Question

Read the assertion and reason carefully and mark the correct option given below. (a) If both assertion and reason are true and the reason is the correct explanation of the assertion. (b) If both assertion and reason are true but reason is not the correct explanation of the assertion. (c) If assertion is true but reason is false. (d) If the assertion and reason both are false. Assertion: The water rises higher in a capillary tube of small diametre than in the capillary tube of large diameter. Reason: Height through which liquid rises in a capillary tube is inversely proportional to the diameter of the capillary tube. (A) a (B) b (C) c (D) d

Short answer

\( \boxed{A} \)

Step-by-step solution

1. Check the correctness of the assertion

To verify the assertion, let's remember the behavior of liquids in capillary tubes. Capillary action is the result of surface tension and adhesion forces between the liquid and the tube walls. The liquid rises or falls inside the tube until the weight of the liquid column is balanced by the adhesive forces acting on the surface. In narrow tubes (small diameter), the water can rise higher than in wide tubes (large diameter) due to these adhesive forces. So, the assertion is true.

2. Check the correctness of the reason

The height to which the liquid rises in a capillary tube can be described using Jurin's law that states: \[ h = \dfrac{2\gamma\cos\theta}{\rho gr} \] where h is the height of the liquid column, γ (gamma) is the surface tension of the liquid, θ (theta) is the contact angle between the liquid and the tube walls, ρ (rho) is the density of the liquid, g is the acceleration due to gravity, and r is the radius of the capillary tube. This shows that the height of the liquid rise in the capillary tube is indeed inversely proportional to the diameter (and radius) of the tube. Thus, the reason is also true.

3. Determine if the reason is the correct explanation of the assertion

Since the reason successfully explains the mathematical relationship between the height of liquid rise and the diameter of the capillary tube, it adequately provides the correct explanation for the assertion. Therefore, the reason supports and correctly explains the assertion. Now, we can choose the correct option based on our analysis of the given statements: (A) a Since both the assertion and reason are true, and the reason is the correct explanation of the assertion, the correct answer is: (A) a.

Key concepts

Capillary Action

Capillary action is the beautiful phenomenon where a liquid moves up or down a narrow tube, defying gravity. It occurs due to the combined effects of surface tension and adhesive forces between the liquid and the tube's walls.
You might have observed how a towel edge absorbs water or how plants draw water from the soil. These are examples of capillary action in action! The liquid moves as adhesion pulls the liquid to the walls of the tube, while cohesion (part of surface tension) keeps the liquid together.
This is central to understanding how liquids travel through tiny spaces. It doesn’t require any external pumping force, relying solely on molecular interactions to lift the liquid.

Surface Tension

Surface tension is like a film that forms on the surface of a liquid, making it resistant to external forces. It is caused by cohesion among the liquid's molecules. These molecules are attracted to each other, which creates tension on the surface.
Imagine a drop of water forming on a leaf or insects walking on water. This is due to surface tension preventing the liquid from spreading out entirely.
For capillary action to occur, surface tension plays a crucial role. It helps the liquid rise in a tube by minimizing the surface area, which aids in raising the liquid column in narrow tubes.

Adhesion Forces

Adhesion forces refer to the attractive forces between different substances, like a liquid and a solid surface. In the context of capillarity, these forces cause the liquid to "stick" to the walls of the capillary tube.
When you notice water climbing up a glass tube, adhesion is pulling the water molecules towards the glass surface.
These forces are essential for capillary action. Without adhesion, the liquid would not be able to climb against gravity, as it wouldn't stick to the tube's surface.

Liquid Rise in Capillaries

The rise of liquid in capillaries is explained by Jurin's Law. According to this law, the height to which a liquid rises within a thin tube is inversely proportional to the tube's radius. Jurin's formula is given by:\[h = \dfrac{2\gamma\cos\theta}{\rho gr}\]

\( h \) (height) — how high the liquid climbs.

\( \gamma \) (surface tension) — the cohesive force holding the liquid molecules together.

\( \theta \) (contact angle) — the angle at which liquid meets the tube surface.

\( \rho \) (density) — how dense the liquid is, influencing its weight.

\( g \) (gravity) — pulls the liquid downwards.

This expression explains why liquids rise higher in tubes with smaller diameters. A narrower tube means a higher adhesive effect relative to gravitational pull on the liquid's weight. This highlights the interdependence of molecular forces and geometrical constraints in fluid dynamics.