Since an erg equals 1 10 ^7 joules, this work is equal to 0.0007275 J. This is
not a lot of work in an absolute sense, but in a relative sense it does have an
effect on the mechanical properties of the liquid (which we will consider
later).Example 22.2
In a zero-gravity environment, a sample of mercury has a spherical shape.
Consider a sample that has a 1.000-cm radius. Determine how much work
must be performed to separate the mercury into 10 equal spheres, assuming
that the only work done is related to the change in surface energy. Recall that
the surface area of a sphere is 4 r^2 and its volume is ^43 r^3. The surface ten-
sion of mercury is 435.5 erg/cm^2.Solution
If the only work that needs to be performed is related to the changing sur-
face area of the drops, then we need to calculate the Avalue for going from
one 1.000-cm drop to 10 equal-volume drops. The single 1.000-cm-radius
drop has a surface area of
A 4 r^2 4 (1.000 cm)^2 12.57 cm^2
Its volume is
V^43 r^3 ^43 (1.000 cm)^3 4.189 cm^3
If 10 equal drops of mercury were formed, then each drop should have a vol-
ume of4. 11089 or 0.4189 cm^3. Using the volume equation, we can determine the
radius of a drop having that volume, and then determine its surface area:
0.4189 cm^3 ^43 r^3
r0.4642 cm
and so the surface area of a smaller drop is
A 4 r^2 4 (0.4642 cm)^2 2.708 cm^2
If there are 10 drops of the same size, then the total area is 10 2.708 cm^2
27.08 cm^2. The change in area for the process is therefore
A(27.08 12.57) cm^2 14.51 cm^2
Again, using the integrated equation 22.5, we find the work as follows:w A435.5
ce
mrg (^2) 14.51 cm
2
w6319 erg 6.319 10 ^4 J
Again, this example shows that only a small amount of work is needed, but
it does suggest that work is needed to turn large drops of liquid into an equal
mass of smaller drops of liquid. The reverse argument is that smaller drops of
liquids will turn into the same mass of a smallernumber oflargerdroplets, and
in doing so work/energy will come outof the system. Since going to a lower
energy is usually (but not always) an indication of a preferred, spontaneous
process, the above example suggests that material will prefer—from an energy
22.2 Liquids: Surface Tension 769