Capillary rise of mercury in a small diameter tube is proportional to (where, d = diameter of the tube, σ = surface tension of mercury)

Answer: Option C

The capillary rise of mercury in a small diameter tube is proportional to 1/d, where d is the diameter of the tube, and σ is the surface tension of mercury. Capillary rise occurs due to the balance between cohesive forces within the liquid and adhesive forces between the liquid and the tube wall. In a small diameter tube, the capillary rise is inversely proportional to the diameter of the tube.

Mathematically, the capillary rise h is given by the capillary rise equation:

h = (2σ cosθ) / (ρgr)

Where:

σ is the surface tension of the liquid (mercury).
θ is the contact angle between the liquid and the tube wall (usually close to 0° for mercury in glass).
ρ is the density of the liquid (mercury).
g is the acceleration due to gravity.
r is the radius of the tube (half of the diameter).

Since the radius r is directly proportional to 1/d, and the capillary rise is proportional to the radius r, the capillary rise is inversely proportional to the diameter d, which is represented by option B.