0.22 Entropy and the second law of thermodynamics  (Page 5/9)

We therefore need a new function S(W), so that, when we combine the two glasses of water, S _total = S ₁ + S ₁ . Since S _total = S(W _total ), S ₁ =S(W ₁ ), and W _total = W ₁ × W ₁ , then our new function S must satisfy the equation

S(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 the property that ln(x × y) = ln(x) + ln(y). We conclude that an appropriate state function that measures the number of microstates in a particular macrostate is

S = k ln W

Observation 2: absolute entropies

It is possible, though exceedingly difficult, to calculate the entropy of any system under any conditions of interest from the equation S = k ln W. It is also possible, using more advanced theoretical thermodynamics, to experimentally determine S by measuring heat capacities and enthalpies of phase transitions. Values of S determined experimentally, often referred to as "absolute" entropies, have been tabulated for many materials at many temperatures. A few examples are given in Table 1 measured at pressure of 1 atm. (The superscript º on S indicates standard pressure. It turns out that, for gases, the entropy depends significantly on the pressure.) Our goal is to analyze these data in the context of Equation (1).

There are several interesting trends observed in Table 1. First, if we compare the entropy of the gaseous form of a substance to either its liquid or solid form at the same temperature, we find that the gas always has a substantially greater entropy. This makes sense from Equation (1): the molecules in the gas phase occupy a very much larger volume. There are many more possible locations for each gas molecule and thus many more arrangements of the molecules in the gas. This means that W should be larger for a gas, and therefore the entropy of a gas is greater than that of the corresponding liquid or solid.

Second, we can see in the table that the entropy of a liquid is always greater than that of the corresponding solid. This is understandable from our kinetic molecular view of liquids and solids. Although the molecules in the liquid occupy a comparable volume to that of the molecules in the solid, each molecule in the liquid is free to move throughout this entire volume. The molecules in the solid are relatively fixed in location. Therefore, the number of arrangements of molecules in the liquid is significantly greater than that in the solid, so the liquid has greater entropy by Equation (1).

Third, the data show that the entropy of a substance increases with increasing temperature. We have previously found that the temperature of a substance is a measure of the average kinetic energy of the molecules in the substance. In a solid or liquid, then, increasing the temperature increases the total kinetic energy available to the molecules. The greater the energy, the more ways there are to distribute this energy amongst the molecules. 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.