0.16 Equilibrium and the second law of thermodynamics  (Page 6/11)

Fourth, the entropy of a substance whose molecules contain many atoms is greater than that of a substancecomposed of smaller molecules. The more atoms there are in a molecule, the more ways there are to arrange those atoms. Withgreater 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 lowmolecular weight. This result is a harder to understand, as it arises from the distribution of the momenta of the molecules ratherthan the positions and energies of the molecules. It is intuitively clear that the number of arrangements of the molecules is not affected by the mass of the molecules. However, even at the same temperature, the range of momentaavailable for a heavier molecule is greater than for a lighter one.To see why, recall that the momentum of a molecule is $p=mv$ and the kinetic energy is $\mathrm{KE}=\frac{mv^2}{2}=\frac{p^2}{2m}$ . Therefore, the maximum momentum available at a fixed total kineticenergy $\mathrm{KE}$ is $p=\sqrt{2m\mathrm{KE}}$ . Since this is larger for larger mass molecules, the range ofmomenta is greater for heavier particles, thus increasing $W$ and the entropy.

Observation 3: condensation and freezing

We have concluded from our observations of spontaneous mixing that a spontaneous process always produces thefinal state of greatest probability. A few simple observations reveal that our deduction needs some thoughtful refinement. Forexample, we have observed that the entropy of liquid water is greater than that of solid water. This makes sense in the contextof [link] , since the kinetic theory indicates that liquid water has a greater value of $W$ . Nevertheless, we observe that liquid water spontaneously freezes attemperatures below 0°C. This process clearly displays a decrease in entropy and therefore evidently a shift from a moreprobable state to a less probable state. This appears to contradict directly our conclusion.

Similarly, we expect to find condensation of water droplets from steam when steam is cooled. On days of highhumidity, water spontaneously liquefies from the air on cold surfaces such as the outside of a glass of ice water or the windowof an air conditioned building. In these cases, the transition from gas to liquid is clearly from a higher entropy phase to a lowerentropy phase, which does not seem to follow our reasoning thus far.

Our previous conclusions concerning entropy and probability increases were compelling, however, and we shouldbe reluctant to abandon them. What we have failed to take into consideration is that these phase transitions involve changes ofenergy and thus heat flow. Condensation of gas to liquid and freezing of liquid to solid both involve evolution of heat. Thisheat flow is of consequence because our observations also revealed that the entropy of a substance can be increased significantly byheating. One way to preserve our conclusions about spontaneity and entropy is to place a condition on their validity: a spontaneousprocess produces the final state of greatest probability and entropy provided that the process does not involve evolution of heat. This is an unsatisfying result, however, sincemost physical and chemical processes involve heat transfer. As an alternative, we can force the process not to evolve heat by isolating the system undergoing the process: no heat can be released if there is no sink to receive the heat, andno heat can be absorbed if there is no source of heat. Therefore, we conclude from our observations that a spontaneous process in an isolated system produces the final state of greatest probability and entropy. This is one statement of the Second Law of Thermodynamics .