Behaviour of heat capacity

Although we have obtained the Boltzmann distribution law for systems in which the values of nt are large, its validity is Irnore general and it can be shown to apply equally well to systems in which the probability of any particular energy state being occupied is very small. 

The behaviour of the heat capacities of substances both in the solid and gaseous phases was a mystery to scientists before the discovery of quantum theory. In classical mechanics where energy is regarded as continuously variable it can be shown that the energy of a system is divided equally between the various modes of motion called degrees of freedom. Furthermore, according to classical physics each degree of freedom contributed RT/ 2 to the 

The classical result is correct for monatomic gases but considerably overestimates the heat capacities of diatomic and polyatomic gases, and 

300 K the value of RT is approximately 2.5 kJ mol 1,) However Aevib is usually much greater than kT and under these circumstances vibrations do not contribute significantly to heat capacity (Fig. 9.7). Thus for nitrogen at room temperature the rotational contribution is approximately iiTand the vibrational contribution almost zero. Thus Cv 5/2)1 and y w 7/5 = 1.40 as opposed to the classical prediction of 1,29. For iodine the vibrational spacings are closer (Table 9.1) and we would predict y 1.29 in accord with the classical value. If the temperature is varied the heat capacity of a diatomic or polyatomic gas may show steps as the contributions from rotations and vibrations rise as the energy separations become comparable to kT, The positions of the steps depend on the moments of inertia and the vibrational frequencies of the molecules. 

The atoms in a monatomic solid can be regarded as three-dimensional oscillators. At high temperatures they contribute the classical value to the molar heat capacity and Cv = 31 or about 25 JK 1mol 1. The observation that the heat capacities of most monatomic solids have this value near room temperature is known as Dulong and Petit s Law. At low temperatures the heat capacity is zero, the temperature dependence having the same shape as 

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There have been published numerous papers reporting the measurements of the heat capacity of water adsorbed on oxide surfaces. The likely important effect of the surface heterogeneity on the behaviour of heat capacity data has never been considered yet in the accompanying theoretical interpretations. It may only surprise us, because in the case of other adsorption systems such studies have already been published, demonstrating the crucial role of surface heterogeneity. 

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