Question

Three liquids of densities ${p_1}$ ​, ${p_2}$ ​ and ${p_3}$ ​ (with ${p_1} > {p_2} > {p_3}$ ​) having the same value of surface tension $T$ , rise to the same height in three identical capillaries The angles of contact ${\theta _1}$ , ${\theta _2}$ and ${\theta _3}$ ​ obey:-
A) $\pi > {\theta _1} > {\theta _2} > {\theta _3} > \dfrac{\pi }{2}$
B) $\dfrac{\pi }{2} > {\theta _1} > {\theta _2} > {\theta _3} \geqslant 0$
C) $0 \leqslant {\theta _1} < {\theta _2} < {\theta _3} < \dfrac{\pi }{2}$
D) $\dfrac{\pi }{2} \leqslant {\theta _1} < {\theta _2} < {\theta _3} < \pi$

Answer 1

Hint The angle of contact is proportional to the product of the rise in the height of the liquid in a capillary tube and the density of the liquids. For increasing densities, the angles of contact will have an inverse order.

Formula used:
$\Rightarrow h = \dfrac{{2T\cos \theta }}{{r\rho g}}$ where $h$ is the rise in height of the liquid inside the capillary, $T$ is the tension in the liquid and $\rho$ is the density of the liquid, $\theta$ is the angle of contact.

Complete step by step answer
We’ve been given that the three liquids of densities ${p_1}$ ​, ${p_2}$ ​ and ${p_3}$ ​ (with ${p_1} > {p_2},{p_3}$ ​) having the same value of surface tension $T$ rise to a rise to the same height in three identical capillaries.
We know that the rise in height of a liquid inside a capillary is given as:
$\Rightarrow h = \dfrac{{2T\cos \theta }}{{r\rho g}}$
On rearranging, we can write:
$\Rightarrow \cos \theta = \dfrac{{hr\rho g}}{{2T}}$
In the above equation, we can see that
$\Rightarrow cos\theta \propto \dfrac{1}{\rho }$
So if ${p_1} > {p_2} > {p_3}$ , we will have
$\Rightarrow \cos {\theta _1} > \cos {\theta _2} > \cos {\theta _3}$
Or equivalently,
$\Rightarrow {\theta _1} < {\theta _2} < {\theta _3}$
Since the liquid is rising in all the three capillaries thus the angle of contact will be between $0^\circ$ and $90^\circ$ . So our answer is
$\Rightarrow 0 \leqslant {\theta _1} < {\theta _2} < {\theta _3} < \dfrac{\pi }{2}$ .

Note
The trick in this question is to realize that since all the three liquids are rising in the capillary, the angle of contact can only lie between $0^\circ$ and $90^\circ$ which also eliminates option (A) and (D). Also, we must notice that we’ve been asked to find the order of the angle of the contact $\theta$ and not $\cos \theta$ since they have reverse orders between $0^\circ$ and $90^\circ$ .