## Wednesday, December 17, 2014

### Surface Energy of a Liquid and Applications

We can express a surface tension in terms of surface energy also. When the shape of the liquid is change and it acquire some energy that energy could be called as a surface energy. We can define surface tension even in terms of surface energy per unit area. We can derive the mathematical equation for surface energy as shown below.

Let us  consider a metal frame who is in a rectangular shape with 3 sides fixed and the 4th side is having a movable piston. You can immerse it in a soap solution and get it out therefore there will be a thin soap film formed in the frame as shown. The soap film because of the surface tension tries to require a minimum surface area. It tried to pull all the sides but being the 3 sides are fixed they are not going to move to the 4th side is going to move.

We can apply an external force on the piston therefore we can get it back to its original shape and during this process we have to do some work. This work is done due to the surface tension and the same work can be stored in the format of the surface energy inside the liquid.

Basing on the definition of the work done as the surface tension multiplied by the change in the area with can solve different problems and we can deal with a different applications. If a liquid drop is rising from the zero radius to a particular radius we can calculate the work done as shown.

If a big drop of known radius splits into an number of identical drops of unknown radius we can calculate the equation for the work done basing on the concept of surface tension as shown below.

We can also calculate the work done when multiple drops of known radius and identical in size are combined together to form a big drop of unknown radius. In solving both this problem is we have to understand that area of the small drops  together will have a larger area than the area of the big drop.

Problem and solution

A water drop of radius 1 cm is broken up into small water drops each of radius 1 mm assuming that at constant temperature what is the work done in this process?

We can solve this problem basing on the concept of work done as shown below. Using the same concept that is explained above to more problems are solved in the given diagram below.

Expression for potential energy

In a capillary tube rain water raises to a certain height due to the capillarity, some work has to be done. This work is stored in the format of potential energy. Here we can derive the equation for the potential energy as shown below. The mass of the water in the capillary tube distributed over its entire height. Its Centre of mass has to be considered at exactly the middle because the mass is uniformly distributed.

In the place of the height we can use the equations that we had derived in the in  the capillary action derivation.

When two plates are attached with each other with the water layer between them we can calculate the force of attraction acting between them as shown below.

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