Modern Thermodynamics
- Chapter 1
5
Since the fundamental quantities in thermodynamics are functions of many variables, thermodynamics
makes extensive use of calculus of many variables. A brief summary of some basic properties of
functions of many variables is given in the appendix A1.1 (at the end of this chapter). Functions of state
variables, such as U and S, are often called state functions.
It is convenient to classify thermodynamic variables into two categories: variables such as
volume and amount of a substance (moles), which indicate the size of the system, are called extensive
variables. Variables such as temperature T and pressure p, which specify a local property, which do not
indicate the system's size, are called intensive variables.
If the temperature is not uniform, heat will flow until the entire system reaches a state of uniform
temperature. Such a state is the state of thermal equilibrium. The state of thermal equilibrium is a
special state towards which all isolated systems will inexorably evolve. A precise description of this state
will be given later in this book. In the state of thermal equilibrium, the values of total internal energy U
and entropy S are completely specified by the temperature T, the volume V and the amounts of the
system's chemical constituents Nk (moles):
U = U(T,V,Nk ) or S = S(T,V,Nk ) (1.1.1)
The values of an extensive variable such as total internal energy U, or entropy S, can also be specified by
other extensive variables:
U = U(S,V,Nk ) or S = S(U,V,Nk ) (1.1.2)
As we shall see in the following chapters, intensive variables can be expressed as derivatives of one
extensive variable with respect to another. For example, we shall see that the temperature
T =
∂U
∂S





V,Nk
.
The laws of thermodynamics and the calculus of many-variable functions give us a rich understanding of
many phenomena we observe in Nature.
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