Partition function general expression

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

The general equation for L is Eq. (10), and the expressions to be used for C in that equation are listed in Table 1 in terms of partition functions. But the entropies of both the activated complex and the reactants, and therefore AS, can also be expressed in terms of partition functions. Therefore, C can be expressed in terms of the entropies of the activated complex and the reactants. As we shall see, it is possible to eliminate partition functions entirely. Also, in all but one case. Step 11, the entropy factor in C can be determined if one knows only the entropy of activation in those cases the entropy of a reactant or the activated complex is not needed. 

S. Adair, H. S. Sinuns, K. Linderstrom-Lang, and, especially, J. Wyman. These treatments, however, were empirical or thermodynamic in content, that is, expressed from the outset in terms of thermodynamic equilibrium constants. The advantage of the explicit use of the actual grand partition function is that it is more general it includes everything in the empirical or thermodynamic approach, plus providing, when needed, the background molecular theory (as statistical mechanics always does). 

The statistical mechanical approach starts from more fundamental ingredients, namely, the molecular properties of all the molecules involved in the binding process. The central quantity of this approach is the partition function (PF) for the entire macroscopic system. In particular, for binding systems in which the adsorbent molecules are independent, the partition function may be expressed as a product of partition functions, each pertaining to a single adsorbent molecule. The latter function has the general form... 

As in Eq. 19-14, the equilibrium between the two concentrations on either side of the boundary shall be expressed by a general partition function ... 

Wiener integrals in general, often useful in statistical mechanics1,4, can be expressed in terms of the propagator in Eq. (4). In particular, putting 0 = 1 /kT, p = ha/kT, we have for the partition function of the particle obeying Eq. (6)... 

It would be more circumspect to say that the partition function cannot in general be naturally or usefully expressed as a canonical average since that average inevitably falls into the category for which canonical sampling is inadequate [13]. 

As seen from our discussion in Chapter 3, which dealt with onedimensional problems, in many relevant cases one actually does not need the knowledge of the behavior of the system in real time to find the rate constant. As a matter of fact, the rate constant is expressible solely in terms of the equilibrium partition function imaginary-time path integrals. This approximation is closely related to the key assumptions of TST, and it is not always valid, as mentioned in Section 2.3. The general real-time description of a particle coupled to a heat bath is the Feynman-Vernon... 

The partition function Q is thus a fundamental quantity which in a way contains all the information about the system under study. Its general expression is ... 

In Eq. (2.9), we have a general expression for mean square fluctuations, if only we can express E — Ts as a function of x. This ordinarily can be done conveniently for the internal energy E. We shall now show that, to a very good approximation, equals the entropy S, so that it also can be expressed in terms of the parameter x, by ordinary thermodynamic means. To do this, we shall compute the partition function Z... 

If we have a mixture of AT molecules of one gas, N2 of another, and so on, the general phase space will first contain a group of coordinates and momenta for the molecules of the first gas, then a group for the second, and so on. The partition function will then be a product of terms like Eq. (3.5), one for each type of gas. The entropy will be a sum of terms like Eq. (1.14), with n in place of n, and Pt, the partial pressure, in place of P. But this is just the same expression for entropy in a mixture of gases which we have assumed thermodynamically in Eq. (2.7). Thus the results of Sec. 2 regarding the thermodynamic functions of a mixture of gases follow also from statistical mechanics. 

The statistical mechanical verification of the adsorption Equation 11 proceeds most conveniently with use of the expression for y given by Equation 5. An identical starting formula is obtained via the virial theorem or by differentiation of the grand partition function (3). We simplify the presentation, without loss of generality, by restricting ourselves to multicomponent classical systems possessing a potential of intermolecular forces of the form... 

In this expression, Qgt and Qr are the partition functions of the generalized transition state and of the reactants, and Vmep(s) is the potential energy of the MEP at s. 

It is desirable to bridge the gap between the general quantal expression for /cet given by Eqs, 35-37 with its explicit summation of vibronic contributions and the simplicity of the classical, harmonic model displayed in Eqs. 33 and 34. As one device for achieving this goal, we switch perspectives from the constrained diabatic free energies, Gi t]) and Gf t]), as in Eq. 22, to an auxiliary (unconstrained) free-energy function, G(t), based on the hybrid partition function Z(r) [36, 98] (cf., Eq. 18) ... 

Eventually, the current considerations will serve to express the partition function Q as a (multidimensional) integral over configuration amd momentum space. It is therefore necessary to investigate the effect of Pk on integrals of the general type... 

First, let us apply the above general scheme to derive the Frumkin isotherm, which corresponds to localized adsorption of interacting molecules. (Expressions corresponding to the Langmuir isotherm can be obtained by setting (3 = 0 in the respective expressions for the Frumkin isotherm.) Let us consider the interface as a two-dimensional lattice having M adsorption sites. The corresponding partition function is"... 

The combinatorial FV expression of the Entropic-FV model is derived from statistical mechanics, using a suitable form of the generalized van der Waals partition function. ... 

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. 

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