Problem Statement
Find the surface tension in a soap bubble of 30 mm diameter when the inside pressure is 1.962 N/m² above atmosphere.
Given Data
- Diameter of bubble, \(d = 30 \, \text{mm}\)
- Excess Pressure, \(p = 1.962 \, \text{N/m}^2\)
Solution
1. Convert Diameter to SI Units (metres)
The pressure is already in SI units, so we only need to convert the diameter.
2. Calculate the Surface Tension (\(\sigma\))
For a soap bubble, there are two surfaces (inner and outer) contributing to the surface tension. Therefore, the formula for excess pressure is different from a liquid droplet.
Rearranging the formula to solve for surface tension, \(\sigma\):
The surface tension in the soap bubble is approximately \( \sigma \approx 0.00736 \, \text{N/m} \).
Explanation: Droplet vs. Soap Bubble
It is crucial to distinguish between a liquid droplet and a soap bubble:
- Liquid Droplet: A solid sphere of liquid with only one surface interface with the surrounding air. The formula is \(p = 4\sigma/d\).
- Soap Bubble: A thin film of liquid with air on both the inside and the outside. This creates two surface interfaces (an inner surface and an outer surface), both of which contribute to the surface tension effect. Therefore, the effect is doubled, and the formula becomes \(p = 8\sigma/d\).
Physical Meaning
The calculated surface tension, \(\sigma \approx 0.00736 \, \text{N/m}\), is the force per unit length acting along the surface of the soap film that holds the bubble together. This value quantifies the strength of the cohesive forces within the soap solution.
This problem demonstrates that for a given surface tension, a soap bubble can contain only half the excess pressure of a water droplet of the same size. This is because the pressure has to support two surfaces instead of one. The delicate balance between the internal air pressure and the inward pull of surface tension is what allows a bubble to exist.
