- Chapter 1
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
The laws of thermodynamics and the calculus of many-variable functions give us a rich understanding of
many phenomena we observe in Nature.