Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

5.8 CRITERIA FORSPONTANEITY 144

l F Fext

x

Figure 5.1 Device to measure the surface tension of a liquid film. The film is

stretched between a bent wire and a sliding rod.

Thesurface tensionor interfacial tension, , is the force exerted by an interfacial sur-
face per unit length. The film shown in Fig.5.1has two surfaces, so we have DF=2l
wherelis the rod length.
To increase the surface area of the film by a practically-reversible process, we slowly
pull the rod to the right in theCxdirection. Thesystemis the liquid. Thexcomponent of
the force exerted by the system on the surroundings at the moving boundary,Fxsys, is equal
toF (Fis positive andFxsysis negative). The displacement of the rod results in surface
work given by Eq.3.1.2:∂w^0 D FxsysdxD2
ldx. The increase in surface area, dAs,
is2ldx, so the surface work is∂w^0 D dAswhere is the work coefficient andAsis the
work coordinate. Equation5.2.7becomes

dUDTdSpdVC dAs (5.7.1)

Substitution into Eq.5.3.6gives

dGDSdTCVdpC dAs (5.7.2)

which is the total differential ofGwithT,p, andAsas the independent variables. Iden-
tifying the coefficient of the last term on the right side as a partial derivative, we find the
following expression for the surface tension:

D



@G

@As



T;p

(5.7.3)

That is, the surface tension is not only a force per unit length, but also a Gibbs energy per
unit area.
From Eq.5.7.2, we obtain the reciprocity relation

@
@T



p;As

D



@S

@As



T;p

(5.7.4)

It is valid to replace the partial derivative on the left side by.@
=@T /pbecause is in-
dependent ofAs. Thus, the variation of surface tension with temperature tells us how the
entropy of the liquid varies with surface area.

5.8 Criteria for Spontaneity

In this section we combine the first and second laws in order to derive some general relations
for changes during a reversible or irreversible process of a closed system. The temperature