The thermoynamic equilibrium in a gas (here a monoatomic perfect gas as you consider only translation and ignore interactions) , as long as the temperature is fixed, is characterized by the Boltzmann distribution of velocities, stating that the density of probability in the space of velocities is proportional to $\exp(-E/k_BT)$ (where $k_B$ is the Boltzmann constant), and here $E$ is simply the translational kinetic energy $E=\frac12 m \vec{v}^2$.
An important consequence is the so called "equipartition of energy" :

$$ \langle E\rangle \equiv \frac12 m \langle \vec{v}^2\rangle \equiv \frac12 m v_{rms}^2 = \frac32 k_B T.$$

where the 3 comes from the 3 directions of the space.

This holds independently of the mass $m$ of the gas constituants, and is still true for a diluted diatomic gas like in air.
Hence, one has actually:

$$\frac12 m_1 v_{rms,1}^2 =
\frac12 m_2 v_{rms,2}^2=\frac32 k_B T,$$

and the relation for the temperature that you give is the "thermodynamic definition of temperature". It is also the fundamental principle of the "gas themometer" used by metrologist for low temperature measurements.

Furthermore, for a perfect gas, and for a given pression and temperature, the number $N$ of molecules depends only on the volume $V$ and not on the molecular weight, and the total *translational* kinetic energy for this volume is the same for any gas.

Nevertheless, this does not imply that the *total* kinetic energy is the same, as for poly atomic gas, you have to consider not only the motion of the barycenter (as we have done above) but also the kinetic energy associated to the relative motion of the atoms in the molecules, making the laws more complex.

## Best Answer

As we decrease the temperature, the vibration decreases and decreases until, at absolute zero, there is a minimum amount of vibration that the atoms can have, but not zero. This minimum amount of motion that atoms can have is not enough to melt a substance, with one exception: helium. Helium merely decreases the atomic motions as much as it can, but even at absolute zero there is still enough motion to keep it from freezing. Helium, even at absolute zero, does not freeze, unless the pressure is made so great as to make the atoms squash together. If we increase the pressure, we can make it solidify.$_1$

So, to your questions, gases can be liquified by lowering the temperature. Liquids may solidify or liquify at absolute zero. Molecules or atoms do have minimum vibration at absolute zero.

Credits: $_1$

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