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KINETIC THEORY 253 where C is the molar specific heat at constant At this point, notice an important feature in p Eq.(12.26). While each translational and pressure. Thus, rotational degree of freedom has contributed only C = 5 R (12.30) one ‘squared term’ in Eq.(12.26), one vibrational p mode contributes two ‘squared terms’ : kinetic 2 and potential energies. C The ratio of specific heatsγ = p = 5 (12.31) Each quadratic term occurring in the C v 3 expression for energy is a mode of absorption of energy by the molecule. We have seen that in 12.6.2 Diatomic Gases thermal equilibrium at absolute temperature T, for each translational mode of motion, the As explained earlier, a diatomic molecule treated as a rigid rotator, like a dumbbell, has 5 degrees average energy is ½ k T. Bhe most elegant of freedom: 3 translational and 2 rotational. principle of classical statistical mechanics (first Using the law of equipartition of energy, the total proved by Maxwell) states that this is so for each internal energy of a mole of such a gas is mode of energy: translational, rotational and 5 5 vibrational. That is, in equilibrium, the total U = kBT × N A = RT (12.32) 2 2 energy is equally distributed in all possible The molar specific heats are then given by energy modes, with each mode having an average energy equal to ½ k T. This is known as the law 5 7 B C (rigid diatomic) = R, C = R (12.33) of equipartition of energy. Accordingly, each v 2 p 2 translational and rotational degree of freedom 7 of a molecule contributes ½ k TBto the energy, γ(rigid diatomic) = (12.34) while each vibrational frequency contributes 5 2 × ½ k T = k T , since a vibrational mode has If the diatomic molecule is not rigid but has B B in addition a vibrational mode both kinetic and potential energy modes. The proof of the law of equipartition of energy U = 5 k T + k T  = 7 RT 2 B B  A 2 is beyond the scope of this book. Here, we shall apply the law to predict the specific heats of gases 7 9 9 theoretically. Later, we shall also discuss briefly, C v R, C p R, γ = R (12.35) 2 2 7 the application to specific heat of solids. 12.6.3 Polyatomic Gases 12.6 SPECIFIC HEAT CAPACITY In general a polyatomic molecule has 3 12.6.1 Monatomic Gases translational, 3 rotational degrees of freedom The molecule of a monatomic gas has only three and a certain number ( f ) of vibrational modes. translational degrees of freedom. Thus, the According to the law of equipartition of energy, it is easily seen that one mole of such a gas has average energy of a molecule at temperature  T is (3/2)B T . The total internal energy of a mole  3 3  of such a gas is U =  kBT + kBT + f B T NA 2 2 3 3 U = kBT × N A = RT (12.27) i.e.,v = (3 + f ) R,pC = (4 + f ) R, 2 2 (4 + f) The molar specific heat at constant volume, γ = (12.36) C , is (3 + ) v Note that C – C = R is true for any ideal dU 3 p v C vmonatomic gas) = = RT (12.28) gas, whether mono, di or polyatomic. dT 2 Table 12.1 summarises the theoretical For an ideal gas, predictions for specific heats of gases ignoring C – C = R (12.29) p v any vibrational modes of motion. The values are Reprint 2026-27