Partition Functions of Atoms and Molecules

at high temperatures the partition function equals the number of energy levels that are available. The average energy of the system follows by applying Eq. (4)  

Example Partition Function of a System with an Infinite Number of Levels 

We will now do the same for a system with an infinite number of equidistant energy levels, separated by Ae. For such a system the partition function becomes the following mathematical series with a well-known sum  

The reader may verify that the limiting values of the partition function for low and high temperature are 1 and respectively. The average energy becomes 

The above two examples illustrate that the value of the partition function is an indicator for how many of the energy levels are occupied at a particular temperature. At T = 0, where the system is in the ground state, the partition function has the value q = 1. In the limit of infinite temperature, entropy demands that all states are equally occupied and the partition function becomes equal to the total number of energy levels. 

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Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

Figure 2.15 Microscopic pictures of the desorption of atoms and molecules via mobile and immobile transition states. If the transition state resembles the ground state, we expect a prefactor of desorption on the order of 1013 s. If the adsorbates are mobile in the transition state, the prefactor goes up by one or two orders of magnitude. In the case of desorbing molecules, free rotation in the transition state increases the prefactor even further. The prefactors are roughly characteristic of atoms such as C, N and O and molecules such as N2, CO, NO and 02. See also the partition functions in Table 2.2 and the prefactors for CO desorption in Table 2.3.

Statistical mechanics provides a bridge between the properties of atoms and molecules (microscopic view) and the thermodynmamic properties of bulk matter (macroscopic view). For example, the thermodynamic properties of ideal gases can be calculated from the atomic masses and vibrational frequencies, bond distances, and the like, of molecules. This is, in general, not possible for biochemical species in aqueous solution because these systems are very complicated from a molecular point of view. Nevertheless, statistical mechanmics does consider thermodynamic systems from a very broad point of view, that is, from the point of view of partition functions. A partition function contains all the thermodynamic information on a system. There is a different partition function... 

For atoms and small molecules both partition functions are similar and therefore v kT/h 1013 s. for typical desorption temperatures. For large molecules, however, the partition function of the free molecules is due to the many rotational and vibrational degrees of freedom much larger than for the adsorbed molecule, where only frustrated rotations and vibrations exist. Therefore v is in this case typically by many orders of magnitude larger than 1013 s 1 [4,5],... 

The most fundamental starting point for any theoretical approach is the quantum mechanical partition function PF), and the fundamental connection between the partition function and the corresponding thermodynamic potential. Once we have a PF, either exact or approximate, we can derive all the thermodynamic quantities by using standard relationships. Statistical mechanics is a general and very powerful tool to connect between microscopic properties of atoms and molecules, such as mass, dipole moment, polarizability, and intermolecular interaction energy, on the one hand, and macroscopic properties of the bulk matter, such as the energy, entropy, heat capacity, and compressibility, on the other. 

Equations (V-18) and (V-19) only differ from the usual expression for the partition function of atoms in the approximation of small quantum corrections, by the fact that the total molecular mass M has been replaced by an effective mass which takes account of the distribution of the total molecular mass among the constituent atoms of the molecule. 

The connection between the quantum mechanical treatment of individual atoms and molecules and macroscopic properties and phenomena is the goal of statistical mechanical analysis. Statistical mechanics is the means for averaging contributions to properties and to energies over a large collection of atoms and molecules. The first part of the analysis is directed to the distribution of particles among available quantum states. An outcome of this analysis is the partition function, which proves to be an essential element in thermodynamics, in reaction kinetics, and in the intensity information of molecular spectra. 

However, if the particles are indistinguishable, as the atoms in a gas, the number of possible configurations is significantly reduced and the partition function for an ensemble of atoms or molecules is therefore... 

This relation indicates that the rate constant can be determined from a knowledge of the partition functions of the activated complex and the reactant species. For stable molecules or atoms... 

Note that gtrans is given per degree of freedom, implying that the total translational partition function for an adsorbed molecule is given by (Qtrms)2- Also, the total partition functions for vibration and rotation are the products of terms for each individual vibration and rotation, respectively. Table 2.2 gives values for the partition functions for adsorbed atoms and molecules at 500 K. Vibrational partition functions are usually close to one, but rotational and translational partition functions have larger values. 

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,... 

For Simple Atoms. If the contribution of single translational, rotational and vibrational degrees of freedom are written as fj,fR and f, the total partition function F for a molecule may be expressedas... 

The partition function of the atomic species consists of the electronic and translational contributions only, but for the diatomic molecule A2 the partition function involves the electronic, translational and rotational factors, and also the contribution of one vibrational mode. The translational partition function is given by equation (16.16) as... 

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