Energy partition ground state molecules

Information from Section 28-5 can be used to calculate the internal energy and heat capacity of a monatomic gas because the complete temperature dependence of Z is accounted for by partition functions for translational motion and the electronic ground state. Molecules exhibit 3 degrees of freedom per atom. Hence, there are no internal degrees of freedom for a monatomic gas (i.e.. He, Ne, Ar, Kr, Xe) because all 3 degrees of freedom are consumed by translational motion in three different coordinate directions. The internal energy is calculated from equation (28-59) ... 

It is instructive to illustrate the relation between the partition function and the equilibrium constant with a simple, entirely hypothetical example. Consider the equilibrium between an ensemble of molecules A and B, each with energy levels as indicated in Fig. 3.5. The ground state of molecule A is the zero of energy, hence the partition function of A vnll be... 

Atoms and molecules have available to them a number of energy levels associated with the allowed values of the quantum numbers for the energy levels of the atom. As atoms are heated, some will gain sufficient energy either from the absorption of photons or by collisions to populate the levels above the ground state. The partitioning of energy between the levels depends on temperature and the atom is then said to be in local thermal equilibrium with the populations of the excited states and so the local temperature can be measured with this atomic thermometer. 

From the known partition function the thermal energy for each mode can be calculated as the sum over all energy levels. Each term consists of the number of molecules in a quantum state Ni multiplied by the number of states gt at that level and the energy of the level above the ground state, tt - e0. If the total energy is represented by the symbol U, the thermal energy is... 

We use this knowledge to derive preexponential factors from (2-20) for a few desorption pathways (see Fig. 2.15). The simplest case arises if the partition functions Q and Q in (2-20) are about equal. This corresponds to a transition state that resembles the ground state of the adsorbed molecule. In order to compare (2-20) with the Arrhenius expression (2-15) we need to apply the definition of the activation energy ... 

The Politzer-Parr partitioning of molecular energies in terms of atomic-bke contributions results in an exact formula for the nonrelativistic ground-state energy of a molecule as a sum of atomic terms that emphasizes the dependence of atomic and molecular energies on the electrostatic potentials at the nuclei. 

Note that in Eq. 8-24 we have used the typical approximation to factorize the partition function [28,32] and hence Qv,s, Qv are the (quantum) vibrational partition functions of the solvent and solute molecule, respectively, and U = + AUvfi with < > the system potential energy (i.e., electronic ground state energy surface) and NUv,o the system vibrational ground state energy shift from a reference value [28,32], typically negligible. 

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