Capillarity in a narrow glass tube dipped in water:\nThe water column rises with which type of free-surface curvature at the meniscus?

Introduction / Context:
Capillary action in narrow tubes depends on the balance of adhesive forces (liquid–solid) and cohesive forces (liquid–liquid), producing a meniscus that determines whether the liquid rises or depresses. Water in clean glass typically wets the surface, creating a concave meniscus and a capillary rise. This phenomenon is central in soil mechanics, porous media flow, and microfluidics.

Given Data / Assumptions:

• Clean glass tube of small internal diameter.
• Liquid is water at ordinary conditions.
• No contamination altering contact angle.

Concept / Approach:
When adhesion (water–glass) exceeds cohesion (water–water), the contact angle is less than 90 degrees, the meniscus curves upward at the walls (concave), and the vertical component of surface tension supports the column against gravity. The height of rise h satisfies h = (4 * σ * cos θ) / (ρ * g * d) for a circular tube, where σ is surface tension, θ is contact angle (θ < 90° for water on glass), ρ is density, g is gravity, and d is tube diameter.

Step-by-Step Solution:

Verification / Alternative check:
In contrast, mercury in glass has θ > 90°, forms a convex meniscus, and experiences capillary depression. Observing both cases side-by-side in a lab demonstrates the role of contact angle and surface tension direction.

Why Other Options Are Wrong:

• Convex: Corresponds to non-wetting liquids like mercury on glass.
• Planar / irregular: Do not describe the stable equilibrium shape in a narrow, clean tube for water.

Common Pitfalls:
Assuming all liquids rise in capillary tubes; some depress. Also, small contamination or surfactants can alter θ and the observed curvature.

Final Answer:
concave