Why is the temperature of a gas proportional to the average kinetic energy of its particles? I'm studying the kinetic theory of gases I managed to derive that pressure is inversely proportional to volume and directly proportional to the average kinetic energy of the gas particles. Similarly, volume is inversely proportional to pressure and also directly proportional to the average kinetic energy. But why does the temperature increase with the average kinetic energy? This is assumed as a postulate but I cannot understand why.
 A: It depends on what temperature is. Two systems are in thermal equilibrium when the fractional change of their multiplicities $\Omega$ with energy $E$, $\frac{1}{\Omega}\frac{{\rm d}\Omega}{{\rm d}E}$, are equal to each other. Let us call this quantity $\beta$.
For a classical ideal gas of $N$ independent particles the number of accessible states $\Omega$ is proportional to the surface of a hypersphere in a phase space with $3N$ dimensions. The radius of that sphere is proportional to the square root of the kinetic energy $\sqrt{E}$, so that $\Omega(E)  \propto  E^\frac{3N-1}{2}.$
This is enough to see that for the ideal classical gas  $\beta=\frac{1}{\Omega}\frac{{\rm d}\Omega}{{\rm d}E} = \frac{3N-1}{2} E^{-1}$ which is equal to $\frac{3N}{2} E^{-1} $ because $N$ is on the order of Avogadro's number.
From kinetic theory, the product $pV= \frac{2}{3}E$.
Combining these two expressions we find the equation of state of the ideal gas $$\beta pV = N.$$
Comparing this with the empirical ideal gas law we see that $\beta = \frac{1}{k_B T}.$
A: To increase the temperture of a specific amount you need energy, this energie  shows itself as kinetic energy , or how would you increase themperature
