The internal partition function

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

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. 

Following van der Waals and Platteeuw (1959, pp. 26ff) the individual particle partition function is related to the product of three factors (1) the cube of the de Broglie wavelength, (2) the internal partition function, and (3) the configurational triple integral, as... 

Here E ( y1 ) stands for the single-particle contribution to the total energy, allowing for molecule interaction with the surface <2 is the heat released in adsorption of molecules z on the /Lh site Fj the internal partition function for the z th molecules adsorbed on the /Lh site F j the internal partition function for the zth molecule in the gas phase the dissociation degree of the z th molecule, and zz the Henry local constant for adsorption of the zth molecule on the /Lh site. Lateral interaction is modeled by E2k([ylj ), and gj (r) allows for interaction between the z th and /Lh particles adsorbed on the /th and gth sites spaced r apart. In the lattice gas model, separations are conveniently measured in coordination-sphere numbers, 1 < r < R. For a homogeneous surface, molecular parameters zz and ej(r) are independent of the site nature, while for heterogeneous, they may depend on it. 

In Eq. (3.3), Z, is the internal partition function of a single molecule. The second way of writing it, in terms of a summation, by analogy with Eq. (5.17) of Chap. Ill, refers to a summation over all cells in a 2s-dimen-sional phase space in which qi p8 are the dimensions. We note, for future reference, that the quantity Zi depends on the temperature, but not on the volume of the gas. 

Equation (4.16) agrees exactly with Eq. (3.6), Chap. VIII, except for the internal partition function Ziy which we are here neglecting for simplicity, and for the extra term N2kTw/2V. This represents the effect of inter-... 

The internal partition function of a molecule, and the conformational distribution function, discussed above on p. 24 and p. 18, respectively, were... 

The internal partition function [ ] does not appear in Eq. (3.35), though it does appear explicitly in Eqs. (3.18) and (3.19), p. 40. Start with Eq. (3.35), and show that this implies Eq. (3.19). Give a physical statement and interpretation of this distinction. 

Why don t the Flory-Huggins approximations for the chemical potentials Eqs. (8.6) require any evaluation of the internal partition function of a single chain molecular,... 

The internal partition function of guest molecules is the same as that of an ideal gas. 

The integration on the rhs of (1.67) extends over all possible locations and orientations of the N particles. We shall refer to the vector XN=Xt,..., XN as the configuration of the system of the N particles. The factor q, referred to as the internal partition function, includes the rotational, vibrational, electronic, and nuclear partition functions of a single molecule. We shall always assume in this book that the internal partition functions are separable from the configurational partition function. Such an assumption cannot always be granted, especially when strong interactions between the particles can perturb the internal degrees of freedom of the particles involved. 

In all of the aforementioned discussions, we left unspecified the internal partition function of a single molecule. This, in general, includes contributions from the rotational, vibrational, and electronic states of the molecule. Assuming that these degrees of freedom are independent, the corresponding internal partition function may be factored into a product of the partition functions for each degree of freedom, namely,... 

We shall never need to use the explicit form of the internal partition function in this book. Such knowledge is needed for the actual calculation, for instance, of the equilibrium constant of a chemical reaction. 

For simplicity, we have assumed that all the pair potentials are spherically symmetrical, and that all the internal partition functions are unity. The general expression for the chemical potential of, say, A in this system is obtained by a simple extension of the one-component expression given in chapter 3. 

We have defined the solvation process as the process of transfer from a fixed position in an ideal gas phase to a fixed position in a liquid phase. We have seen that if we can neglect the effect of the solvent on the internal partition function of the solvaton s, the Gibbs or the Helmholtz energy of solvation is equal to the coupling work of the solvaton to the solvent (the latter may be a mixture of any number of component, including any concentration of the solute s). In actual calculations, or in some theoretical considerations, it is often convenient to carry out the coupling work in steps. The specific steps chosen to carry out the coupling work depend on the way we choose to write the solute-solvent interaction. 

Note that the rotational partition function of the entire molecule, as well as the internal partition functions of s, are included in the pseudo-chemical potential. In classical systems, the momentum partition function is independent of the environment, whether it is a gas or a liquid phase. 

Here we cannot separate the internal partition function from the coupling work. The reason is that each conformation has a different binding energy to the solvent. In a formal way, we can use the definition of qfnt from (7.168) to rewrite (7.169) as... 

For his calculations, Burton chose the simplest possible material—a cluster of atoms interacting with nearest-neighbor harmonic forces and with the atoms packed onto lattice positions of a close-packed cubic material. He then calculated the partition functions in Eq. (43) and ultimately the cluster concentrations and nucleation rates. The major problem in this calculation was the internal partition function Zmt> which was calculated by diagonalizing the 3i X 3i dynamical matrices of i atom clusters to obtain normal mode vibrational frequencies and ultimately harmonic oscillator partition functions. This calculation was very expensive and could not be done for i larger than about 100. 

The internal partition function is a sum of Boltzmann factors, multiplied by their degeneracies, Pj, over the vibrational and rotational states of the molecule. In the summation over the rotational states it is necessary to consider symmetry selection rules. This leads to a factor S in and + RlnS... 

The second approximation is the separability of the internal partition function from the configurational PF. Liquid water can be viewed as a mixture of hydrogens and oxygens these can be chosen either as charged or as neutral atoms. In any case, it is very difficult to include interactions among all atoms in an explicit form in writing the classical analog of the partition functions ... 

The model is a simplified version of a model first developed by Lovett and Ben-Naim (1969). The idea is to define a sequence of n HBed particles as an n-cluster (mimicking the ice-like clusters of HBed molecules in liquid water). In doing so, we include all the HBing in the cluster as part of the internal partition function of the -cluster. The interaction potential between any pair of clusters (including the 1-cluster, i.e. the monomers, or non-HBed particle), is now the hard-rod HR) potential. (In the original model, this part was chosen to be a square-well potential. As with the primitive model discussed in Sec. 2.5.2, the... 

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