The → study of pressure as part of the kinetic theory gave the following result: $P=\frac{mNu^2}{3V}$ where $P$ is the pressure of the gas $V$ is the volume of the gas $m$ is the mass of one molecule $N$ is the number of molecules Considering that the average kinetic energy of a molecule is $E_{kin}=\frac{1}{2}mu^2$ we have: $P$= $\frac{mNu^2}{3V}$ = $\frac{2N}{3V}\frac{1}{2}mu^2$ = $\frac{2N}{3V}E_{kin}$
Pressure of an ideal gas: $P$ $=$ $\frac{2}{3}\frac{N}{V}E_{kin}$ where $\frac{N}{V} $ is the molecular density (number of molecules per $m^3$) $E_{kin}$ is the average kinetic energy of the molecules ( $J$) The pressure of an ideal gas is proportional to the molecular density and the average kinetic energy of molecules
and so: $P\cdot V $ $=$ $\frac{2}{3}N E_{kin}$ $=$ $\frac{2}{3}N\frac{1}{2}mu^2\,(1)$
The well-known → ideal gas law is: $P\cdot V=n\cdot R\cdot T$ where: $n=\frac{N}{6,023\cdot 10^{23}}$ is the quantity (number of moles) $T$ is the absolute (Kelvin) temperature $R= 8.314\frac{J}{K\cdot mol}$ is the real gas constant Let's assume $k$ $=$ $\frac{R}{6,023\cdot 10^{23}}$ $=$ $\frac{8.314}{6,023\cdot 10^{23}}$ $=$ $1,38\cdot 10^{-23}\frac{J}{K\cdot molecule}$ then the ideal gas law gives: $P\cdot V $ $=$ $\frac{N}{6,023\cdot 10^{23}}k\cdot 6,023\cdot 10^{23} T=$ $N\cdot k\cdot T\,(2)$ Comparing (1) and (2) : $E_{kin}=\frac{3}{2}kT$ which gives us (finally) a reasonable interpretation of this quantity we call temperature:
Average kinetic energy of a molecule of an ideal gas: $E_{kin}=\frac{3}{2}kT$ where $k=1,38\cdot 10^{-23}$ is Boltzmann's constant $T$ is the absolute temperature ($K$) The temperature is a measure of the average kinetic energy of the molecules
The previous law also allows calculation of the mean square speed of the molecules in an ideal gas: $E_{kin}=\frac{3}{2}kT$ $\frac{1}{2}mu^2=\frac{3}{2}kT$ $u=\sqrt{\frac{3kT}{m}}$
Mean square speed of a molecule in an ideal gas: $u=\sqrt{\frac{3kT}{m}}$ where $T$ is the Kelvin temperature $k=1.38\cdot 10^{-23}$ is Boltzmann's constant $m$ is the mass of one molecule