Capillarity and Surface Forces (Liquids) - Surface and Colloid Chemistry: Principles and Applications - page 15

surface of a liquid. The phenomena is also called capillary force . One may ask then
if a similar consideration is also required in the case of solids. It has been found that
the answer to the latter is positive. For example, in order to remove a liquid that is
inside a porous media such as a sponge, force equivalent to these capillary forces
would be needed.
As seen in Figure 2.4, the bubble is produced by applying a suitable pressure, Δ P ,
to obtain a bubble of radius R , where the surface tension of the liquid is γ. Let us
consider a situation the bubble is expanded by applying pressure P inside . The surface
area of the bubble will increase by d A , and the volume by d V . This means that there
are two opposing actions: expansion of volume and of area. The work done can be
expressed in terms of that done against the forces of surface tension and that done
in increasing the volume. At equilibrium, the following condition will exist between
these two kinds of work:

γ d A = (P g − P liquid ) dV (2.15)

where d A = 8 π R dR ( A = 4 π R 2 ), and dV = 4 π R 2 dR (V = 4/3 π R 3 ). Combining
these relations gives the following:

γ 8 π R dR = Δ P 4 π R 2 d R (2.16)

and

Δ P = 2 γ/ R ( 2.17)

where Δ P = (P g − P liquid ). Since the free energy of the system at equilibrium is
constant, dG = 0, and then these two changes in the system are equal. If the same
consideration is applied to the soap bubble, then the expression for Δ P bubble
will be

Δ P bubble = P inside − P outside

= 4 γ/ R (2.18)

because now there exists two surfaces, and the factor 2 is needed to consider this
state.
The pressure applied produces work on the system, and the creation of the bub-
ble leads to the creation of a surface area increase in the fluid. The Laplace equa-
tion relates the pressure difference across any curved fluid surface to the curvature,
1/ radius and its surface tension γ. In those cases where nonspherical curvatures are
present, the more universal equation is obtained:

Δ P = γ (1/R 1 + 1/R 2 ) (2.19)

It is also seen that, in the case of spherical bubbles, since R 1 = R 2 , this equation
becomes identical to Equation 2.18. Thus, in the case of a liquid drop in air (or gas