Modern Engineering Thermodynamics

58 CHAPTER 3: Thermodynamic Properties
3.2 WHY ARE THERMODYNAMIC PROPERTY VALUES IMPORTANT?
Since all of the basic laws of thermodynamics have terms containing key thermodynamic properties, we have to
determine numerical values for these properties before the laws can be used to solve a thermodynamics
problem. In other words, you cannot solve thermodynamic problems without accurate numerical values for the
system’s thermodynamic properties.
Thermodynamic property values can be determined from five sources:
1. Thermodynamic equations of state.
2. Thermodynamic tables.
3. Thermodynamic charts.
4. Direct experimental measurements.
5. The formulae of statistical thermodynamics.
This chapter deals with the first three sources. Source 4, the techniques of direct property measurement, are not
discussed in this text, but the information given in many of the thermodynamic problem statements can be
assumed to have come from such measurements. The last source, the formula of statistical thermodynamics, is
covered in Chapter 18 of this textbook.
Property values are often given in thermodynamic problem statements in the process path designation. For example,
if a system changes its state by an isothermal process at 250.°C, then we know that T 1 =250.°C =T 2 . Thus,
the process path statement gives us the value of a thermodynamic property (temperature, in this case) in each
of the two states. Process path statements that imply that some property is held constant during a change of
state are quite common in thermodynamics.
3.3 FUN WITH MATHEMATICS
In the previous chapter, you were told that the values of any two thermodynamic properties are sufficient to fix
the state of a homogeneous (single-phase) pure substance subjected to only one work mode. This means that
each thermodynamic property of the a pure substance can be written as a function of any two independent thermodynamic
properties. Thus, if x, y, andz are all intensive properties, we can write
or
f ðx, y, zÞ = 0 (3.1)
x = xðy, zÞ
y = yðx, zÞ
z = zðx, yÞ
Using the chain rule for differentiating the composite functions in the previous equations yields
dx =
∂x
dy +
∂x
dz
∂y
z
∂z y
dy =
∂y
dx +
∂y
dz
∂x
z
∂z
x
dz =
∂z
dx +
∂z
dy
∂x y ∂y
x
where the notation (∂x/∂y) z means the partial derivative of the function x with respect to the variable y while holding
the variable z constant. Substituting the expression for dy into the expression for dx and rearranging gives
1 − ∂x
∂y
z
∂y
∂x
z
dx =
∂x
∂y
z
∂y
∂z
x
+ ∂x
∂z
Normally, the partial differential notation (∂x/∂y) automatically implies that all the other variables of x are held
constant while differentiation with respect to y is carried out. However, in thermodynamics, we always have a
wide choice of variables with which to construct the function x, but when we change variables, we do not
always change the functional notation. For example, we can write x = x(y, z) =x(y, w) =x(y, q), where each of
y
dz