Calculating the Radius of Liquid Droplets from a Capillary Tube Using Surface Tension and Density

An interesting question in fluid dynamics is how to determine the radius of a liquid drop that falls from a capillary tube. This can be calculated using a combination of surface tension and the density of the liquid. In this article, we will explore the method and provide a detailed solution to a specific problem involving a liquid with given surface tension and density.

Introduction to the Stalagmometric Method

The stalagmometric method is often used in the laboratory to determine the surface tension of liquids. It involves observing the behavior of a liquid droplet as it hangs from a capillary tube. When the droplet is about to fall, it is in equilibrium, with the weight of the droplet balanced by the surface tension force. This balance allows us to derive the formula for calculating the radius of the droplet.

Mechanics Involved

The key components in this calculation are the surface tension (gamma;), the density (rho;) of the liquid, the acceleration due to gravity (g), and the radius of the capillary tube (R). The formula for the radius of the droplet is given by:

r sqrt{frac{3 gamma;}{rho; g}}

This formula is derived from the equilibrium condition where the weight of the droplet is equal to the surface tension force.

Gathering the Known Values

In our specific case, we have the following known values:

Diameter of the capillary tube: 1 mm → Radius of the tube R 0.5 mm 0.0005 m Surface tension gamma; 65 times 10-3 N/m Density rho; 1.3 g/cm3 1300 kg/m3 Acceleration due to gravity g 9.81 m/s2

Performing the Calculation

The first step is to substitute the known values into the formula:

r sqrt{frac{3 times 65 times 10^(-3)}{1300 times 9.81}}

Calculating the Denominator

First, calculate the denominator:

1300 times 9.81 12753

Substituting and Solving

Now substitute this back into the equation:

r sqrt{frac{3 times 65 times 10^(-3)}{12753}}

Next, calculate the numerator:

3 times 65 times 10^(-3) 0.195

The final equation becomes:

r sqrt{frac{0.195}{12753}} approx sqrt{1.528 times 10^(-5)} approx 0.00391 m

Converting to Millimeters

To convert the radius from meters to millimeters:

r approx 0.00391 m times 1000 approx 3.91 mm

Conclusion

The radius of the drop of liquid falling from the capillary tube is approximately 3.91 mm.

Summary of Key Concepts

Stalagmometric Method: A technique for measuring the surface tension of liquids by analyzing the behavior of droplets from a capillary tube. Surface Tension: The energy per unit area needed to increase the surface area of a liquid. Density: The mass per unit volume of a substance. Radius of Droplet: The calculation involves understanding the equilibrium of forces acting on a liquid droplet.

By understanding these concepts, you can accurately calculate the radius of a droplet from a capillary tube using the provided surface tension and density values.