Question

Three liquids have same surface tension and have densities \(ρ_1,ρ_2\) and \(ρ_3\) \((ρ_1 > ρ_2 > ρ_3)\). In three identical capillaries rise of liquid is same. The corresponding angles of contact \(θ_1,θ_2\) and \(θ_3\) are related as

• A. \(θ_1 > θ_2 > θ_3\)
• B. \(θ_1 < θ_2 < θ_3\)
• C. \(θ_1 = θ_2 = θ_3\)
• D. \(θ_1 > θ_2 < θ_3\)

Hint:

Use the capillary rise formula h = 2T cos(theta) / (r rho g) with h, T and r equal.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Capillary rise is given by \(h = \dfrac{2T\cos\theta}{r\rho g}\).

Step 2: Apply the condition.
The surface tension \(T\), the capillary radius \(r\) and the rise \(h\) are the same for all three. So \(\dfrac{\cos\theta}{\rho}\) must be the same, which means \(\cos\theta \propto \rho\).

Step 3: Compare.
Given \(\rho_1 > \rho_2 > \rho_3\), we get \(\cos\theta_1 > \cos\theta_2 > \cos\theta_3\). On the range \(0\) to \(90^\circ\), the cosine is smaller for a larger angle. So \(\theta_1 < \theta_2 < \theta_3\).

Step 4: Check the options.
Option (A) is the reverse. Option (C) would need equal densities. Option (D) is not a consistent ordering.

Final Answer:
The relation is \(\theta_1 < \theta_2 < \theta_3\), option (B). \[ \boxed{\theta_1 < \theta_2 < \theta_3} \]