Average kinetic energy in ideal gas
Recall: ideal gas law
`PV = NkT`
where `k` is Boltzmann's constant
`k = 1.38 times 10^(-23)\ J//K`
`P = F/A`
`P =(N(moverline(v^2))/(3L))/A`
`P =1/3(Nmoverline(v^2))/(AL)` where area `A` times `L` is the volume `V`
`P =1/3(Nmoverline(v^2))/V`
`=>`
`PV =1/3(Nmoverline(v^2)) = NkT`
`=>`
`1/3(moverline(v^2)) = kT`
`2/3(1/2moverline(v^2)) = kT` where `overline(KE) = 1/2moverline(v^2)`
=>`
`overline(KE)=1/2moverline(v^2)=3/2kT`
This equations tells us that the average translational kinetic energy of molecules in random motion in an ideal gas is directly proportional to the absolute temperature of the gas.