Modern Thermodynamics

- Chapter 1

8

u[T(x), nk(x)] = Internal energy per unit volume, (1.2.1)

can be defined in terms of the local temperature T(x) and the concentration,

nk (x) = Moles of constituent k per unit volume. (1.2.2)

Similarly an entropy density, s(T, nk), can be defined. The atmosphere of the earth, shown in Box 1.2, is

an example of a non-equilibrium system in which both nk and T are functions of position. The total

energy U, the total entropy S and the total amount of the substance are:

S = s[T(x),nk (x)]

dV,

∫V

(1.2.3)

U = u[T(x),nk (x)] dV

V

∫ (1.2.4)

Nk = nk (x) dV

V

∫ (1.2.5)

In nonequilibrium (non-uniform) systems, the total energy U is no longer a function of other

extensive variables such as S, V and Nk (as in (1.1.2)) and obviously one cannot define a single

temperature for the entire system because it may not uniform. In general, each of the variables, the total

energy U, entropy S, the amount of substance Nk and the volume V is no longer a function of the other

three variables, as in (1.1.2). But this does not restrict in any way our ability to assign an entropy to a

system that is not in thermodynamic equilibrium, as long as the temperature is locally well defined.

In texts on classical thermodynamics, when it is sometimes stated that entropy of a

nonequilibrium system is not defined, it is meant that S is not a function of the variables U, V and Nk. If

the temperature of the system is locally well defined, then indeed the entropy of a nonequilibrium system

can be defined in terms of an entropy density as in (1.2.3).