Carbon monoxide residual entropy

For most substances, the Third Law and statistical calculations of the entropy of the ideal gas are in agreement, but there are exceptions, some of which are summarized in Table 4.2. The difference results from residual entropy, So, left in the solid at 0 Kelvin because of disorder so that St - So calculated from Cp/TdT is less than the St calculated from statistical methods. In carbon monoxide the residual disorder results from a random arrangement of the CO molecules in the solid. Complete order in the solid can be represented schematically (in two-dimensions) by... 

As the temperature is reduced, the thermal energy available to overcome kinetic barriers to the lowest-energy state is reduced and, in some cases, residual entropy is difficult to remove at low temperatures. In other words, in the low-temperature cooling of these substances, reversibility cannot be approached. An example of a substance with residual entropy is solid CO. Carbon monoxide has a very small dipole moment, which indicates that there is a preferential orientation of molecules at low temperature. The magnitude of the dipole is so small, however, that at temperatures at which the preference becomes appreciable, there is insufficient thermal energy to overcome kinetic barriers for rotation of the molecules in the solid. 

Example 1. Calculate the residual entropy at T = 0 for carbon monoxide. 

Solution Carbon monoxide has a small electric dipole moment (approx 0.1 Debye), which gives the molecules an energetically preferred orientation as T — 0. However, this dipole moment is so small that the preference is not appreciable until very low temperatures, and the random orientation of the molecules (the dipole has equal probability of pointing in one direction or its opposite) remains as the temperature is lowered. For a mole of CO, each molecule can point in either of two directions and there are 2Na configurations that are about equally probable. This model predicts a residual entropy of... 

A smaller but entirely similar effect is found in the carbon monoxide, CO, lattice at 0 K and this material displays a residual entropy of 4.5 J K-1 mol-1. This case is discussed in Frame 17, section 17.1. 

The actual value for the residual entropy of crystalline carbon monoxide is 0.55R = 4.6 J/K mol the mixing is apparently not quite half and half. In the case of NO, the residual entropy is 033R = 2.8 J/K mol, which is about one-half of 5.76 J/K mol this has been explained by the observation that the molecules in the crystal of NO are dimers, (N0)2. Thus one mole of NO contains only double molecules this reduces the residual entropy by a factor of two. 

Figure 8.8 (a) Two-dimensional representation of the defect-free arrangement of crystalline carbon monoxide. The third law will hold for this crystal, and 5 = 0 at 0 K. (b) Two-dimensional arrangement of real crystalline carbon monoxide. Because of defects in orientation, this crystal will have a residual entropy, and 5 > 0 at 0 K. 

At 0 K, the entropy of carbon monoxide crystal is not zero but has a value of 4.2 J/K mol, called the residual entropy. According to the third law of thermodynamics, this means that the crystal does not have a perfect arrangement of the CO molecules, (a) What would be the residual entropy if the arrangement were totally random (b) Comment on the difference between the result in (a) and... 

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. 

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