 # Kinetic Theory Of Gases

## 8. Specific Heat Capacity

(i) Monoatomic gases :
• 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
U= (3/2)KBTN
= (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
U= (5/2)KBTN
= (5/2) RT
(16)
• Specific heats are thus
CV =(5/2)R
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
and CV=(7/2)R
CP=(9/2)R
and γ=CP/CV=9/7

## 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
U= 3NKBT
= 3RT

• At constant pressure ΔQ =ΔU+PΔV=ΔU since for solids ΔV is negligible hence
C=ΔQ/ΔT=ΔU/ΔT=3R
• 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.