Do molecules of the ideal gas at a particular temperature have the same kinetic energy?
1 Answer
No. If that were true, then these Maxwell-Boltzmann distributions of speeds would be vertical lines:
But since this speed distribution is just that---a distribution... there exist a myriad of speeds for a given temperature, and thus a myriad of different kinetic energies for a given temperature (but only a single average kinetic energy).
Molecules of an ideal gas at the same temperature have potentially different kinetic energies... but the same average kinetic energy.
As in the equipartition theorem , at high enough temperatures, the average molar kinetic energy is given by:
⟨κ⟩≡Kn=N2RT in units of
J/mol , where
N is the number of degrees of freedom, i.e. the number of coordinates for each type of motion, basically.This number
N has contributions of:
Ntr=3 for translation (linear motion),
Nrot=2 for rotational motion of linear molecules orNrot=3 for rotational motion of nonlinear polyatomics, andUp to
Nvib=1 for vibration of polyatomics, but typically very small at room temperature. For simple molecules, likeN2 andCl2 , the contribution to vibration is usually ignored due to negligibility.
n is themols of ideal gas.R=8.314472 J/mol⋅K is the universal gas constant.T is the temperature inK .
It is the average, in the sense that we take a sample of molecules that ALL have somehow DIFFERENT speeds AND traveling directions... and through observing all of them, the ensemble average leads to a distribution of speeds for a given temperature, which corresponds to a single observed (average) kinetic energy based on the temperature.
It is strictly NOT the same as observing a single molecule's velocity and using that to calculate the kinetic energy from
