Absolute Entropies

Author: John Hutchinson

It is possible, though exceedingly difficult, to calculate the entropy of any system under any conditions of interest from the equation S=klnW. It is also possible, using more advanced theoretical thermodynamics, to determine S experimentally 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, and a few examples are given in table 1. We treat these values as observations and attempt to understand these in the context of equation 1.

Absolute Entropies of Specific Substances

	T [°C]	S [J/mol°C]
H2O(g)	25	188.8
H2O(l)	25	69.9
H2O(l)	0	63.3
H2O(s)	0	41.3
NH3(g)	25	192.4
HN3(l)	25	140.6
HN3(g)	25	239.0
O2(g)	25	205.1
O2(g)	50	207.4
O2(g)	100	211.7
CO(g)	25	197.7
CO(g)	50	200.0
CO2(g)	24	213.7
CO2(g)	50	216.9
Br2(l)	25	152.2
Br2(g)	25	245.5
I2(s)	25	116.1
I2(g)	25	260.7
CaF2(s)	25	68.9
CaCl2(s)	25	104.6
CaBr2(s)	25	130
C8H18(s)	25	361.1

There are several interesting generalities observed in table 1. First, in comparing 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 is easy to understand from equation 1: the molecules in the gas phase occupy a very much larger volume. There are very many more possible locations for each gas molecule and thus very many more arrangements of the molecules in the gas. It is intuitively clear 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 observe 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 entropy of a substance increases with increasing temperature. The temperature is, of course, a measure of the average kinetic energy of the molecules. 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.

Fourth, the entropy of a substance whose molecules contain many atoms is greater than that of a substance composed of smaller molecules. The more atoms there are in a molecule, the more ways there are to arrange those atoms. With 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. This result is a harder to understand, as it arises from the distribution of the momenta of the molecules rather than 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 momenta available 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 KE = mv²/2 = p²/2m. Therefore, the maximum momentum available at a fixed total kinetic energy KE is . Since this is larger for larger mass molecules, the range of momenta is greater for heavier particles, thus increasing W and the entropy.