BioPHYSICAL chemistry

80 PARTI THERMODYNAMICS AND KINETICS

resulting in spheres due to their small surface area/volume ratio. For a

curved surface with a radiusr, the pressure on the concave side of the

interface,Pin, is always greater than the pressure on the convex side,Pout.

The outward force is given by the product of the pressure and the area,

4 πr^2 Pin. The inward force is given by two contributions. The first con-

tribution is from the pressure, 4πr^2 Pout. The second is due to the surface

tension,γ, defined as the proportionality constant relating the work needed

to change the surface area, A, of a sample by an infinitesimal amount.

For the case of a drop, the work done by changing a drop from a radius

r to r +dr is given by:

γ[A(r+dr) −A(r)] =γ[4π(r+dr)^2 − 4 πr^2 ] = 8 πγrdr (4.11)

(r+dr)^2 −r^2 =r^2 + 2 rdr+dr^2 −r^2 = 2 rdr+dr^2 ≈ 2 rdr dr << 1

Since work is equal to force times distance, the force opposing the move-

ment through the distance dris given by 8πγdr. At equilibrium the inward

and outward forces balance and so:

4 πr^2 Pin= 4 πr^2 Pout+ 8 πγdr (4.12)

This relationship shows that the pressure on the concave surface is always

greater than the pressure on the convex surface by an amount given by

the surface tension.

The tendency of liquids to move up a capillary is called capillary action

and is a consequence of surface tension. When a glass capillary is initi-

ally immersed in a liquid, the liquid will adhere to the walls (Figure 4.9).

As the film climbs up, the surface curves and the pressure beneath

the meniscus is less than the atmospheric pressure by 2γ/r. This

pressure difference drives the liquid upward until hydrostatic

equilibrium is reached:

(4.13)

where ρis the density of the liquid andgis the gravitational

force constant.The surface tension of water at 293 K is equal to

72.7 mN m−^1. This value is strongly dependent upon the prop-

erties of the solutions, as methanol has a much lower value,

28.9 mN m−^1 , whereas mercury has a much higher value,

472 mN m−^1.

ΔP

r

gh h

gr

== =

22 γ

ρ

γ

ρ

or

PP

in out r

=+

2 γ

Pouter

Pouter h

Pinner  pgh

Figure 4.9Water
will travel up a small
capillary until the
pressure difference
is equal to the
hydrostatic pressure
arising from the
water in the
capillary.