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
  • If CV is melar specific heat at constant volume then
         CVv = dU/dT
          = (3/2)R
    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                    
  • 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
    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
  • 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.

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