Problem Statement
Find the surface tension in a soap bubble of 40 mm diameter when the inside pressure is \(2.5 \, \text{N/m}^2\) above atmospheric pressure.
Given Data
- Diameter of bubble, \(d = 40 \, \text{mm} = 40 \times 10^{-3} \, \text{m} = 0.040 \, \text{m}\)
- Pressure in excess of outside (gauge pressure), \(p = 2.5 \, \text{N/m}^2\)
Solution
1. Apply the Formula for Pressure Inside a Soap Bubble
For a soap bubble, there are two free surfaces (an inner and an outer surface), so the excess pressure inside is given by the formula:
Where:
- \(p\) = Excess pressure inside the bubble (N/m\(^2\))
- \(\sigma\) = Surface tension (N/m)
- \(d\) = Diameter of the bubble (m)
2. Rearrange the Formula to Solve for Surface Tension (\(\sigma\))
3. Substitute the Given Values and Calculate \(\sigma\)
The surface tension of the soap bubble is \(0.0125 \, \text{N/m}\).
Explanation
1. Surface Tension:
Surface tension is a property of liquid surfaces that causes them to behave like an elastic membrane. It is the force per unit length acting parallel to the surface, which resists an increase in the surface area.
2. Pressure Difference in Bubbles:
For a spherical interface, a pressure difference exists across the curved surface. For a single curved surface (like a liquid droplet), the excess pressure is \( \frac{4\sigma}{d} \). However, a soap bubble has two air-liquid interfaces (an inner and an outer film), effectively doubling the surface tension effect, which leads to the formula \( p = \frac{8\sigma}{d} \).
3. Calculation:
By rearranging the formula, we can directly calculate the surface tension using the given excess pressure and the diameter of the bubble. The units are consistent, leading to surface tension in N/m.
Physical Meaning
1. Stability of Bubbles:
The calculated surface tension is what allows the soap bubble to maintain its spherical shape and resist the internal pressure. Without surface tension, the bubble would immediately burst. The higher the surface tension, the more stable the bubble for a given internal pressure.
2. Minimizing Surface Area:
Liquids naturally tend to minimize their surface area due to surface tension. This is why bubbles and droplets tend to be spherical, as a sphere has the smallest surface area for a given volume.
3. Practical Applications:
Understanding surface tension is crucial in various fields, including:
- Detergents and Soaps: Surfactants in soaps reduce the surface tension of water, allowing it to spread more easily and penetrate fabrics for effective cleaning.
- Medical Applications: Lung surfactants reduce the surface tension in the alveoli, preventing them from collapsing.
- Industrial Processes: In processes like painting, coating, and printing, surface tension plays a significant role in how liquids spread and adhere to surfaces.
