1. Surface tension: Thermodynamic properties of the interface between two phases are
described by a state function called the surface tension S. It is defined in terms of the
work required to increase the surface area by an amount dA through d̄W = SdA.
(a) By considering the work done against surface tension in an infinitesimal change in
radius, show that the pressure inside a spherical drop of water of radius R is larger than
outside pressure by 2S/R. What is the air pressure inside a soap bubble of radius R?
• The work done by a water droplet on the outside world, needed to increase the radius
from R to R + ∆R is
∆W = (P − Po ) · 4πR2 · ∆R,
where P is the pressure inside the drop and Po is the atmospheric pressure. In equilibrium,
this should be equal to the increase in the surface energy S∆A = S · 8πR · ∆R, where S
is the surface tension, and
∆Wtotal = 0, =⇒ ∆Wpressure = −∆Wsurface ,
resulting in
2S
(P − Po ) · 4πR2 · ∆R = S · 8πR · ∆R, =⇒ (P − Po ) = .
R
In a soap bubble, there are two air-soap surfaces with almost equal radii of curvatures,
and
2S
Pfilm − Po = Pinterior − Pfilm = ,
R
leading to
4S
Pinterior − Po = .
R
Hence, the air pressure inside the bubble is larger than atmospheric pressure by 4S/R.
(b) A water droplet condenses on a solid surface. There are three surface tensions involved
S aw , S sw , and S sa , where a, s, and w refer to air, solid and water respectively. Calculate
the angle of contact, and find the condition for the appearance of a water film (complete
wetting).
• When steam condenses on a solid surface, water either forms a droplet, or spreads on
the surface. There are two ways to consider this problem:
Method 1: Energy associated with the interfaces
1
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