Questions: Surface Tension and Capillary Phenomena

Question 1

Two glass tubes are dipped vertically into water — one with radius 0.5 mm and one with radius 2 mm. Compared to the wider tube, the narrower tube will show:

• A. Lower capillary rise, because less water fits in the narrower tube
• B. The same capillary rise, because surface tension is a property of the liquid, not the tube
• C. Higher capillary rise, because h ∝ 1/r and smaller radius means greater rise
• D. Higher capillary rise only if the tube material has a smaller contact angle

Answer: (C)

The capillary rise formula h = 2σ cosθ / (ρgr) shows rise height is inversely proportional to tube radius r. The narrower tube (r = 0.5 mm) rises four times higher than the 2 mm tube. The physical reason: surface tension force acts around the tube perimeter (2πr) while the weight of liquid resisting it scales with the cross-sectional area (πr²). As r shrinks, perimeter shrinks linearly but area shrinks quadratically, so the surface tension force wins more and more — smaller tubes pull liquid higher.

Question 2

Mercury is poured into a narrow glass tube. What happens, and why?

• A. Mercury rises above the external level, just like water, because surface tension acts the same way in all liquids
• B. Mercury stays level with the external surface because its high density cancels the surface tension effect
• C. Mercury depresses below the external level, forming a convex meniscus, because its contact angle with glass exceeds 90°
• D. Mercury rises slightly then falls back to equilibrium after the gas dissolved in mercury escapes

Answer: (C)

Mercury is a non-wetting liquid on glass: its contact angle θ ≈ 140°, making cosθ negative. The capillary formula h = 2σ cosθ / (ρgr) gives a negative h — depression rather than rise. Physically, mercury's cohesive forces (liquid-to-liquid) greatly exceed its adhesive forces to glass (liquid-to-solid), so the liquid curves away from the wall, forming a convex meniscus. The Young-Laplace pressure jump pushes mercury down inside the tube rather than pulling it up.

Question 3

Surface tension is a bulk property of a liquid — it depends on the total volume of liquid present, so a larger body of liquid has higher surface tension than a small droplet.

• T. True
• F. False

Answer: False

Surface tension σ is an intensive property — it is a property of the interface itself, characterizing the energy per unit area (J/m²) required to create new surface. Its value depends on the molecular identity of the two phases in contact and temperature, not on the total volume of liquid. A large tank of water and a tiny droplet have the same σ ≈ 0.072 N/m at room temperature. What changes with size is the total surface energy (σ × area), not the surface tension itself.

Question 4

Capillary rise height is inversely proportional to tube radius — halving the radius doubles the rise height.

• T. True
• F. False

Answer: True

This follows directly from h = 2σ cosθ / (ρgr): h ∝ 1/r, so halving r doubles h. This relationship is why capillary action is so significant in fine-pored materials (soil, paper, plant xylem) and negligible in large pipes. It also explains why engineers designing microfluidic devices must account for capillary effects that would be completely irrelevant at larger scales — the physics is the same, but the relative importance shifts dramatically with scale.

Question 5

Explain why the contact angle θ determines whether a liquid rises or falls in a capillary tube, and what physical property of the system it encodes.

Model answer: The contact angle encodes the balance between adhesive forces (liquid-to-solid) and cohesive forces (liquid-to-liquid). When adhesion dominates (θ < 90°, e.g., water on glass), the liquid spreads along the wall and curves concave upward at the meniscus. The Young-Laplace pressure discontinuity across this curved interface creates lower pressure inside the liquid than in the gas above, pulling the column upward. When cohesion dominates (θ > 90°, e.g., mercury on glass), the liquid resists contact with the wall, curves convex downward, and the pressure inside the liquid exceeds atmospheric, pushing the column down. At θ = 90°, cosθ = 0 and there is no net capillary effect.

The contact angle is the observable signature of the surface energy balance at the three-phase contact line (solid-liquid-gas). The capillary rise formula makes this explicit: h = 2σ cosθ / (ρgr). For θ < 90°, cosθ > 0 and h > 0 (rise); for θ > 90°, cosθ < 0 and h < 0 (depression). The cos factor precisely quantifies how much of the surface tension force is directed vertically — at θ = 0° (perfect wetting), the full surface tension acts upward; at θ = 180° (perfect non-wetting), it acts fully downward.