Excess Pressure in Drops and Bubbles

Understanding the concept of excess pressure in drops and bubbles is crucial for various fields such as physics, engineering, and even biology. This concept explains the pressure difference across the curved surface of a liquid drop or a gas bubble.

Surface Tension and Curvature

Surface tension is a property of the liquid that arises due to the cohesive forces between the molecules at the surface. These forces cause the liquid to minimize its surface area, leading to the formation of drops and bubbles with curved surfaces.

The curvature of these surfaces creates a pressure difference between the inside and the outside of the drop or bubble. This pressure difference is known as excess pressure.

Excess Pressure in Liquid Drops

For a liquid drop, the excess pressure inside the drop is higher than the pressure outside. This is because the molecules on the surface of the drop are pulled inward by the surface tension, leading to a compressed state inside the drop.

The formula for excess pressure inside a spherical liquid drop is given by:

$$ P_{excess} = \frac{2\sigma}{r} $$

where:

• ( P_{excess} ) is the excess pressure inside the drop,
• ( \sigma ) is the surface tension of the liquid,
• ( r ) is the radius of the drop.

Excess Pressure in Gas Bubbles

In the case of a gas bubble in a liquid, there are two interfaces: the inner interface between the gas and the liquid, and the outer interface between the liquid and the surrounding medium. Each interface contributes to the excess pressure.

The formula for excess pressure inside a spherical gas bubble is:

$$ P_{excess} = \frac{4\sigma}{r} $$

Here, the factor of 4 arises because both the inner and outer surfaces of the bubble contribute to the excess pressure.

Differences Between Drops and Bubbles

Here is a table summarizing the differences and important points regarding excess pressure in drops and bubbles:

Aspect	Liquid Drops	Gas Bubbles
Interfaces	Single interface (liquid-air)	Double interface (gas-liquid and liquid-air)
Excess Pressure	( \frac{2\sigma}{r} )	( \frac{4\sigma}{r} )
Inside Pressure	Higher than outside	Higher than outside
Surface Tension	Causes inward pull	Causes inward pull on both interfaces
Formula Derivation	Consider balance of forces on half of the drop	Consider balance of forces on both interfaces

Examples

Example 1: Excess Pressure in a Water Drop

Consider a water drop with a radius of 1 mm and a surface tension of ( 0.0728 \, \text{N/m} ). The excess pressure inside the drop can be calculated as follows:

$$ P_{excess} = \frac{2\sigma}{r} = \frac{2 \times 0.0728 \, \text{N/m}}{0.001 \, \text{m}} = 145.6 \, \text{Pa} $$

Example 2: Excess Pressure in a Soap Bubble

Assume a soap bubble has a radius of 2 cm and the surface tension of the soap solution is ( 0.025 \, \text{N/m} ). The excess pressure inside the bubble is:

$$ P_{excess} = \frac{4\sigma}{r} = \frac{4 \times 0.025 \, \text{N/m}}{0.02 \, \text{m}} = 5 \, \text{Pa} $$

Conclusion

Excess pressure in drops and bubbles is a fundamental concept in fluid mechanics, which is governed by the interplay between surface tension and curvature. The excess pressure is responsible for various phenomena, such as the spherical shape of drops and bubbles, and plays a significant role in processes like capillarity, emulsification, and the behavior of foams and aerosols. Understanding these principles is essential for students and professionals working with fluids and their interfaces.