Modern Thermodynamics

- Chapter 1

7

s(T,n) u(T,n)

n(x) T(x)

diS/dt0

deS/dt

(a)

T1

T2

T(x)

(b)

Fig. 1.2. (a) In a nonequilibrium system, the temperature number density nk(x) may vary with position.

The entropy and energy of such a system may be described by an entropy density s(T, nk) and an energy

density u(T, nk). The total entropy,

S = s[T(x),nk

(x)]dV,

∫V

the total energy

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

∫v

and the total mole number Nk = nk (x)dV.

∫V

For such a nonequilibrium system, the total entropy S is

not a function of U, N and the total volume V. The term diS /dt is the rate of change of entropy due to

chemical reactions, diffusion, heat conduction and other such irreversible processes; according to the

second law diS/dt can only be positive. In open system, entropy can also change due to exchange of

energy and matter; this is indicated by the term deS/dt, which can be either positive or negative. (b) A

system in contact with thermal reservoirs of unequal temperatures is a simple example of a

nonequilibrium system. The temperature is not uniform and there is a flow of heat due to the temperature

gradient. The term deS/dt is related to the exchange of heat at the boundaries in contact with the heat

reservoirs while diS/dt is due to the irreversible flow of heat within the system.