The ratio of specific to molar heat capacity of a body

Explanation

Monoatomic molecule : Average kinetic energy of a molecule
`=[3/2kT]`
(i) Total energy of a mole of gas
`=3/2 kT xxN_A = 3/2 RT`
(ii) For one mole , the molar specific heat at constant volume
`C_V = (dU)/(dT) =d/(dT) [3/2RT]`
`C_V = [3/2R]`
`C_P = C_V + R =3/2 R+R`
`=5/2R`
(iii) The ratio of specific heats,
`gamma=C_P/C_V=(5//2R)/(3//2R)=5/3=1.67`
Diatomic molecule : Average kinetic energy of a diatomic molecule at low temperature `5/2kT`.
(i) Total energy of one mole of gas
`=5/2kT xxN_A = 5/2RT`
(Here, the total energy is purely kinetic)
(ii) For one mole specific heat at constant volume
`C_V=(dU)/(dT)=[5/2RT]=5/2R`
But, `C_P = C_V + R = 5/2R + R = 7/2R`
`:. gamma=C_P/C_V=(7//2R)/(5//2R)=7/5=1.40`
(iii) Energy of a diatomic molecule at high temperature is equal to `7/2` RT.
`C_V=(dU)/(dT)=[7/2RT]=7/2R`
`:. C_P = C_v + R = 7/2 R+R`
`C_P=9/2R`
It is noted that the `C_V and C_P` are higher for diatomic molecules than the monoatomic molecules. It is implied that to increase the temperature of diatomic gas molecules by `1^@C` it require more heat energy than monoatomic molecules.
`:.gamma=C_P/C_V=(9//2R)/(7//2R)=9/7=1.28`
Triatomic molecule :
(i) Linear molecule :
Energy of one mole
`=7/2 kTxxN_A = 7/2RT`
`C_V=(dU)/(dT)=d/(dT)[7/2RT]`
`C_V=[7/2R]`
`C_P=C_V+R=7/2R+R`
`=9/2R`
`gamma=C_P/C_V=(9//2R)/(7//2R)=9/7=1.28`
(ii) Non - Linear molecule :
Energy of one mole
`=6/2kTxxN_A=6/2RT=3RT`
`C_V=(dU)/(dT)=3R`
`C_P=C_V+R`
`=3R+R =4R`
`gamma=C_P/C_V=(4R)/(3R)=4/3=1.33`