Problem Statement
The surface tension of water in contact with air at 20°C is given as 0.0716 N/m. The pressure inside a droplet of water is to be 0.0147 N/cm² greater than the outside pressure. Calculate the diameter of the droplet of water.
Given Data
- Surface Tension, \(\sigma = 0.0716 \, \text{N/m}\)
- Excess Pressure, \(p = 0.0147 \, \text{N/cm}^2\)
Solution
1. Convert Pressure to SI Units (N/m²)
To ensure our units are consistent, we convert the pressure from N/cm² to N/m².
2. Calculate the Droplet Diameter (\(d\))
The relationship between excess pressure, surface tension, and diameter for a spherical droplet is given by the Young-Laplace equation.
3. Convert Diameter to Millimeters
It is often more convenient to express such a small diameter in millimeters.
The diameter of the droplet of water is approximately \( d \approx 1.95 \, \text{mm} \).
Explanation of Surface Tension
Surface Tension (\(\sigma\)) is a property of a liquid’s surface that causes it to behave like a stretched elastic membrane. It arises from the cohesive forces between liquid molecules. Molecules at the surface have fewer neighboring molecules compared to those in the bulk, resulting in a net inward force that pulls the surface molecules together.
This force tries to minimize the surface area, which is why free liquid droplets are spherical. To maintain this curved surface, the pressure inside the droplet must be greater than the pressure outside. This pressure difference is known as excess pressure.
Physical Meaning
The calculation demonstrates the inverse relationship between a droplet’s size and the excess pressure inside it. A smaller droplet has a more tightly curved surface, which requires a higher internal pressure to be maintained against the constant pull of surface tension.
This principle is crucial in many natural and engineering applications, such as the formation of rain, the function of atomizers and spray nozzles, and the stability of emulsions. For a very small droplet, the internal pressure can become surprisingly large.
