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.