Find the capillary rise of water in a tube 0.03 cm in diameter. The surface tension of water is 0.0735 N/m.

Capillary Rise of Water in a Tube

Problem Statement

Find the capillary rise of water in a tube 0.03 cm in diameter. The surface tension of water is 0.0735 N/m. (SI Units)

Given Data

Tube Diameter, $d = 0.03 \, \text{cm}$
Surface Tension, $\sigma = 0.0735 \, \text{N/m}$
Fluid: Water in contact with a glass tube
Density of water, $\rho = 1000 \, \text{kg/m}^3$
Angle of contact for water-glass, $\theta = 0^\circ$
Acceleration due to gravity, $g = 9.81 \, \text{m/s}^2$

Solution

1. Convert Units to SI

The tube diameter must be converted from centimeters to meters.

$$ d = 0.03 \, \text{cm} $$ $$ d = 0.03 \times 10^{-2} \, \text{m} $$ $$ d = 0.0003 \, \text{m} $$

2. Apply the Capillary Rise Formula

The formula for capillary rise $h$ is given below. For water in a clean glass tube, the contact angle $\theta$ is assumed to be $0^\circ$, so $\cos\theta = 1$.

$$ h = \frac{4\sigma \cos\theta}{\rho g d} $$

3. Substitute Values and Calculate

Now we substitute the known values into the formula.

$$ h = \frac{4 \times 0.0735 \, \text{N/m} \times \cos(0^\circ)}{1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.0003 \, \text{m}} $$ $$ h = \frac{0.294}{2.943} \, \text{m} $$ $$ h \approx 0.0999 \, \text{m} $$

4. Convert Height to a More Convenient Unit

The calculated height can be expressed in centimeters.

$$ h = 0.0999 \, \text{m} \times 100 \, \frac{\text{cm}}{\text{m}} $$ $$ h \approx 10.0 \, \text{cm} $$

Final Result:

The capillary rise of water in the tube is approximately $ h = 10.0 \, \text{cm} $.

Explanation of Capillary Action

Capillary Action is the phenomenon where a liquid spontaneously rises or falls in a narrow space such as a thin tube. This effect is driven by the balance between adhesion (the attraction of liquid molecules to the surface of the tube) and cohesion (the attraction of liquid molecules to each other). When adhesion is stronger than cohesion, the liquid wets the surface and is pulled upwards by surface tension, resulting in a capillary rise.

Physical Meaning

The result shows that for a very narrow tube (0.03 cm diameter), water will rise to a significant height (about 10 cm). This illustrates the strong effect of capillary action in small-diameter tubes.

This principle is fundamental to many natural and engineering processes:

In nature: It's how water is transported from the roots to the leaves of plants.
In technology: It is used in wicks for lamps and candles, the absorption of ink in paper, and in certain medical diagnostic tests.
In engineering: The effect must be accounted for in fluid-measuring devices like manometers to ensure accuracy. A smaller tube diameter leads to a larger capillary rise, which could introduce significant measurement error if not corrected.