# 18: Partition Functions and Ideal Gases

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

In chemistry, we are typically concerned with a collection of molecules. However, 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. The present chapter deals with systems in which intermolecular interactions are ignored. 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 molecular partition function enables us to calculate the probability of finding a collection of molecules with a given energy in a system. 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).

• 18.1: Translational Partition Functions of Monotonic Gases
The energy levels of translation are very closely spaced, so a large number of translational states are accessible and available for occupation by the molecules of a gas. This result is very similar to the result of the classical kinetic gas theory.
• 18.2: Most Atoms are in the Ground Electronic State
The energy difference between the ground electronic state of a system and its first excited state are typically much larger the thermal energy, $$kT$$. This means that most atoms are in their ground electronic state, unless the temperature of the system is very high.
• 18.3: The Energy of a Diatomic/Polyatomic Molecule Can Be Approximated as a Sum of Separate Terms
A reasonable partition function of a diatomic/polyatomic molecule is the product of the partition function for the translational, vibrational, rotational, and electronic degrees of freedom. The total energy of the molecule then becomes the sum of the translational, vibrational, rotational, and electronic energies.
• 18.4: Most Molecules are in the Ground Vibrational State
At room temperature, most molecules are in the ground vibrational state. This is because the vibrational energies of molecules are larger than the average thermal energy available.
• 18.5: Most Molecules are Rotationally Excited at Ordinary Temperatures
At room temperature, many rotational states will be populated. This is due to the smaller rotational energies compared to vibrational or electronic energies.
• 18.6: Rotational Partition Functions of Diatomic Gases Contain a Symmetry Number
Homonuclear diatomic molecules have a high degree of symmetry and rotating the molecule by 180° brings the molecule into a configuration which is indistinguishable from the original configuration. This leads to an overcounting of the accessible states. To correct for these symmetry factors, we divide the partition function by $$σ$$, which is called the symmetry number.
• 18.7: Vibrational Partition Functions of Polyatomic Molecules Include the Partition Function for Each Normal Coordinate
The partition function for polyatomic molecules include the partition functions for translational, electronic, vibrational, and rotational states. For translational states, the number of states available is far greater than the number of molecules. For electronic states, we only consider the ground electronic state due to the large gap between electronic states. For vibrational states, we include all the normal modes of vibration.
• 18.8: Rotational Partition Functions of Polyatomic Molecules Depend on the Sphar of the Molecule
For a polyatomic molecule containing NNN atoms, the total number of degrees of freedom is 3N3N3N. Out of these, three degrees of freedom are taken up for the translational motion of the molecule as a whole. The translational partition function was discussed previously and now we have to consider the three rotational degrees of freedom and the 3N–63N–63N–6 vibrational degrees.
• 18.9: Molar Heat Capacities
The heat capacity of a substance is a measure of how much heat is required to raise the temperature of that substance by one degree Kelvin. For a simple molecular gas, the molecules can simultaneously store kinetic energy in the translational, vibrational, and rotational motions associated with the individual molecules. In this case, the heat capacity of the substance can be broken down into translational, vibrational, and rotational contributions.
• 18.10: Ortho and Para Hydrogen
The molecules of hydrogen can exist in two forms depending on the spins on the two hydrogen nuclei. If both the nuclear spins are parallel, the molecule is called ortho and if the spins are antiparallel, it is referred to as para (In disubstituted benzene, para refers to the two groups at two opposite ends, while in ortho, they are adjacent or “parallel” to each other).
• 18.11: The Equipartition Principle
The equipartition theorem states that every degree of freedom that appears only quadratically in the total energy has an average energy of ½kT in thermal equilibrium and contributes ½k to the system's heat capacity. Here, k is the Boltzmann constant, and T is the temperature in Kelvin. The law of equipartition of energy states that each quadratic term in the classical expression for the energy contributes ½kBT to the average energy.
• 18.E: Partition Functions and Ideal Gases (Exercises)

18: Partition Functions and Ideal Gases is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.