Kinetic Theory Of Gases
8. Specific Heat Capacity
(i) Monoatomic gases :
(ii) Diatomic 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
= (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)
- 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
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- 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.