0.16 Equilibrium and the second law of thermodynamics  (Page 5/11)

We therefore need a new function $S(W)$ , so that, when we combine the two glasses of water, $S_{\mathrm{total}}=S_1+S_1$ . Since $S_{\mathrm{total}}=S(W_{\mathrm{total}})$ , $S_1=S(W_1)$ , and $W_{\mathrm{total}}=\times (W_1, W_1)$ , then our new function $S$ must satisfy the equation $$S(\times (W_1, W_1))=S(W_1)+S(W_1)$$ The only function $S$ which will satisfy this equation is the logarithm function, which has theproperty that $\ln \times (x, y)=\ln x+\ln y$ . We conclude that an appropriate state function which measures thenumber of microstates in a particular macrostate is [link] .

Observation 2: absolute entropies

It is possible, though exceedingly difficult, to calculate the entropy of any system under any conditions ofinterest from the equation $S=k\ln W$ . It is also possible, using more advanced theoreticalthermodynamics, to determine $S$ experimentally by measuring heat capacities and enthalpies of phase transitions. Values of $S$ determined experimentally, often referred to as "absolute" entropies, havebeen tabulated for many materials at many temperatures, and a few examples are given in [link] . We treat these values as observations and attempt to understand thesein the context of [link] .

There are several interesting generalities observed in [link] . First, in comparing the entropy of the gaseous form of a substance to eitherits liquid or solid form at the same temperature, we find that the gas always has a substantially greater entropy. This is easy tounderstand from [link] : the molecules in the gas phase occupy a very much larger volume. Thereare very many more possible locations for each gas molecule and thus very many more arrangements of the molecules in the gas. It isintuitively clear that $W$ should be larger for a gas, and therefore the entropy of a gas is greaterthan that of the corresponding liquid or solid.

Second, we observe that the entropy of a liquid is always greater than that of the corresponding solid. Thisis understandable from our kinetic molecular view of liquids and solids. Although the molecules in the liquid occupy a comparablevolume to that of the molecules in the solid, each molecule in the liquid is free to move throughout this entire volume. The moleculesin the solid are relatively fixed in location. Therefore, the number of arrangements of molecules in the liquid is significantlygreater than that in the solid, so the liquid has greater entropy by [link] .

Third, the entropy of a substance increases with increasing temperature. The temperature is, of course, ameasure of the average kinetic energy of the molecules. In a solid or liquid, then, increasing the temperature increases the totalkinetic energy available to the molecules. The greater the energy, the more ways there are to distribute this energy amongst themolecules. Although we have previously only referred to the range of positions for a molecule as affecting $W$ , the range of energies available for each molecule similarly affects $W$ . As a result, as we increase the total energy of a substance, we increase $W$ and thus the entropy.