Example of Surface Tension and Capillary Action

 Example 1

a) Develop an expression to calculate the pressure inside a droplet of water,

b) Determine the pressure inside a 1cm diameter (D = 1cm) water droplet exposed to atmospheric pressure (P₀ = 101,300 N/m²) {Assume water surface tension, σ = 71.97 * 10^-3 N/m)

Solution

a) The difference in pressure on the inside and the outside tend to expand the droplet and the surface tension constrains the pressure.

Carrying out a force balance on the droplet we obtain:

(Pressure difference) x (Cross sectional area) = (Surface tension) x (Surface length)

(P_i – P₀)πR² = σ(2πR)

simplifying the above expression we get:

b) From the above equation, we can substitute the known variables to solve for the unknown Pressure inside the droplet, P_i  (where: R = D/2 = 0.5cm = 0.005m)

P_i = (101,300 N/m²) + ((2*71.97 * 10^-3 N/m)/0.005m)

P_i = 101,330 N/m²

Example 2

A capillary tube has a very small inner diameter and when it is immersed slightly in a liquid, the liquid will rise within the tube to a height that is proportional to its surface tension. This phenomenon is referred to as a capillary action.

The figure below depicts a glass capillary tube slightly immersed in water. The water rises by a height, h, within the tube and the angle between the meniscus and glass tube is θ. Perform a force balance on the system and develop a relationship between the capillary rise ,h, and surface tension ,σ.

Solution

Let us consider a cross section of the liquid column with a height h and diameter 2R with the weight of the column, W and the surface tension force, T. 

The weight can be calculated as:

The force due to surface tension acts at the circumference where the liquid touches the wall. The surface tension force is, therefore, the product of surface tension and circumference:

T = (2πR)σ

Addition of the forces in the vertical direction show that the vertical component of the surface tension force must equal the weight of the liquid column:

W = T cos(θ) or ρπR²hg = 2πRσcos(θ)

Thus, rearranging we can solve for the height, h.