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CHAPTER
4

Material Equilibrium

CHAPTER OUTLINE

The zeroth, first, and second laws of thermodynamics give us the state functions T, U, 4.1 Material Equilibrium
and S. The second law enables us to determine whether a given process is possible. A
process that decreases S univ is impossible; one that increases S univ is possible and irre- 4.2 Entropy and Equilibrium
versible. Reversible processes have S univ 0. Such processes are possible in principle 4.3 The Gibbs and Helmholtz
but hard to achieve in practice. Our aim in this chapter is to use this entropy criterion Energies
to derive specific conditions for material equilibrium in a nonisolated system. These
conditions will be formulated in terms of state functions of the system. 4.4 Thermodynamic Relations for
a System in Equilibrium

4.5 Calculation of Changes in
4.1 MATERIAL EQUILIBRIUM
State Functions
Material equilibrium (Sec. 1.2) means that in each phase of the closed system, the
number of moles of each substance present remains constant in time. Material equilib- 4.6 Chemical Potentials and
rium is subdivided into (a) reaction equilibrium, which is equilibrium with respect to Material Equilibrium
conversion of one set of chemical species to another set, and (b) phase equilibrium, 4.7 Phase Equilibrium
which is equilibrium with respect to transport of matter between phases of the system
without conversion of one species to another. (Recall from Sec. 1.2 that a phase is a ho- 4.8 Reaction Equilibrium
mogeneous portion of a system.) The condition for material equilibrium will be de-
rived in Sec. 4.6 and will be applied to phase equilibrium in Sec. 4.7 and to reaction 4.9 Entropy and Life
equilibrium in Sec. 4.8.
To aid in discussing material equilibrium, we shall introduce two new state func- 4.10 Summary
tions in Sec. 4.3, the Helmholtz energy A U TS and the Gibbs energy G
H TS. It turns out that the conditions for reaction equilibrium and phase equilibrium
are most conveniently formulated in terms of state functions called the chemical
potentials (Sec. 4.6), which are closely related to G.
A second theme of this chapter is the use of the combined first and second laws to
derive expressions for thermodynamic quantities in terms of readily measured proper-
ties (Secs. 4.4 and 4.5).
Chapter 4 has lots of equations and is rather abstract and not so easy to master.
Later chapters, such as 5, 6, and 7, apply the general results of Chapter 4 to specific
chemical systems, and these chapters are not as intimidating as Chapter 4.

The initial application of the laws of thermodynamics to material equilibrium is largely the
work of Josiah Willard Gibbs (1839–1903). Gibbs received his doctorate in engineering
from Yale in 1863. From 1866 to 1869 Gibbs studied mathematics and physics in Europe.
In 1871 he was appointed Professor of Mathematical Physics, without salary, at Yale. At
that time his only published work was a railway brake patent. In 1876–1878 he published
in the Transactions of the Connecticut Academy of Arts and Sciences a 300-page mono-
graph titled “On the Equilibrium of Heterogeneous Substances.” This work used the first
and second laws of thermodynamics to deduce the conditions of material equilibrium.
Gibbs’ second major contribution was his book Elementary Principles in Statistical
Mechanics (1902), which laid much of the foundation of statistical mechanics. Gibbs also