## S k log W

Implicit in Boltzmann's equation is the Third law of thermodynamics, which states that the entropy of crystalline solids is zero at 0 Kelvin. This due to the perfectly-ordered arrangement of the atoms and the cessation of their vibration. The W in Boltzmann's equation (Fig. 1.6) is the number of ways that a large number of indistinguishable particles can be arranged. If all atoms in the solid are in their equilibrium positions and their lowest vibrational state, only one arrangement is possible, or W = 1. The consequence is S = 0.

### 1.1.6 Gibbs and chemical thermodynamics

Between 1876 and 1878, J. Willard Gibbs, a professor of mathematical physics at Yale University, published a series of three papers entitled "On the Equilibrium of Heterogeneous Substances", which was destined to stand with Carnot's "Reflections..." as an indispensable foundation of thermodynamics. The first two papers dealt with the graphical representation of thermodynamic properties and processes (movement of a system from one state to another). Particularly significant was the introduction of a graph representing the entropy and temperature at various stages of a steam engine. Given the conditions at the four key points of the engine's cycle, this plot enabled the heat terms in Fig. 1.4 to be expressed quantitatively. Along with the analogous pressure Vs volume graph, which gave the work of the four processes, Carnot's heat-engine was placed on a firm quantitative basis.

Gibbs was the first to represent the thermodynamic properties of matter on three-dimensional plots. As shown in his second paper, these depicted a surface of a thermodynamic property, say pressure, as a function of two other properties, usually temperature and volume (although Gibbs' favorite was entropy as a function of internal energy and volume). Such diagrams represent the equation of state of a substance in a manner that remains to this day.

Far and away the most important of Gibbs' contributions to thermodynamics was his third paper. In it, he developed the concept of equilibrium, not just for multiple phases of a pure substance, but for systems that contained two or more chemical species, or components, as Gibbs Fig. 1.6 Statue of Bolt/mann on his tomb, on which is carved his famous equation called them. He showed that a new thermodynamic property originating from a particular combination of internal energy, pressure, temperature, volume and entropy was the determining factor in expressing all types of thermodynamic equilibria. This new property is not called the Gibbs free energy. In essence, Gibbs succeeded in transforming the portion of physical chemistry that dealt with thermodynamics from a qualitative collection of observations to the quantitative, theoretical science it is today. Figure 1.7 lists some of the other thermodynamic terms that bear his name.

Fig. 1.7 J. Willard Gibbs and some of his important contributions to the thermodynamics of chemical and nonreacting multicomponent systems

### 1.1.7 Historical summary

The fundamentals of thermodynamics were essentially fully established by the beginning of the 20th century. More precisely, this discipline is termed classical thermodynamics to distinguish it from the later arrival of statistical thermodynamics (to which Gibbs contributed). The latter variant incorporates the motion and interactions of the atoms, molecules and electrons in a system in order to calculate its macroscopic thermodynamic properties. Delving into the submicroscopic behavior of the particles of a body is beyond the ken of classical thermodynamics.

Statistical thermodynamics relies on quantum mechanics to permit statistical averaging of the energy states of individual particles that ultimately leads to prediction of macroscopic properties. The intermediate in this process is a quantity called the partition function, which is directly related to a classical property called the Helmholz free energy. For example, the particles of an ideal-gas do not (by definition) attract or repel each other and are very small relative to the volume available to them. With only this information, statistical thermodynamics is able to produce the ideal gas law, a relation that classical thermodynamics simply accepts as a fact of nature. Other insights provided by statistical thermodynamics include the behavior of the specific heat of solids as the temperature approaches absolute zero and the motions of atoms responsible for the internal energy of gases (translation, rotation) and of solids (vibration).  