![]() ![]() ![]() Here, the summation is over the electronic, vibrational and rotational states can be done separately since they are assumed to be independent. Assuming the first three to be independent and neglecting the last term, the molecular partition function (ie, a sum over the molecular energy states) is given by The term E others includes nuclear spin energy levels and may also be used later to include the interactions between the first four. The total energy is a sum of all these energies and is given byĮtotal = Eelectronic + Evibrational + Erotational + Etranslational + Eothers (3.3) This is a simplified and useful model to start with. In each electronic level, there are several vibrational levels and for each vibrational level, there are several rotational states. The electronic energy levels are generally very widely separated in energy compared to the thermal energy kT at room temperature. The rotational quantum numbers are shown between the vibrational levels.Įlectronic, Vibrational, Rotational and Translational Partition Functions Vibrational quantum numbers are to shown in the extreme left. The electronic quantum umbers are shown to the extreme right. We will consider the simpler problem of molecular energy levels which are pictorially shown in the Fig 3.1.įigure 3.1 A schematic diagram showing electronic (bold lines), vibrational and rotational energy levels. The energy ("levels") of a system can be built up from the molecular energy levels. To calculate P(Ei)s we need the energy levels of a system. #POISSON RELATIONS THERMODYNAMICS CALCULATOR FREE#But when there is no ambiguity, we will simply write k. Once we know the probability distribution for energy, we can calculate thermodynamic properties like the energy, entropy, free energies and heat capacities, which are all average quantities. Often the Boltzmann constant is written as kB. Here, b = 1/ kT and e – βEi is called the Boltzmann factor. We have already seen that in the canonical ensemble, the probability of a system having energy Ei is proportional to the Boltzmann factor and is given in terms of the canonical partition function q by A schematic energy level diagram is shown in Fig. While atoms have only electronic energy levels, molecules have quantized energy levels arising from electronic, vibrational and rotational motion. The energies of atoms and molecules are quantized. Since we are dealing with number of particles of the order of Avogadro number, the ensemble description and the molecular descriptions are equivalent. ![]() The equivalence of the ensemble approach and a molecular approach may be easily realized if we treat part of the molecular system to be in equilibrium with the rest of it and consider the probability distribution of molecules in this subsystem (which is actually quite large compared to systems containing a small number of molecules of the order of tens or hundreds). The molecular partition function enables us to calculate the probability of finding a collection of molecules with a given energy in a system. In ensemble theory, we are concerned with the ensemble probability density, i.e., the fraction of members of the ensemble possessing certain characteristics such as a total energy E, volume V, number of particles N or a given chemical potential μ and so on. The next chapters will include detailed consideration of intermolecular forces. The present chapter deals with systems in which intermolecular interactions are ignored. If the molecules are reasonably far apart as in the case of a dilute gas, we can approximately treat the system as an ideal gas system and ignore the intermolecular forces. In chemistry, we are concerned with a collection of molecules. We have been introduced to the three main ensembles used in statistical mechanics and some examples of calculations of partition functions were also given. ![]()
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