Average Translational Kinetic Energy per Molecule of Gas
In Ideal gases, the molecules are considered as point particles. For point particles, there is no internal excitation, no vibration and no rotation. The point particles can have only translational motion and thus only translational energy. For an ideal gas the internal energy can only be translational kinetic energy.
From microscopic consideration based on kinetic theory of gases,
P = ⅓ (Nm/V) v²rms or PV = ⅓ Nmv²rms … (1)
From macroscopic consideration based on experimental observation of gases,
PV = nRT … (2)
From equations (1) and (2):
⅓ Nmv²rms = nRT
½ mv²rms = 3/2 nRT/ N
= 3/2 (R/ NA)T
Where, NA = Number of molecules per mole = N/n.
While dealing with energies of molecules, we denote the ratio R/NA by the symbol kB which is called Boltzmann constant.
Hence, we can write:
½ mv²rms = 3/2 kBT
Where, ½ mv²rms = Average kinetic energy of translation Kֿ̄ of a gas molecule.
Thus, the average kinetic energy of translation of a gas molecule (½mv²rms = k̄) depends on its temperature and it’s independent of pressure, volume or nature of the gas and is given by 3/2 kBT.
Obviously, molecules of different gases (Say He, H₂O, etc) at the same temperature possess same value of K̄ through their rms speed are different as vrms α 1/√ρ. In fact, for an ideal gas, K̄ is directly proportional to the absolute temperature.