Young-Laplace and Kelvin Equations
(Chapter 4, pp. 67-69 in Shaw and pp. 10-11 in Osipow)
Phenomena at Curved Surfaces:
• As a consequence of interfacial tension, there is
a balancing pressure across any curved
surface.
• Assume we deform a soap bubble by a small
amount by applying a pressure:
radius = r
radius = r - dr
DA = 2 x (4pr2 – 4p(r-dr)2)
DA = 2 x (4pr2 – 4p(r2-dr2+2rdr))
DA = -16prdr
Two types of work are done:
•
Surface Energy Work
DWS= 16 pgrdr
•
Pressure Work
DWP= DP x 4pr2dr
When the two are equal we get for the
pressure across the curved surface:
DP x 4pr2dr = 16pgrdr
Hence:
DP = 4g/r
Known as the Young-Laplace equation
for a hollow bubble.
Other curved surfaces:
Young-Laplace equation for a spherical
droplet:
DP = 2g/r
Young-Laplace equation for any curved
surface with principle radii of curvature r1
and r2:
DP = g(1/r1 + 1/r2)
1/r2~0
1/r1
DP = g/r1
The Young-Laplace equation has been
experimentally validated in the
following ways:
1) Two connected soap bubbles.
2) Measurement of capillary rise for
liquids with known surface tensions.
h
3) Through a validation of the Kelvin
equation which is derived from the
Young-Laplace equation.
The Kelvin Equation
(Chapter 4, pp. 67-69 in Shaw)
If the radius of a droplet increases from r
to r+dr, the increase in surface free
energy will be:
DGi = 8pgrdr
Pr
Pr+dr
Po
If the process involves the transfer of
dn moles of liquid from the plane
surface with vapor pressure Po to the
droplet with vapor pressure Pr, the
free energy increase is also equal to:
dn RT ln(Pr/Po)
Thus:
dn RT ln(Pr/Po) = 8pgrdr
And:
dn = 4pr2 dr r/M
Then:
Pr 2gM 2gVm
RT ln 

Po
rr
r
r is the density of the fuid;
M is the molar mass of the fluid; and
Vm is the molar volume.
The above relationship is known as the
Kelvin equation for liquid droplets.
For water droplets:
r, nm
Pr/Po
100
10
1
1.01
1.1
3.0
The Kelvin equation has been
experimentally verified with crossed
mica cylinders (see Heimenz, p. 263):
Pr
For bubbles in a liquid the Kelvin
equation is given as:
Pr
2gM
2gVm
RT ln  

Po
rr
r
When the curvature is negative
(i.e. concave) then a lowering of
the vapor pressure is predicted.
The Kelvin equation can easily explain
the following processes:
• Vapor condensing in small capillaries
• Capillary rise phenomena
• Supersaturation phenomena
(scratching organic chemists)
• Emulsion breakdown mechanism
known as Ostwald ripening
Next lectures: Measurement of Surface
Tensions and Surfactants