Calculate the capillary effect in millimetres in a glass tube of 4 mm diameter, when immersed in (i) water, and (ii) mercury. The temperature of the liquid is 20°C and the values of the surface tension of water and mercury at 20°C in contact with air are 0.073575 N/m and 0.51 N/m respectively. The angle of contact for water is zero and that for mercury is 130°. Take density of water at 20°C as equal to 998 kg/m³.

Capillary Effect Calculation

Problem Statement

Given Data

Tube Diameter, $d = 4 \, \text{mm} = 4 \times 10^{-3} \, \text{m}$
Surface Tension of Water, $\sigma_w = 0.073575 \, \text{N/m}$
Surface Tension of Mercury, $\sigma_m = 0.51 \, \text{N/m}$
Density of Water at 20°C, $\rho_w = 998 \, \text{kg/m}^3$
Specific Gravity of Mercury, $S.G._m = 13.6$
Contact Angle for Water, $\theta_w = 0^\circ$
Contact Angle for Mercury, $\theta_m = 130^\circ$

Solution

The height of capillary rise or fall ($h$) is given by Jurin's Law:

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

Where $\rho$ is the density of the liquid, $g$ is the acceleration due to gravity ($9.81 \, \text{m/s}^2$), and other variables are as defined above.

(i) Capillary Rise for Water

For water, with $\theta_w = 0^\circ$, we have $\cos(0^\circ) = 1$.

$$ h_w = \frac{4 \times 0.073575 \, \text{N/m} \times \cos(0^\circ)}{998 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 4 \times 10^{-3} \, \text{m}} $$ $$ h_w = \frac{0.2943}{39161.52} \approx 0.007515 \, \text{m} $$

(ii) Capillary Depression for Mercury

First, find the density of mercury: $\rho_m = S.G._m \times 1000 = 13.6 \times 1000 = 13600 \, \text{kg/m}^3$. The contact angle $\theta_m = 130^\circ$.

$$ h_m = \frac{4 \times 0.51 \, \text{N/m} \times \cos(130^\circ)}{13600 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 4 \times 10^{-3} \, \text{m}} $$ $$ h_m = \frac{2.04 \times (-0.6428)}{533664} \approx \frac{-1.311}{533664} \approx -0.002457 \, \text{m} $$

The negative sign indicates a capillary depression.

Final Results:

Capillary Rise in Water: $ h_w \approx 7.51 \, \text{mm} $

Capillary Depression in Mercury: $ h_m \approx -2.46 \, \text{mm} $

Explanation of Capillarity

Capillary action is the tendency of a liquid to rise or fall in a narrow tube. This phenomenon arises from the balance between cohesive forces (attraction between liquid molecules) and adhesive forces (attraction between liquid molecules and the tube material).

Cohesive Forces: Keep the liquid molecules together.
Adhesive Forces: Make the liquid molecules stick to the solid surface.

The contact angle ($\theta$) at the liquid-solid interface determines the outcome. It's a direct measure of whether cohesive or adhesive forces are dominant.

Physical Meaning of Results

(i) Water: Water has strong adhesive forces with glass ($\theta = 0^\circ$), meaning it "wets" the surface. This adhesion pulls the liquid up the sides of the tube, causing the liquid level inside to be higher than outside—a capillary rise.

(ii) Mercury: Mercury has extremely strong cohesive forces, which are much greater than its adhesive forces with glass. The mercury molecules pull strongly on each other, forming a convex meniscus and pulling away from the glass walls. This results in the liquid level inside the tube being lower than outside—a capillary depression. The obtuse contact angle ($\theta = 130^\circ$) yields a negative height, confirming this depression.