Molecular partition function canonical

We have derived a formula for the molecular partition function by considering a system containing many molecules at equilibrium with a heat bath. We can generalize our statistical mechanics by a gedanken experiment of considering a large number of identical systems, each with volume V and number of particles N at equilibrium with the heat bath at temperature T. Such a supersystem is called a canonical ensemble. Our derivation is the same the fraction of systems that are in a state with energy Et is... 

It is also possible to express the thermodynamic properties in terms of what is called a canonical partition function, Z. This is related to z by Z = zN for an ideal solid and by Z — zN/N for a perfect gas, where z is the molecular partition function. 

Statistical thermod5rnatnics enables us to express the entropy as a function of the canonical partition function Zc (relation [A2.39], see Appendix 2). This partition function is expressed by relation [A2.36], on the basis of the molecular partition functions. These molecular partition functions are expressed, in relation [A2.21], by the partition functions of translation, vibration and rotation. These are calculated on the basis of the molecule mass and relation [A2.26] for a perfect gas, the vibration frequencies (relation [A2.30]) of its bonds and of its moments of inertia (expression [A2.29]). These data are determined by stud5dng the spectra of the molecules - particularly the absorption spectra in the iirffared. Hence, at least for simple molecules, we are able to calculate an absolute value for the entropy - i.e. with no frame of reference, and in particular without the aid of Planck s hypothesis. 

A2.3.3. Canonical partition function and molecular partition functions... 

Microscopic modeUng is based on the statistics of collections of discernible or indiscernible objects, the introduction of canonical and molecular partition functions and the use of data about the molecule obtained from spectroscopy. Chapters 3, 4, 5 and 6 will develop these concepts and follow onto microscopic modeling. 

Quantum mechanics helps determine the energies of molecules and the molecular partition functions, as we will see in Chapter 6, can be calculated from the characteristics of molecules (mass, moments of inertia, vibrational frequencies, etc.). The aim of this section is to determine the value of the canonical partition functions, since this is useful to link them to molecular partition functions. 

So, for ensembles of discemable molecules, the canonical partition function can be calculated from the molecular partition functions using relations [5.15] or [5.16] depending on whether the system has one or several constituents. These relations will be used for crystals in which the atoms (or molecules) are localized to network nodes. 

Now we will move onto indistinguishable particles but in the approximation of the classical hmiting case. The canonical partition function is given by expression [5.25] using the molecular partition functions. By applying equation [5.32], we calculate for the internal energy ... 

The formulas for the thermodynamic functions in the previous section apply to any kind of system. They can be applied to a dilute gas by using Eq. (27.1-27) to express the canonical partition function in terms of the molecular partition function. 

As previously stated, the classical molecular partition function has units of kg s raised to some power, so a divisor with units must be included to make the argument of the logarithm dimensionless. If a divisor of lkgm s is used, values are obtained for the entropy and the Helmholtz energy that differ from the experimental values. However, when the classical canonical translational partition function is divided by h A and Stirling s approximation is used for ln(iV ), the same formulas are obtained as Chapter 26. For a dilute monatomic gas the corrected classical formula is... 

There are several model theories that treat a liquid like a disordered solid. In the cell modefi each atom of a monatomic fluid such as liquid argon is assumed to be confined in a cell whose walls are made up of its nearest neighbors. In the simplest version, this cell is approximated as a spherical cavity inside which the potential energy of the moving atom is constant and outside of which the potential energy is infinite. Because each atom moves independently, the classical canonical partition function can be written as a product of molecular partition functions. The classical canonical partition function is... 

The energy and entropy, which are related in this equation to the molecular partition function, are statistical entities, defined by particles which do not interact except to maintain the equilibrium conditions. To obtain similar relationships for real systems, it is necessary to apply statistical mechanics to the calculation of the thermodynamic entities, which correspond to the molar quantities of particles, or that is N approaches 1 this treatment it is convenient to use the canonical ensemble already discussed and presented in Figure n.l. This ensemble consists of a very large number of systems, N, each containing 1 mol of molecules and separated from the others by diathermic walls, which allow heat conduction but do not allow particles to pass. The set of all the systans is isolated from the outside and has a fixed energy E, which is the energy of the canonical ensemble. 

The standard state Helmholtz free energy difference, 8AA°, was introduced in Equations 5.9 and 5.11 to show the connection between VPIE and molecular structure and dynamics. Molecular properties are conveniently expressed using standard state canonical partition functions for the condensed and vapor phases, Qc° and Qv° remember A0 = —RT In Q°. The Q s are 3nN dimensional, n is the number of atoms per molecule and N is Avogadro s number. For convenience we have now dropped the superscript o s on the Q s. The o s specify standard state conditions, now to be implicitly understood. For VPIE and a respectively, not too close to the critical region,... 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

As is briefly described in the Introduction, an exact equation referred to as the Ornstein-Zernike equation, which relates h(r, r ) with another correlation function called the direct correlation function c(r, r/), can be derived from the grand canonical partition function by means of the functional derivatives. Our theory to describe the molecular recognition starts from the Ornstein-Zernike equation generalized to a solution of polyatomic molecules, or the molecular Ornstein-Zernike (MOZ) equation [12],... 

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 )... 

It is interesting to consider the relationship between the reaction degeneracy and the molecular symmetries in canonical transition state theories. In the latter, the rate constants are expressed in terms of the partition functions, including the rotational partition functions, so that the molecular symmetries are automatically included. On the other hand, in the microcanonical TST, the rotational density of states is often not part of the rate constant expression (see discussion of rotational effects in the following chapter). Thus, the reaction degeneracy must be included separately. 

In the canonical ensemble, that is, assuming the electrical neutrality of each molecule, the molecular electronic canonical partition function (now renamed Z) is given by... 

---