× ½ k B T = k B T , since a vibrational mode has both kinetic and potential energy modes. The proof of the law of equipartition of energy is beyond the scope of this book. Here, we shall apply the law to predict the specific heats of gases theoretically. Later, we shall also discuss briefly, the application to specific heat of solids.
. SPECIFIC HEAT CAPACITY . . Monatomic Gases The molecule of a monatomic gas has only three translational degrees of freedom.
Thus, the average energy of a molecule at temperature T is ( / ) k B T . The total internal energy of a mole of such a gas is U ( . ) The molar specific heat at constant volume, C v , is C v (monatomic gas) = d d U T = RT ( . ) For an ideal gas, C p – C v = R ( .
) where C p is the molar specific heat at constant pressure. Thus, C p = R ( . ) The ratio of specific heats p C C γ = ( . ) .
. Diatomic Gases As explained earlier, a diatomic molecule treated as a rigid rotator, like a dumbbell, has degrees of freedom: translational and rotational. Using the law of equipartition of energy, the total internal energy of a mole of such a gas is U ( . ) The molar specific heats are then given by C v (rigid diatomic) = R , C p = R ( .
) γ (rigid diatomic) = ( . ) If the diatomic molecule is not rigid but has in addition a vibrational mode U k T N , , p C R C R γ R ( . ) . .
Polyatomic Gases In general a polyatomic molecule has translational, rotational degrees of freedom and a certain number ( f ) of vibrational modes. According to the law of equipartition of energy, it is easily seen that one mole of such a gas has U =