Carbon monoxide absolute entropy

A carbon monoxide (CO) crystal is found to have entropy greater than zero at absolute zero of temperature. Give two possible explanations for this observation. 

Since any impure crystal has at least the entropy of mixing at the absolute zero, its entropy cannot be zero such a substance does not follow the third law of thermodynamics. Some substances that are chemically pure do not fulfill the requirement that the crystal be perfectly ordered at the absolute zero of temperature. Carbon monoxide, CO, and nitric oxide, NO, are classic examples. In the crystals of CO and NO, some molecules are oriented differently than others. In a perfect crystal of CO, all the molecules should be lined up with the oxygen pointing north and the carbon pointing south, for example. In the actual crystal, the two ends of the molecule are oriented randomly it is as if two kinds of carbon monoxide were mixed, half and half. The molar entropy of mixing would be... 

Generate a graph of the temperature dependence of the absolute entropy of carbon monoxide from 50 to 2500 K at 1 atm total pressure. It is reasonable to assume that CO behaves ideally over this temperature range. Do not neglect the contribution from vibrational motion. 

The existence of a residual entropy at the temperature of 0 K means, by application of Boltzmann s equation, that the number of complexions Q is not necessarily equal to 1 at that temperature. Thus, absolute order is not always reached, and the value of the number of complexions must be able to be greater than 1 (never less than 1, of course, which would be absmd. This explains why the residual entropy value is always positive). By quantitative study, we have been able to quantify the initial state of disorder and evaluate the residual entropy. To exhibit the method, we shall consider a crystal of carbon monoxide, in which the two oxygen and carbon atoms are differentiated, but are nonetheless very similar. In particular, they have the same weak electric dipole moment. 

As mentioned earlier, there are some substances such as carbon monoxide that do not obey the third law of thermodynamics in their ordinary forms. The absolute entropy of these substances determined by an integration such as in Eq. (3.5-1) turned out to be too small to agree with values inferred from entropy changes of chemical reactions and absolute entropies of other substances. Carbon monoxide molecules have only a small dipole moment (small partial charges at the ends of the molecule) and the two ends of the molecule are nearly the same size, so a carbon monoxide molecule fits into the crystal lattice almost as well with its ends reversed as in its equilibrium position. Metastable crystals can easily form with part of the molecules in the reversed position. If we assume that the occurrence of reversed molecules is independent of the rest of the state of the crystal, we can write... 

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