24: Indistinguishable Molecules - Statistical Thermodynamics of Ideal Gases
- Page ID
- 151977
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The ensemble analysis shows that the thermodynamic functions for an \(N\)-molecule system can be developed from the principles of statistical mechanics whether the molecules of the system interact or not. The theory is valid irrespective of the strengths of inter-molecular attractions and repulsions. However, to carry out numerical calculations, it is necessary to know the energy levels for the \(N\)-molecule system. For systems in which the molecules interact, obtaining useful approximations to these levels is a difficult problem. As a result, many applications assume that the molecules do not interact with one another. In this chapter we apply the results from the ensemble theory to the particular case of ideal gases.
- 24.2: The Partition Function for N Indistinguishable, Non-interacting Molecules
- Experiment demonstrates that the partition function for a system of indistinguishable molecules is different from that of an otherwise-identical system of distinguishable molecules. The reason for this becomes evident when we compare the microstates available to a system of distinguishable molecules to those available to a system of otherwise-identical indistinguishable molecules.
- 24.3: Occupancy Probabilities for Translational Energy Levels
- A particle in a 3D rectangular box is a quantum mechanical model for an ideal gas molecule. The molecule moves in three dimensions, but the component of its motion parallel to any one coordinate axis is independent of its motion parallel to the others. This being the case, the kinetic energy of a particle in a three-dimensional box can be modeled as the sum of the energies for motion along each of the three independent coordinate axes that describe the translational motion of the particle.
- 24.4: The Separable-modes molecular Model
- The approximation that a molecule’s translational motion is independent of its rotational, vibrational, and electronic motions is usually excellent. The approximation that its intramolecular rotational, vibrational and electronic motions are also independent proves to be surprisingly good. Moreover, the very simple quantum mechanical systems described previously prove to be surprisingly good models for the individual kinds of intramolecular motion. The much of this chapter illustrates these poin
- 24.7: The Electronic Partition Function of an Ideal Gas
- The energy of the electronic ground state that we obtain by direct quantum mechanical calculation includes the energy effects of the motions of the electrons and the energy effects from the electrical interactions among the electrons and the stationary nuclei. Because we calculate it for stationary nuclei, the electronic energy does not include the energy of nuclear motions.
- 24.8: The Vibrational Partition Function of A Diatomic Ideal Gas
- We base the electronic potential energy for a diatomic molecule on a model in which the nuclei are stationary at the bottom of the electronic potential energy well. We now want to expand this model to include vibrational motion of the atoms along the line connecting their nuclei. It is simple, logical, and effective to model this motion using the quantum mechanical treatment of the classical (Hooke’s law) harmonic oscillator.
- 24.9: The Rotational Partition Function of A Diatomic Ideal Gas
- For a diatomic molecule that is free to rotate in three dimensions, we can distinguish two rotational motions.
- 24.10: The Gibbs Free Energy for One Mole of An Ideal Gas
- In our discussion of ensembles, we find that the thermodynamic functions for a system can be expressed as functions of the system’s partition function. Now that we have found the molecular partition function for a diatomic ideal gas molecule, we can find the partition function for a gas of N such molecules. From this system partition function, we can find all of the thermodynamic functions for this N-molecule ideal-gas system.