Three-dimensional partition function

The sum is from n = 1 to oo so the limits of integration are from 0 to oo in three dimensions. We may consider the molecule to reside in a rectangular box with dimensions, a, b, and c. The three-dimensional partition function then becomes... 

The quantity K defined by Equation (31) may easily be expanded somewhat further. First, we write the two-dimensional partition function by analogy with its three-dimensional counterpart, Equation (16). To do this, replace V by the area accessible to the adsorbed molecule a and the exponent by 2/2 (= 1) in place of 3/2 since two rather than three degrees of freedom are involved. Therefore we obtain... 

Obviously, the above derivation can be repeated for the other two Cartesian directions. As the energies are additive, the partition function for the three-dimensional translation of the molecule can be written as a product, viz. 

In Equation 4.71 the individual qvib s have been specified qVib(v ) to indicate that these partition functions depend on the normal mode frequencies. It is interesting to note that the partition function for translation, which is usually considered in terms of the problem of the particle in a three dimensional rectangular box, is, itself a product of three partition functions one for motion in the x dimension, one for y, etc. 

We will now present some of the results of the recent Monte Carlo computations of the chain partition function and the related thermodynamic functions for some three-dimensional lattices performed recently by McCrackin and Mazur.2... 

Using the previously derived expressions for q, we can now obtain expressions for each of the entropy terms. Equation 8.59 gives the molecular partition function for three-dimensional translational motion of a gas. Substituting this qtrans into Eq. 8.102, we obtain... 

In such a case the motion in the x — y plane is treated with the particle-in-a-box approach used to derive the partition function for gas-phase one- and three-dimensional motion. In the case of the 2D gas of mobile surface species, the partition function is... 

In the three dimensional chain on a cubical lattice, N/3 elements are considered to be directed along the three directions x, y and z. The analysis of Krigbaum and Kaneko shows that the partition function A for the three dimensional chain can be written as a product AXAVAX, in which each of the terms are given by the solution of a one-dimensional Markov chain of Nj3 steps ... 

If the guest molecule is assumed to be confined, in the thermodynamic sense, within a particular cavity, if the internal vibrational and rotational states of the molecule are unaltered by occlusion, and if the potential field within a cavity is sufficiently uniform so that the movement of the molecule within the cavity can be represented as three-dimensional translation within the free volume of the cavity (), then the appropriate expression for the ratio of partition functions becomes simply... 

In these expressions, UB is the number of ways of adding (me segment to an essentially infinite chain or one loop in the three-dimensional bulk phase, Us is the corresponding number for a two-dimensional train on the surface, aB and a5 are exduded-volume parameters in the bulk and on the surface, respectively, and yB and ys the probability of transforming a sequence of adsorbed segments to a loop and that of the reverse process, respectively. The partition function q°(P, T) of the system is obtained from Eq. (B-38) as follows ... 

The energy levels for a particle in a three-dimensional box are given as the sum of the energies for each dimension, and the partition function for the three-dimensional box is simply a product of the partition functions for each dimension that is,... 

Once the structure is partitioned into three-dimensional hyperplanes of simultaneity further features can be added. One such feature is a distance function d(p,q) defined on any two points within the same hyper-plane. This distance function provides the hyper-plane with a metric. And with such a metric, the space will exhibit what contempo-... 

More specifically, let us set ourselves the task of improving the Volmer equation for mobile adsorbates to include lateral interaction. The logic is in the analogy with the three-dimensional Van der Waals equation (1.5.23) is already a two-dimensional Van der Waals equation of state but without the interaction term. In the three-dimensional case (1.2.18.26( molecular interaction was seml-emplrically accounted for by replacing p by [p + an /V ]. Let us continue the analogy and introduce the parameter a° as the two-dimensional Van der Waals constant. For such an adsorbate, the canonical partition function can be shown to be )... 

The latter free energy can be represented as a surface integral over the solvent accessible surface of the molecule on the basis of a local free energy surface density (FESD) p. This surface density function is represented in terms of a three-dimensional scalar field which is comprised of a sum of atomic increment functions to describe lipophilicity in the molecular environment.The empirical model parameters are obtained by a least squares procedure with experimental log P values as reference data. It is found that the procedure works not only for the prediction of unknown partition coefficients but also for the localization and quantification of the contribution of arbitrary fragments to this quantity. In addition, the... 

Despite the absence of capillary condensation, the one-dimensional hard-rod fluid is still so useful because we have an analytic expression for its partition function [see Eq. (3.12)] that permits us to derive closed expressions for any thormophysical property of interest. One such quantity that is closely related to the Isothermal compressibility discussed in the preceding section is the particle-number distribution P (N), whidi one may also employ to compute thermomechanical properties [see, for example, Eqs. (3.65) and (3.68)]. Moreover, in a three-dimensional system P ) is useful to investigate the sj stem-size dependence of density fluctuations as we shall demonstrate in Section 5.4.2 [see Eq. (5.80)]. 

Note that in the one-dimensional case, the canonical partition function has the form of Q = Vj1 /N A where is the free volume. In this case, the quantity Vf is indeed the volume unoccupied by particles. In the free volume theories of liquids, this form of the partition function was assumed to hold for a three-dimensional liquid. 

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