Modern Engineering Thermodynamics

3.7 Phase Diagrams 65
WHO WAS EMMY NOETHER?
PART 2
Emmy Noether never became a language teacher; instead she decided to attend the University of Erlangen to study mathematics.
Unfortunately, at that time, women were not allowed to enroll because the faculty felt that allowing female students
would “overthrow all academic order.” She could only audit classes with the permission of each professor whose
lectures she wished to attend. Nonetheless, on July 14, 1903, she passed the graduation exam.
During the winter of 1903–1904, she studied at the University of Göttingen, attending lectures by astronomer Karl
Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert. By then, restrictions
on women’s rights in Erlangen were rescinded and she returned there. She officially reentered the university on October
24, 1904, and declared her intention to focus solely on mathematics. In 1907, she received a doctorate in mathematics.
Other thermodynamic properties, such as entropy and availability, are introduced later in this text when they are
needed. It must be remembered, however, that not all thermodynamic properties are directly measurable. A pressure
gauge and a thermometer give us numerical values for p and T, but there are no instruments that give us
values of u and h directly. It takes much more sophisticated measurements to allow us to calculate accurate
values for u and h. More complex mathematical relations between thermodynamic properties are developed
after the reader is thoroughly familiar with the concept of entropy discussed in Chapter 7.
3.7 PHASE DIAGRAMS
A pure substance is composed of a single chemical compound, which may itself be composed of a variety of
chemical elements. Water (H 2 O), ammonia (NH 3 ), and carbon dioxide (CO 2 ) are all pure substances, but air is
not because it is a mixture of N 2 ,O 2 ,H 2 O, CO 2 , and so forth. All substances can exist in one or more of the
gaseous (or vapor), liquid, or solid physical states, and some solids can have a variety of molecular structures.
In 1875, the American physicist Josiah Willard Gibbs (1839–1903) introduced the term phase to describe the different
forms in which a pure substance can exist. We now speak of the gaseous, liquid, and solid phases of a
pure substance, and we recognize that a pure substance may have a number of different solid phases. 2 Multiple
solid phases are called allotropic, a term that comes from the Greek words allos, meaning“related to,” and trope
meaning “forms of the same substance.” For example, graphite and diamond are allotropic forms of carbon.
A substance made up of only one physical phase is called homogeneous; if it is composed of two or more phases
it is called heterogeneous. Coexistent phases are separated by an interface, called the phase boundary, of finite thickness
across which the property values change uniformly. A system in which two phases coexist in equilibrium is
called saturated.
The number of degrees of freedom within a heterogeneous mixture of pure substances is given by Gibbs’s phaseruleas
f = C − P + 2
where f is the number of degrees of freedom, C is the number of components (pure substances) in the mixture, and
P is the number of phases. Also, f can be interpreted to be the number of intensive properties of the individual
phases required to fix the state of the individual phases. For example, a homogeneous (P = 1) pure substance (C =1)
requires f =1− 1+2=2intensivepropertiestofixitsstate.Similarly,ahomogeneous(P = 1) mixture of two pure
substances (C = 2) requires f =2− 1+2=3intensive properties to fix its state, and so forth. The case of a two-phase
(P = 2) pure substance (C = 1), however, is misleading, because f =1− 2+2=1,butthissimplymeansthateach
SATURATED WITH WHAT?
The term saturated comes from the 18th century, when heat was thought to be a fluid. At that time, it was thought that a
substance could be saturated with heat, just like water can become saturated with salt or sugar. Today we recognize that
heat is not a fluid, and therefore the use of the word saturation in reference to a thermodynamic phase change is really a
misnomer. However, this term is now completely entrenched in modern thermodynamic literature and cannot be changed.
2 Actually, matter can exist in a bewildering variety of phases beyond the common solid, liquid, and vapor forms. Ferromagnetic,
antiferromagnetic, ferroelectric, superconducting, superfluid, nematic, smectic, and so on are all valid phases.