What is the value of the gas constant derived from the kinetic model of gases?

I used this resource (<http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1151&context=physicskatz>) to derive the ideal gas law from the kinetic model of gases:


$$PV=n(2/3)N\_oT$$
with T being equal to $(1/2)mv\_{average}^2$.


And now I'm super confused. Does this mean that the universal gas constant R is simply equal to (2/3)\*avogadro's number?




$T$ here does not refer to temperature but rather the average kinetic energy of a molecule.


I can see why there is some confusion. Sadly we only have so many symbols and kinetic energy also happens to be commonly denoted with $T$.


To relate the average kinetic energy to the temperature, you need to bring in equipartition theory


$$\frac{1}{2}m\langle v^2\rangle = \frac{3}{2}kT$$


(assuming three translational degrees of freedom only). If you substitute that into your above equation


$$pV = \frac{2}{3}nN\_\mathrm{A}\cdot m\langle v^2\rangle$$


then you will find that $pV = n(N\_\mathrm{A}k\_\mathrm{B})T \Longrightarrow R = N\_\mathrm{A}k\_\mathrm{B}$, hopefully a familiar expression. The question of how the value of $R$ is derived then becomes a question of how equipartition is derived - and that deserves another question itself.


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