Page 130 - Physical Chemistry

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and thermal equilibrium. System and surroundings can exchange energy (as heat and Section 4.2
work) but not matter. Let chemical reaction or transport of matter between phases or Entropy and Equilibrium
both be occurring in the system at rates small enough to maintain thermal and me-
chanical equilibrium. Let heat dq syst flow into the system as a result of the changes that
occur in the system during an infinitesimal time period. For example, if an endother-
mic chemical reaction is occurring, dq syst is positive. Since system and surroundings
are isolated from the rest of the world, we have
dq surr dq syst (4.2)

Since the chemical reaction or matter transport within the nonequilibrium system is
irreversible, dS univ must be positive [Eq. (3.39)]:

dS univ dS syst dS surr 7 0 (4.3)
for the process. The surroundings are in thermodynamic equilibrium throughout the
process. Therefore, as far as the surroundings are concerned, the heat transfer is re-
versible, and [Eq. (3.20)]

dS surr dq surr >T (4.4)
However, the system is not in thermodynamic equilibrium, and the process involves an
irreversible change in the system. Therefore dS syst dq syst /T. Equations (4.2) to (4.4)
give dS syst dS surr dq surr /T dq syst /T. Therefore

dS syst 7 dq syst >T
dS 7 dq irrev >T closed syst. in therm. and mech. equilib. (4.5)

where we dropped the subscript syst from S and q since, by convention, unsubscripted
symbols refer to the system. [Note that the thermal- and mechanical-equilibrium
condition in (4.5) does not necessarily mean that T and P are held constant. For
example, an exothermic reaction can raise the temperature of the system and the
surroundings, but thermal equilibrium can be maintained provided the reaction is
extremely slow.]
When the system has reached material equilibrium, any infinitesimal process is a
change from a system at equilibrium to one infinitesimally close to equilibrium and
hence is a reversible process. Thus, at material equilibrium we have
dS dq >T (4.6)
rev
Combining (4.6) and (4.5), we have

dq material change, closed syst. in
dS (4.7)
T mech. and therm. equilib.

where the equality sign holds only when the system is in material equilibrium. For a re-
versible process, dS equals dq/T. For an irreversible chemical reaction or phase change,
dS is greater than dq/T because of the extra disorder created in the system by the irre-
versible material change.
The first law for a closed system is dq dU dw. Multiplication of (4.7) by T
(which is positive) gives dq T dS. Hence for a closed system in mechanical and ther-
mal equilibrium, we have dU dw TdS, or
material change, closed syst. in
dU T dS dw (4.8)
mech. and therm. equilib.
where the equality sign applies only at material equilibrium.