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 $ .