Physical Chemistry of Foods

explained in the figure. The equality gives for the height of the column

h¼

2 g

rrg

ð 10 : 8 Þ

Forr¼0.5 mm,g¼72 mN?m
^1 ,r¼ 103 kg?m
^3 , andg¼9.8 m^2 ?s
^1 , this
results inh¼29.4 mm. It provides a method of determining surface tension.
The meniscus at the wall of a wider vessel has to be explained in the same
way, but the mathematics is complicated, as is the shape of such a meniscus.
If the wall of the capillary is not at all wetted by the liquid, capillary
depression results, as shown in Figure 20.22b; Eq. (10.8) also holds. We will
discuss contact angles (as in Figure 10.22c) and capillary displacement in
Section 10.6.

10.5.3 The Kelvin Equation

As discussed, the gas in a small bubble in a liquid is at increased pressure.
According to Henry’s law, which states that the solubility of a gas in a liquid
is proportional to its pressure, its solubility is increased. From the Laplace
equation (10.8) we can derive

sðrÞ

s?

¼exp

x^0

r



x^0 ¼

2 gVD

RT

ð 10 : 9 Þ

which is known as the Kelvin equation. Heresmeans solubility, which is
thus larger for a sphere of radiusrthan in the absence of curvature (r¼?).
x^0 is an auxiliary parameter of dimension length,VDthe molar volume of
the disperse phase (in m^3 ?mol
^1 , it equals molar mass over mass density),
andRThas its usual meaning.
Unlike the Laplace equation, which can only be applied to fluid
surfaces, the Kelvin equation is valid for any phase boundary, also solid–gas,
solid–liquid, and even solid–solid. As for the Laplace equation, Eq. (10.9)
can also be modified to accommodate nonspherical curved surfaces;
then 2/(1/R 1 þ1/R 2 ) should be used instead ofr. The values for the
solubility should give the thermodynamic activity of the substance involved,
for the equation to be generally valid; for ideal or ideally dilute solutions,
concentrations can be used. Most gases in water show ideal behavior. It
should further be noted that Eq. (10.9) applies for one component; if a
particle contains several components, it should be applied to each of them
separately. Finally, there are some conditions that may interfere with the