30. Temperature - a measure of the average kinetic energy of the chaotic movement of molecules

Let now turn to the most important consequences arising from the basic equation of molecular-kinetic theory of gases. It can be used to establish that temperature is a measure of the average kinetic energy of the chaotic movement of molecules.

Let the quantity of gas be equal to one mole and it occupies volume \(V_{\mu}\). Production of molecules concentration on the molar volume is equal to Avogadro number

\( N_A ~= ~nV_{\mu} \)

Multiply equation \((3-8)\) by \(V_{\mu}\) and use equation \(\large\overline{E}\) for average kinetic energy. Then we get

\(pV_{\mu} ~= ~\frac{1}{3}mN_A\overline\upsilon^2 ~= ~\frac{2}{3} N_A\overline{E} \)(3-9)

We found a connection between two thermodynamic parameters of pressure \(p\) and volume \(V\) with the average kinetic energy of the translational motion of molecules.

On the other hand, the experimentally found ideal gas equation of state for one mole has the form

\(pV_{\mu} ~= ~RT \)(3-10)

Comparing equation \((3-9)\) and \((3-10)\), we get the relationship between the average kinetic energy and temperature

\(\overline{E} ~= ~\frac{3}{2}\frac{R}{N_A}T \)(3-11)

The average kinetic energy of the chaotic movement of gas molecules is proportional to the absolute temperature. The faster the molecules move, then the higher the temperature. This opens up a deep physical meaning of the concept of temperature.

The connection between temperature and average kinetic energy of molecules \((3-11)\) is established by us for rarefied gases. However it appears fair for any substances, movement of atoms or molecules which submits to laws of classical mechanics of Newton. This connection is true for liquids and solids, in which atoms can only oscillate near the positions of equilibrium in the nodes of the crystal lattice. It follows that if the gas and the walls of the vessel have the same temperature, then the average kinetic energy of the atoms of gas and the wall are the same.

At very low temperatures, the model of ideal gas becomes unacceptable, the motion of atoms does not obey the laws of classical mechanics and the ratio \((3-11)\) is no longer applied. The relationship between temperature and average kinetic energy is more complex.