2016 Phase II Mechanical Properties of FluidsCapillary rise and angle of contact Medium Question 28 of 34 in Mechanical Properties of Fluids

Three liquids of densities $\rho_1$, $\rho_2$ and $\rho_3$ (with $\rho_1 > \rho_2 > \rho_3$), 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 $\frac{\pi}{2} < \theta_1 < \theta_2 < \theta_3 < \pi$
B $\pi > \theta_1 > \theta_2 > \theta_3 > \frac{\pi}{2}$
C $\frac{\pi}{2} > \theta_1 > \theta_2 > \theta_3 \ge 0$
D $0 \le \theta_1 < \theta_2 < \theta_3 < \frac{\pi}{2}$

The answer is D.

Solution

Equal rise needs $\cos\theta \propto \rho$.

$h = \frac{2T\cos\theta}{\rho gr}$; as r, h and T are the same, $\frac{\cos\theta}{\rho}$ = constant
$\frac{\cos\theta_1}{\rho_1} = \frac{\cos\theta_2}{\rho_2} = \frac{\cos\theta_3}{\rho_3}$
As $\rho_1 > \rho_2 > \rho_3$: $\cos\theta_1 > \cos\theta_2 > \cos\theta_3 \Rightarrow \theta_1 < \theta_2 < \theta_3$
As the liquid rises, θ must be acute, so $0 \le \theta_1 < \theta_2 < \theta_3 < \frac{\pi}{2}$

Concept capillary-rise-contact-angle

Mechanical Properties of Fluids in past papers

34 questions from this chapter have appeared across 14 exam years.