concentration of 10 M, a diameter of 300 pm, a molecular mass of 30 g, and a temperature of 25°C. 

Calculate the internal energy and entropy of the hard-sphere system described in problem 6. Assume that the atomic weight of the hard sphere is 39.95 g moU. ... 

In as much as the configurational internal energy of a hard sphere system is zero, we can also write... 

The equation of state and free energy of a system of hard spheres with surface adhesion are calculated from the internal energy of the fluid as given by the Percus-Yevick theory. A first-order phase change occurs. Further, the liquid-vapor coexistence curve, which cannot be found by the more usual routes to the equation of state, is calculated. It is found that the equation of state exhibits van der Waals type sigmoid isotherms in the region in which the Percus-Yevick theory has solutions. 

S(s) = polynomial in the Laplace transform of the Percus-Yevick RDF of a hard-sphere fluid SH = Snider-Herrington s = Laplace transform variable T = temperature Tc = critical temperature T — kT/c, reduced temperature U = internal energy... 

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