Problem Statement
The surface tension of water in contact with air at 20°C is 0.0725 N/m. The pressure inside a droplet of water is to be 0.02 N/cm² greater than the outside pressure. Calculate the diameter of the droplet of water.
Given Data
- Surface Tension, \(\sigma = 0.0725 \, \text{N/m}\)
- Excess Pressure, \(p = 0.02 \, \text{N/cm}^2\)
Solution
1. Convert Pressure to SI Units
The surface tension is given in N/m, so we must convert the pressure to N/m² (Pascals) for consistency.
2. Apply the Pressure in a Droplet Formula
The excess pressure \(p\) inside a spherical liquid droplet is related to its surface tension \(\sigma\) and diameter \(d\) by the Young-Laplace equation for a sphere:
We can rearrange this formula to solve for the diameter \(d\):
3. Convert Diameter to Millimeters
For small dimensions, it is often more convenient to express the result in millimeters.
The diameter of the droplet of water is \( d = 1.45 \, \text{mm} \).
Explanation of Surface Tension
Surface Tension (\(\sigma\)) is a property of a liquid's surface that allows it to resist an external force. It is caused by the cohesive forces between liquid molecules. Molecules within the bulk of the liquid are pulled equally in all directions, but molecules at the surface experience a net inward pull. This creates a "skin" on the surface.
This "skin" tries to minimize its surface area, which is why free liquid droplets naturally form a spherical shape (a sphere has the smallest surface area for a given volume). To support this curved shape, the pressure inside the droplet must be higher than the pressure outside. This pressure difference is known as excess pressure or Laplace pressure.
Physical Meaning
The result shows an inverse relationship between the droplet's diameter and the excess pressure inside it. A smaller droplet requires a much higher internal pressure to maintain its highly curved surface against the force of surface tension.
This principle is fundamental to many natural and industrial phenomena, including:
- Capillary Action: The rise or fall of liquids in narrow tubes.
- Atomization: The process of forming fine sprays, where energy is used to create a vast amount of surface area in tiny droplets.
- Emulsions: The stability of mixtures like milk or mayonnaise, where tiny droplets of one liquid are suspended in another.
