The Macroscopic Description of Nonequilibrium
In a one-phase simple system at equilibrium, the intensive macroscopic state is
specified by s + 1 variables, where s is the number of independent chemical substances.
These variables could be T, P, and s-
1 concentrations or mole fractions. (We use the
letter s now rather than the letter c for the number of substances because we will use c
for concentrations.) Nonequilibrium states are more complicated than equilibrium states
and require more variables to specify them. The discussion of this chapter is limited to a
one-phase simple system containing several substances in which no chemical reactions
can occur and in which the deviation from equilibrium is not very large. Processes that
take place far from equilibrium, including such things as explosions and turbulent flow,
are difficult to describe mathematically, and we do not attempt to describe them.
The thermodynamic variables of a nonequilibrium system can depend on position,
although the definitions of these variables require measurements at equilibrium. In order
to define these variables in a nonequilibrium system, we visualize the following
process: A small portion of the system (a subsystem) is suddenly removed from the
system and allowed to relax adiabatically to equilibrium at fixed volume. Once
equilibrium is reached, variables such as the temperature, pressure, density, and
concentrations in this subsystem are measured. These measured values are assigned
to a point inside the volume originally occupied by the subsystem and to the time at
which the subsystem was removed. This procedure is performed repeatedly at different
times and different locations in the system, and interpolation procedures are carried out
to obtain smooth functions of position and time to represent the temperature,
T = T(x,
y, z, t) = T(r, t)
y, z, t) = P(r, t)
c i -- ci(x, y, z, t) =-
(i -- 1,2 .
is the concentration of substance i, measured in mol m -3 or mol L -1. The
symbol r stands for the position vector with components x, y, and z.
The intensive variables are measured after each subsystem comes to equilibrium, so
they obey the same relations among themselves as they would in an equilibrium system.