Monoatemic gas moleules has three translational degrees of freedom.
From law of equipartition of energy average energy of an molecule at temperature T is (3/2)KBT
Total internal energy of one mole of such gas is
= (3/2) RT (12)
If CV is melar specific heat at constant volume then
CVv = dU/dT
= (3/2)R (13)
now for an ideal gas
CP - CV = R
CP - molar specific heat capacity at constant presseve
CP = 5/2 R (14)
Thus for a monoatomic gas ratio of specific heats is
γmono = CP/CV= 5/3 (15)
(ii) Diatomic gases :
A diatomic gas molecule is treated as a rigid rotator like dumb-bell and has 5 degrees of freedom out of which three degrees of freedom are translatoinal and two degrees of freedom are rotational.
Using law of equipartition of energy the total internal energy of one mole of diatomic gas is
= (5/2) RT (16)
Specific heats are thus
γdia= 5/7 (rigid rotater)
If diatomic molecule is not only rigid but also has an vibrational mode in addition, then
U = (7/2) RT
9. Specific heat Capacity of Solids
From law of equipartation of energy we can can also determine specific heats of solids.
Consider that atoms in a solid are vibrating about their mean position at some temperature T.
Oscillation in one dimension has average energy equals 2(1/2)KBT=KBT, as (1/2)KBT is PE and (1/2)KBT is KE of the atom.
In three dimensions average kinetic energy is 3KBT.
For one mole of solid total energy is
At constant pressure ΔQ =ΔU+PΔV=ΔU since for solids ΔV is negligible hence
This is Dulang and Petit law.
Here we note that predictions of specific heats of solids on the basis of law of equipetation of energy are independent of temperature.
As we go towards low temperatures T→0 there is a pronounced departure from the value of specific heat of solids as calculated.
It is seen that specific heats of substance aproaches to zero as T→0.
This result can further be explained using the principles of quantum mechanics which is beyond our scope.