The formula for determining the radius of a capillary tube, often derived from considerations of surface tension and hydrostatic pressure in capillary action, is given by r = 2σ / hρg.
Understanding the Capillary Tube Radius Formula
This formula is derived from the principles governing capillary rise or fall, where the surface tension of the liquid, the density of the liquid, and the gravitational force all play a role. It allows you to calculate the radius of a narrow tube based on how high (or low) a liquid rises or falls within it.
The formula incorporates the following key physical properties:
- r: The radius of the capillary tube you want to determine.
- σ (sigma): The surface tension of the liquid. This is a measure of the energy required to increase the surface area of a liquid.
- h: The height the liquid column rises or falls within the capillary tube. This is the measured difference in liquid level inside the tube compared to the outside.
- ρ (rho): The density of the liquid. This indicates the mass per unit volume of the liquid.
- g: The acceleration due to gravity. This is the standard value (approximately 9.8 m/s²) representing the force of gravity.
In essence, the formula balances the upward force due to surface tension (which causes the liquid to rise) with the downward force due to gravity acting on the liquid column (which limits the rise).
Practical Example
Based on the provided reference, an example calculation using this formula is shown:
r = 2σ / hρgUsing specific values:
r = 2 × 0.073 / 0.35 × 1000 × 9.8This calculation yields a radius of:
r = 4.3 × 10⁻⁵ mThis example demonstrates how the measured height (h = 0.35 m), along with known values for the liquid's surface tension (σ = 0.073), liquid's density (ρ = 1000 - likely water in kg/m³), and acceleration due to gravity (g = 9.8 m/s²), can be used to calculate the radius of the capillary tube (r).
