Thermal Energy of The Ideal Gas
Thermal energy is most easily defined in the context of the ideal gas, which is well approximated by a monatomic gas at low pressure. The ideal gas is a gas of particles considered as point objects of perfect spherical symmetry that interact only by elastic collisions and fill a volume such that their mean free path between collisions is much larger than their diameter.
The mechanical kinetic energy of a single particle is
where m is the particle's mass and v is its velocity. The thermal energy of the gas sample consisting of N atoms is given by the sum of these energies, assuming no losses to the container or the environment:
where the line over the velocity term indicates that the average value is calculated over the entire ensemble. The total thermal energy of the sample is proportional to the macroscopic temperature by a constant factor accounting for the three translational degrees of freedom of each particle and the Boltzmann constant. The Boltzmann constant converts units between the microscopic model and the macroscopic temperature. This formalism is the basic assumption that directly yields the ideal gas law and it shows that for the ideal gas, the internal energy U consists only of its thermal energy:
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