Specific heat rotational contribution

Differences in specific heats can be obtained in a similar fashion. Since translational and rotational contributions to Cp at elevated temperatures are minor, the differences to be accounted for are entirely due to vibrational effects. The most effective way to accomplish this is to identify the incremental contribution of each atom or group to Cp, and add or subtract this value from... 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

But according to the quantum theory even the rotational movements of the molecule cannot take place as was supposed by the older kinetic theory. They, too, must become negligible at sufficiently low temperatures, i.e. must contribute no appreciable amount to the specific heat, for otherwise the thermal motion would be in disagreement with the radiation. 

In the gas phase, each molecule has three degrees of translational plus two of rotational freedom so that cp iR, plus a small contribution from vibration which will increase to R at higher temperatures. At low temperatures the atoms in the adsorbed layer will be localized and vibrationally unexcited. In this temperature range the isosteric heat therefore increases as iR. As the temperature is raised, however, and surface vibrations begin to contribute, the specific heat of the adatoms will approach that of the gas and finally exceed it, causing a diminution of the heat. This trend reverses once the adsorbed layer approaches the two dimensional gas. Presumably the vibration perpendicular to the surface contributes somewhat before the vibrations in the gas phase become important. The specific heat of the adsorbed layer will therefore continue to exceed that of the gas by a quantity of the order of iR, until all vibrational degrees of freedom are excited and the gas again dominates. 

Since E — RT d a.fjdT, is computable, and its differential coefficient with respect to T gives the rotational contribution to the specific heat. 

In liquid water there are a number of eoUeetive inter-molecular vibrational modes, such as the HB excitation around 200 cm, which is like a breathing mode involving displacement of many molecules. Another example is the libration mode at around 600 cm, which is a restricted rotation, due to hydrogen-bonding. These low-frequency modes contribute to the entropy and specific heat of liquid water. But these modes are highly anharmonic, with a short lifetime, so the above description based on a harmonic oscillator cannot be used to describe them. 

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