Modern Engineering Thermodynamics

3.9 Thermodynamic Equations of State 77
When World War I ended, significant social changes occurred in Germany, including more rights for women. In 1919, the
University of Göttingen allowed Noether to proceed with her habilitation (eligibility for tenure), and she was given tenure
in June 1919.
She had a very successful career in advanced mathematics, and one of her most important but unheralded discoveries is
Noether’s theorem, which proves a relationship between symmetries and conservation principles. This basic result was
praised by Albert Einstein in a letter to David Hilbert, when he referred to Noether’s “penetrating mathematical thinking.”
It was her work that led to formulations for several concepts of Einstein’s general theory of relativity.
3 The actual term is privatdozent, which means an unsalaried university “private” lecturer or teacher paid directly by the students.
CRITICAL THINKING
If an equation of state is an equation that relates thermodynamic properties of the system when it is in different thermodynamic
states, then what can you say about a system that has an equation of state of the form pT = constant, where p is the
absolute pressure and T is the corresponding absolute temperature of the system at any time?
Most materials have very complex thermodynamic equations of state, which are not given in textbooks. However,
these complex equations are easily solved by computer programs like spreadsheets, Engineering Equation
Solver, 4 Matlab, and so forth, which are readily available today. Thermodynamic property programs can also be
found for PDAs and smart cell phones. 5
While engineers in the 21st century use computer programs to find the thermodynamic property values they
need, thermodynamic courses still rely on the use of printed property tables and charts in textbooks. However,
students are encouraged to search out and use modern computer programs to solve problems or to verify that
their table or chart solutions are correct.
In this section, we focus on the relatively simple equations of state of incompressible substances, ideal gases,
and a few variations on the ideal gas equation. A more comprehensive discussion of the behavior of real gases
is given in a subsequent chapter.
3.9.1 Incompressible Materials
The simplest equation of state is that for an incompressible material. One of its equations of state is merely v =
constant. This equation can be used for either a solid or a liquid, but it cannot be used for a vapor or a gas.
Incompressible substances also have one other state equation, which specifies that the specific internal energy of
an incompressible material is a function of only one variable, temperature. Thus, the full set of state equations
that characterize an incompressible material are
and
v = constant (3.28)
u = uT ð Þ (3.29)
The constant volume specific heat of an incompressible material is given by Eq. (3.15) as
c v =
∂u
= du ðsince u is independent of v hereÞ (3.30)
∂T v dT
Because the specific enthalpy is defined as h = u + pv, Eq. (3.19) gives the constant pressure specific heat of an
incompressible material as
c p =
∂h
∂T p
= du
dT +
∂ðpvÞ
∂T
p
= du
dT + pvβ = c v = c ðwhen β = 0Þ (3.31)
4 F-Chart Software, Box 44042, Madison, WI 53744 (info@fchart.com).
5 For example, see enggtools.com, processacesoftware.com, and appstorehq.com/engineeringtables.