Rewriting the above result for the pressure as , we may combine it with the ideal gas law
where  is the Boltzmann constant and  the absolute temperature defined by the ideal gas law, to obtain
which leads to a simplified expression of the average translational kinetic energy per molecule,
The translational kinetic energy of the system is  times that of a molecule, namely . The temperature,  is related to the translational kinetic energy by the description above, resulting in
which becomes
Equation () is one important result of the kinetic theory:
The average molecular kinetic energy is proportional to the ideal gas laws absolute temperature'.
From equations () and (), we have
Thus, the product of pressure and volume per mole is proportional to the average
translational molecular kinetic energy.
Equations () and () are called the "classical results", which could also be derived from statistical mechanics;
for more details, see:
The equipartition theorem requires that kinetic energy is partitioned equally between all kinetic degrees of freedom, D. A monotatomic gas is axially symmetric about each spatial axis, so that D = 3 comprising translational motion along each axis. A diatomic gas is axially symmetric about only one axis, so that D = 5, comprising translational motion along three axes and rotational motion along two axes. A polyatomic gas, like water, is not radially symmetric about any axis, resulting in D = 6, comprising 3 translational and 3 rotational degrees of freedom.
Because the equipartition theorem requires that kinetic energy is partitioned equally, the total kinetic energy is
Thus, the energy added to the system per gas particle kinetic degree of freedom is
Therefore, the kinetic energy per kelvin of one mole of monatomic ideal gas (D = 3) is
where  is the Avogadro constant, and R is the ideal gas constant.
Thus, the kinetic energy per unit kelvin of an ideal monoatomic gas can be calculated easily:
* per mole: 12.47 J / K
* per molecule: 20.7 yJ / K = 129 μeV / K
At standard temperature (273.15 K), the kinetic energy can also be obtained:
* per mole: 3406 J
* per molecule: 5.65 zJ = 35.2 meV.
At higher temperatures (typically thousands of kelvins), vibrational modes become active to provide additional degrees of freedom, creating a temperature-dependence on D and the total molecular energy. Quantum statistical mechanics is needed to accurately compute these contributions.