Page 250 - Physical Chemistry

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Actual system Model system Section 7.8
Curved Interfaces

Volume V bulk

Volume V

Interphase region Gibbs dividing surface
Figure 7.20
Volume V bulk Volume V
(a) A two-phase system. (b) The
corresponding Gibbs model
(a) (b) system.

presence of the interphase region is allowed for by a two-dimensional surface phase
that has zero volume but nonzero values of other thermodynamic properties. The
actual system of Fig. 7.20a (which consists of the bulk phases a and b and the inter-
phase region) is replaced by the model system of Fig. 7.20b. In the model system,
phases a and b are separated by a surface of zero thickness, the Gibbs dividing sur-
face. Phases a and b on either side of the dividing surface are defined to have the same
intensive properties as the bulk phases a and b in the actual system. The location of
the dividing surface in the model system is somewhat arbitrary but usually corre-
sponds to a location within or very close to the interphase region of the actual system.
Experimentally measurable properties must be independent of the location of the di-
viding surface, which is just a mental construct. The Gibbs model ascribes to the
dividing surface whatever values of thermodynamic properties are needed to make the
model system have the same total volume, internal energy, entropy, and amounts of
components that the actual system has. For a detailed treatment of the Gibbs model,
see Defay, Prigogine, Bellemans, and Everett.

7.8 CURVED INTERFACES
When the interface between phases a and b is curved, the surface tension causes
the equilibrium pressures in the bulk phases a and b to differ. This can be seen from
Fig. 7.21a. If the lower piston is reversibly pushed in to force more of phase a into the
conical region (while some of phase b is pushed out of the conical region through
the top channel), the curved interface moves upward, thereby increasing the area of
the interface between a and b. Since it requires work to increase , it requires a
greater force to push in the lower piston than to push in the upper piston (which would
a
b
decrease ). We have shown that P P , where a is the phase on the concave side
of the curved interface. (Alternatively, if we imagine phases a and b to be separated
by a thin membrane under tension, this hypothetical membrane would exert a net
b
a
downward force on phase a, making P exceed P .)

P b P b

P a R
Figure 7.21
Two-phase systems with a curved
(a) (b) interface.