**Gary L. Bertrand**
**University of Missouri-Rolla**

**
The early interpretation of capillarity considered the liquid to "climb"
the walls of the tube to the point that the "grip" of the ring of the surface
around the inner wall could just barely support the weight of the column
of liquid beneath it. This surface tension was calculated as the
weight of the column divided by the inner circumference of the tube.
This was later refined to units of force/length. Thus, if a liquid
of density (D)
0.800 g/cm3 rose to a height (H)
of 5.00 cm in a capillary tube with an inner radius (r)
of 0.0100 cm (in a gravitational field, g
=
980 cm/sec ^{2}), the surface tension (g)
was calculated:**

**Volume = Area x Height
, Mass = Volume x Density , Force = Mass x Acceleration**

**V = P
r^{2}
H , M
= P r^{2}HD ,
F = P r^{2}HDg**

**Circumference = 2 P
r**

**Surface Tension = g
= Force/Circumference = P r2HDg / 2P
r**

**g
= rHDg / 2**

**g
= (0.0100 x 5.00 x 0.800 x 980)/2 = 19.6 dyne/cm = 19.6 erg / cm ^{2}.**

**In SI units, this is expressed
as: g = 19.6 mN / m = 19.6
mJ / m ^{2 }.**

** This idea
of the surface tension supporting the weight of the liquid carried over
to the detachment methods which provide alternative ways of determining
the surface tension. In 1805, Young and LaPlace independently related
surface tension to a pressure difference across a curved surface, providing
the modern definition of surface tension. They used two radii of
curvature (R1 and R2) in mutually perpendicular planes containing a line
normal to the surface to describe the curvature at any point on a surface.
These radii can be difficult to visualize, but are clear for two simple
cases, a spherical surface and a cylindrical surface.**

** For a spherical
surface, the two radii are identical R _{1}
= R_{2}:**

** For a cylindrical
surface, one radius is infinite R _{2}
= infinity, 1/R_{2} = 0 :**

**The equation of Young and LaPlace
is:**

**DP
= g (1/R _{1} + 1/R_{2}) .**

**The application to capillary
rise uses two relationships:**

** In a continuous fluid (a
liquid or air), the pressure is everywhere**
**the same at the same height
in the gravitational field.**

**At different heights in the gravitational
field, the pressure within the**
**fluid differs (DP)
by the density of the fluid (D) multiplied by the**
**difference in height (H) and
the gravitational constant (g).**

**The height (H) at the bottom
of the meniscus is related to the pressure difference through the following
analysis:**

**DP
= P _{1} - P_{2}**

**P _{2} = P_{3}
- D_{liq}gH**

**P _{1} = P_{4}
- D_{air}gH**

**P _{3} = P_{4}**

**DP
= (D _{liq} - D_{air})gH**

**and the equation of Young and
LaPlace becomes:**

**g (1/R _{1}
+ 1/R_{2}) = (D_{liq} - D_{air})gH .**

**At the bottom of the meniscus,
the two radii are equal to each other R _{1}
= R_{2} = R ,**

**and if the meniscus is a hemisphere,
the radii are equal to the radius of the capillary R
= r ,**

**allowing an approximate solution:**

**g
= rHg (D _{liq} - D_{air})/2 .**

**This is a good approximation
for a capillary with a very small radius, but the error increases with
the radius of the capillary because the meniscus tends to flatten at the
bottom and the radius becomes greater than the hemisphere approximation.
In 1915, Lord Rayleigh (see Physical Chemistry
of Surfaces 5th Ed by Arthur
W. Adamson, 1990, John Wiley and Sons, Ch
2) developed a better approximation:**

**g
= (rg/2) (D _{liq} - D_{air}) (H + r/3 - 0.1288r^{2}/H
+ 0.1312r^{3}/H^{2} + ...) .**

**This provides a good approximation
for small capillaries (such that H/r is large), provided that the height
can be measured accurately. The location of the bottom of the meniscus
can be measured very accurately, but the liquid level outside of the capillary
is difficult to observe due to the thickness of the meniscus at the wall
of the container. The twin capillary
technique was developed to avoid this problem, using two capillaries of
different radii (r _{1} , r_{2})
and measuring the difference in the heights of the liquid (DH),
giving a reasonable first approximation:**

**g
= (gDH/2) (D _{liq} - D_{air})
/ (1/r_{1} - 1/r_{2}) .**

**This usually is accurate to within
0.1 dyne/cm for most liquids in a capillary with an inner diameter less
than 1 mm.**