Let's start by defining internal energy of a molecule:
The bond energy usually dominates the thermal energy with a much bigger value, so we often neglect the thermal energy. However, heat capacity depends only on thermal energy, NOT bond energy, so it is important for something.
Translational Energy
The translational energy is defined as
I will offer an explanation as to where this definition came from later, but until I update this post, I suggest checking both http://www.chem.queensu.ca/people/faculty/mombourquette/firstyrchem/GasLaws/KMT.htm
and section 1.9 from the text book. (My laziness is because I am incredibly hungry and tired.)
The translational energy is partitioned among three degrees of freedom along the three cartesian axes.
Rotational Energy
The rotational energy also is split up equally between three degrees of freedom. However, with a linear molecule, rotating it about the z-axis has insignificant energy as the mass of the molecule is almost completely in the atomic nuclei, so we only consider its rotation about the x- and y-axis.
A linear molecule has only two degrees of rotational freedom.
A nonlinear molecule has three degrees of rotational freedom.
Vibrational Energy
The vibrational energy can be resolved into a certain number of normal modes of vibration, known as its degrees of freedom.
Summary of Degrees of Freedom
If a molecule has N atoms, 3N degrees of freedom must be specified. We have already defined the translational and rotational degrees of freedom for nonlinear and linear molecules. The vibrational degrees of freedom are merely the leftovers from the total. This is summarized in the table below:
Heat Capacities Further Defined
Equiparition of energy states that each translational and rotational degree of freedom contains 1/2RT energy, of kinetic energy only. Each vibrational degree of freedom contains RT energy, which includes both the potential energy and kinetic energy.
The heat capacities are given by multiplying the energy factor by the gas constant times the number of degrees of freedom. This gives a value in J/(mol*K), which is the unit for heat capacity.
These formulae can be used to estimate the molar heat capacities of gases. The molar heat capacity of monatomic gases only depends on translational motion. For diatomic molecules, the translational and rotational components are more dominant than the vibrational components for determining the heat capacities. For anything larger, the calculated values are very far off the actual heat capicities. So instead of throwing our hands up in the air and praying to the quantum mechanic gods, mortals immortalized (Einstein and Stern) suggest that this extreme variation from the actual values is because equiparitioning breaks down at lower temperatures, especially for vibrational states.
Why is that? Remember that vibrational levels are much more widely spaced than the rotational and translational levels. The larger gaps require a large quantum of energy to be supplied to move from one vibrational level to the next. This would sufficiently muck up our calculated values. However, at higher temperatures the calculated molar heat capacity values would have much better agreement with actual values, as more energy is supplied to the molecules.
Summary:
Molecules have various degrees of freedom which compose their thermal energy. These degrees of freedom are related to the heat capacity. By knowing the number of degrees of freedom and how energy is distributed between the degrees of freedom (translational, rotational and vibrational), we then can construct the heat capacities of the substance.
At lower temperatures, the heat capacities calculated in this manner will not correspond with the actual heat capacities.
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