Animated Solution for Physics - Properties of Solids and Liquids: A capillary tube (A) is dipped in water. Another identical tube (B) is dipped in a soap-water solution. Which of the following shows the relative nature of the liquid columns in the two tubes?

You dip the first tube, let's call it Tube A, into a beaker of pure water. Immediately, you observe the water creeping up the tube against gravity. This is capillary action at work!

Now, you take the second tube, Tube B, and dip it into a beaker containing a soap-water solution. Again, the liquid rises. But the critical question is: will it rise to the same height as the pure water? To answer this, we need to dive into the physics of capillary rise.

Jurin's Law The height `$h$` to which a liquid rises in a capillary tube is governed by Jurin's Law, which is expressed as: Here, `$T$` represents the surface tension of the liquid, `$\theta$` is the angle of contact between the liquid and the glass, `$\rho$` is the density of the liquid, `$r$` is the radius of the capillary tube, and `$g$` is the acceleration due to gravity.

Let's compare the parameters for our two tubes. Since the tubes are identical, the radius `$r$` is the same for both. The density `$\rho$` of a soap solution is practically the same as that of pure water. Furthermore, for both pure water and soap solution against clean glass, the angle of contact `$\theta$` is very close to `$0^{\circ }$`, meaning `$\cos \theta \approx 1$`.

With `$r$`, `$\rho$`, and `$\theta$` being constant, our equation simplifies to a direct proportionality: This tells us that the capillary rise is directly proportional to the surface tension of the liquid.

So, what happens when we add soap to water? Soap molecules are surfactants. They accumulate at the surface and disrupt the strong cohesive hydrogen bonds between the water molecules. This disruption significantly lowers the surface tension of the liquid. Therefore, the surface tension of the soap solution is less than that of pure water (`$T_{\text{soap}}<T_{\text{water}}$`).

Consequently, the height of the liquid column in Tube A (pure water) will be strictly greater than the height in Tube B (soap solution).

Mathematically, since `$T_A>T_B$`, it directly follows that: Looking at the given visual options, Option (c) perfectly illustrates this relative nature, showing a higher liquid column in Tube A compared to Tube B.