Explain in detail the kinetic interpretation of temperature .

Explanation

Kinetic interpretation of temperature :
(i) To understand the microscopic origin of temperature in the same way,
Rewrite the equation `P=nmbar(v_(x)^(2))`
`P=(1)/(3)(N)/(V)mbar(v^(2))`
`PV=(1)/(3)mbar(v^(2))" "...(1)`
(ii) Comparing equation (1) with ideal gas equation PV = NkT,
`NkT=(1)/(3)Nmbar(v^(2))`
`kT=(1)/(3)mbar(v^(2))" "...(2)`
(iii) Multiply the above equation by 3/2 on both sides,
`(3)/(2)kT=(1)/(2)mbar(v^(2))" "...(3)`
(iv) R.H.S. of equation (3) is called average kinetic energy of a single molecule `(bar(KE))`.
(v) The average kinetic energy per molecule
`bar(KE)=in=(3)/(2)kT" "...(4)`
(vi) Equation (3) implies that the temperature of a gas is a measure of the average translational kinetic energy per molecule of the gas.
(vii) Equation (4) is a very important result from kinetic theory of gas. We can infer the following from this equation.
(viii) The average kinetic energy of the molecule is directly proportional to absolute temperature of the gas. Equation (3) gives the connection between the macroscopic world (temperature) to microscopic world (motion of molecules).
(ix) The average kinetic energy of each molecule depends only on temperature of the gas not on mass of the molecule.
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)/(2)mbar(v^(2)))` By using equation (3)
`U=(3)/(2)NkT" "...(5)`
From equation (5), we understand that the internal energy of an ideal gas depends only on absolute temperature and is independent of pressure and volume.