0.22 Entropy and the second law of thermodynamics  (Page 6/9)

Fourth, the entropy of a substance whose molecules contain many atoms is greater than that of a substance composed of smaller molecules. This is somewhat new to us. We have discussed the structures of molecules in terms of specific geometries. However, these structures are not rigid. The atoms within a molecule vibrate, or wiggle, back and forth about the geometry the molecule. The more atoms there are in a molecule, the more ways there are for those atoms to wiggle with respect to each other. With this greater internal flexibility, W is larger when there are more atoms, so the entropy is greater.

Fifth, the entropy of a substance with a high molecular weight is greater than that of substance with a low molecular weight, even if the number of atoms is the same or nearly the same. This result is a harder to understand because it does not relate to the positions that the atoms can occupy. It is clear that the number of arrangements of the molecules is not affected by the masses of the molecules. However, one way to describe the state of a molecule would be to specify its position and its momentum. (In quantum mechanics, we know that it is not possible to specify both the position and the momentum, but this non-quantum explanation is easier to visualize than the quantum explanation.) The range of momenta available for a heavier molecule is greater than for a lighter one, even at the same temperature. To see why, recall that the momentum of a molecule is p = mv and the kinetic energy is KE = (1/2)mv ² = (1/2m)p ² . Therefore, the maximum momentum available at a fixed temperature and a fixed total kinetic energy KE is p = √2mKE . When m is larger for larger mass molecules, the range of momenta is greater for heavier particles. With more possible states for each molecule, W and the entropy are both larger.

Observation 3: condensation and freezing

We have concluded from our observations of spontaneous mixing that a spontaneous process always produces the final state of greatest probability and greatest entropy. A few simple observations reveal that this is not entirely correct and that our conclusion needs some careful refinement. For example, we have observed that the entropy of liquid water is greater than that of solid water. This makes sense in the context of Equation (1), since the kinetic theory indicates that liquid water has a greater value of W. Nevertheless, we observe that liquid water spontaneously freezes at temperatures below 0 ºC. This process clearly displays a decrease in entropy and therefore evidently a shift from a more probable state to a less probable state. This appears to directly contradict our conclusion.

Similarly, we expect to find condensation of water droplets from steam when steam is cooled. On days of high humidity, water spontaneously liquefies from the air on cold surfaces such as the outside of a glass of ice water or the window of an air conditioned building. In these cases, the transition from gas to liquid is clearly from a higher entropy phase to a lower entropy phase, which does not seem to follow our reasoning thus far. In fact, our reasoning would imply that ice should always melt and water should always evaporate, no matter what the conditions are. We have either left something out of our reasoning, or we have made a serious mistake. Before abandoning our reasoning, let’s recall that our previous conclusions concerning entropy and probability increases were quite compelling. We should be reluctant to abandon them. What did we overlook?