Explain in detail the kinetic interpretation of temperature .

Explanation

To understand the microsropic origin of temperature in the same way , from perssure exerted by a gas,
`P=1/3N/Vmnu^bar(2)`
`PV=1/3Nmnu^bar(2)` .....(1)
Comparing the equation (1) with ideal gas equation PV=NkT,
`NkT=1/3Nmnu^bar(2)`
`kT=1/3mnu^bar(2)` ......(2)
Multiply the above equation by `3/2` on both sides,
`3/2kT=1/2mbarnu^(2)` .....(3)
R.H.S. of the eqution (3) is called average kinetic energy of a single molecule `(bar(KE))`.
`bar(KE)=epsilon=3/2kT` ......(4)
Equation (3) implies that the temperature of a gas is a measure of the average translational kinetic energy per molecule of the gas.
Equation (4) is a very important result from kinetic theory of gas. We can infer the following from this equation,
(i) The average kinetic energy of the molecule is directly proportional to absolute temperature of the gas. The equation (3) gives the connection between the macroscopic world (temperature) to microscopic world (motion of molecules).
(ii) The average kinetic energy of each molecule depends only on temperature of the gas not on mass of the molecule. In other words, if the temperature of an ideal gas is measured using thermometer, the average kinetic energy of each molecule can be calculated without seeing the molecule through naked eye.
By multiplying the total number of gas molecules with average kinetic energy of each molecule, the internal energy of the gas is obtained.
Internal energy of ideal gas `U=N(1/2mnu^bar(2))`
By using equation (3), `U=3/2NkT .... (5)`
Here, we understand that the internal energy of an ideal gas depends only on absolute temperature and is independent of pressure and volume.