Calculate the capillary rise in a glass tube of 2.5 mm diameter when immersed vertically in (a) water and (b) mercury.

Capillary Rise and Depression Calculation

Problem Statement

Calculate the capillary rise in a glass tube of 2.5 mm diameter when immersed vertically in (a) water and (b) mercury. Take surface tensions $\sigma = 0.0725$ N/m for water and $\sigma = 0.52$ N/m for mercury in contact with air. The specific gravity for mercury is 13.6 and the angle of contact is 130°.

Given Data

Tube Diameter, $d = 2.5 \, \text{mm} = 2.5 \times 10^{-3} \, \text{m}$
Surface Tension of Water, $\sigma_w = 0.0725 \, \text{N/m}$
Surface Tension of Mercury, $\sigma_m = 0.52 \, \text{N/m}$
Specific Gravity of Mercury, $S.G._m = 13.6$
Contact Angle for Water, $\theta_w \approx 0^\circ$ (assumed for clean glass)
Contact Angle for Mercury, $\theta_m = 130^\circ$

Solution

The height of capillary rise or fall ($h$) is given by the formula:

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

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.

(a) Capillary Rise for Water

For water, the density $\rho_w = 1000 \, \text{kg/m}^3$ and the contact angle $\theta_w$ is assumed to be $0^\circ$, so $\cos(0^\circ) = 1$.

$$ h_w = \frac{4 \times 0.0725 \, \text{N/m} \times \cos(0^\circ)}{1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 2.5 \times 10^{-3} \, \text{m}} $$ $$ h_w = \frac{0.29}{24525} \approx 0.0118 \, \text{m} $$

(b) Capillary Depression for Mercury

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

$$ h_m = \frac{4 \times 0.52 \, \text{N/m} \times \cos(130^\circ)}{13600 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 2.5 \times 10^{-3} \, \text{m}} $$ $$ h_m = \frac{2.08 \times (-0.6428)}{333540} \approx \frac{-1.337}{333540} \approx -0.0040 \, \text{m} $$

The negative sign indicates a capillary depression, meaning the mercury level inside the tube is lower than the level outside.

Final Results:

Capillary Rise in Water: $ h_w \approx 0.0118 \, \text{m} = 1.18 \, \text{cm} $

Capillary Depression in Mercury: $ h_m \approx -0.004 \, \text{m} = -0.4 \, \text{cm} $

Explanation of Capillarity

Capillary action (or capillarity) is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity. The effect is a result of the interplay between two types of forces:

Cohesive Forces: The intermolecular attraction between like-molecules (e.g., between one water molecule and another).
Adhesive Forces: The intermolecular attraction between unlike-molecules (e.g., between a water molecule and a glass molecule).

The contact angle ($\theta$) is the angle where the liquid surface meets the solid surface. It is a measure of the relative strength of these forces.

Physical Meaning of Results

(a) Water: For water in a glass tube, the adhesive forces (water-to-glass) are stronger than the cohesive forces (water-to-water). This causes the water to "climb" the glass walls, resulting in a concave meniscus and a capillary rise. The contact angle is acute ($\theta < 90^\circ$).

(b) Mercury: For mercury in a glass tube, the cohesive forces (mercury-to-mercury) are much stronger than the adhesive forces (mercury-to-glass). The liquid pulls itself inward and away from the glass walls. This results in a convex meniscus and a capillary depression (or fall). The contact angle is obtuse ($\theta > 90^\circ$), which makes $\cos\theta$ negative, leading to a negative value for $h$.