Notes : Capillary Rise: Derivation, Formula, Factors, Applications ,Numericals | Class 11 Physics Notes (NCERT, JEE, NEET)

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Learn Capillary Rise with complete derivation, formula, factors affecting rise, capillary depression, numericals, MCQs and NCERT Class 11 Physics notes chapter 9 Mechanical Properties of Fluids - Physicskund 

Capillary Rise

One consequence of the pressure difference across a curved liquid-air interface is the phenomenon of capillary rise. When a narrow tube is dipped vertically into water, the water rises inside the tube against gravity. This phenomenon is called capillary rise.

Definition of Capillary Rise

Capillary rise is the phenomenon in which a liquid rises in a narrow tube due to the pressure difference created by surface tension at the curved liquid surface.

Capillary Tube

A capillary tube is a very narrow tube of small radius.

  • Water rises inside the tube.
  • The liquid surface becomes curved.
  • A pressure difference develops across the curved surface.
  • The liquid rises until equilibrium is established.

Meniscus

The curved surface of a liquid in contact with a solid surface is called a meniscus.

Types of Meniscus

1. Concave Meniscus

  • Formed by water in a glass tube.
  • Angle of contact is less than 90°.
  • Produces capillary rise.

2. Convex Meniscus

  • Formed by mercury in a glass tube.
  • Angle of contact is greater than 90°.
  • Produces capillary depression.

Why Does Water Rise in a Capillary Tube?

  • Adhesive force between water and glass is greater than the cohesive force between water molecules.
  • Water wets the glass surface.
  • A concave meniscus is formed.
  • Pressure below the meniscus becomes less than atmospheric pressure.
  • Atmospheric pressure pushes the liquid upward.
  • Water rises until equilibrium is reached.

Geometry of the Meniscus

Consider a capillary tube of radius a.

Let:

  • Radius of curvature of meniscus = r
  • Angle of contact = θ

From geometry,

$\cos\theta=\frac{a}{r}$

Therefore,

$r=\frac{a}{\cos\theta}$

or

$r=a\sec\theta$

Pressure Difference Across the Meniscus

For a curved liquid surface,

$(P_i-P_0)=\frac{2S}{r}$

where:

  • $P_i$ = Atmospheric pressure
  • $P_0$ = Pressure just below the meniscus
  • $S$ = Surface tension
  • $r$ = Radius of curvature of meniscus

Substituting $r=a\sec\theta$,

Equation (1)

$(P_i-P_0)=\frac{2S}{a\sec\theta}$

$(P_i-P_0)=\frac{2S\cos\theta}{a}$

Pressure Balance

Consider points A and B at the same horizontal level.

Since pressure at the same horizontal level in a liquid is equal,

$P_A=P_B$

Pressure at point B is:

$P_0+h\rho g$

Pressure at point A is:

$P_i$

Equation (2)

$P_0+h\rho g=P_i=P_A$

Derivation of Capillary Rise Formula

From Equation (2),

$h\rho g=P_i-P_0$

Using Equation (1),

$P_i-P_0=\frac{2S\cos\theta}{a}$

Equation (3)

$h\rho g=(P_i-P_0)$

$h\rho g=\frac{2S\cos\theta}{a}$

Formula for Capillary Rise

Dividing by $\rho g$,

Equation (4)

$\boxed{h=\frac{2S\cos\theta}{\rho ga}}$

where:

  • $h$ = Height of capillary rise
  • $S$ = Surface tension
  • $\theta$ = Angle of contact
  • $\rho$ = Density of liquid
  • $g$ = Acceleration due to gravity
  • $a$ = Radius of capillary tube

Numerical Example

Given:

  • $a=0.05\,cm=5\times10^{-4}\,m$
  • $S=0.073\,N\,m^{-1}$
  • $\rho=10^3\,kg\,m^{-3}$
  • $g=9.8\,m\,s^{-2}$
  • $\cos\theta\approx1$

Using Equation (4),

$h=\frac{2S}{\rho ga}$

$h=\frac{2\times0.073}{(10^3)(9.8)(5\times10^{-4})}$

$h=\frac{0.146}{4.9}$

$h=0.0298\,m$

$\boxed{h=2.98\times10^{-2}\,m}$

$\boxed{h=2.98\,cm}$

Capillary Depression

If the liquid meniscus is convex, as in mercury:

  • Angle of contact is greater than 90°.
  • $\cos\theta$ becomes negative.

From Equation (4),

$h=\frac{2S\cos\theta}{\rho ga}$

Since $\cos\theta<0$, therefore $h<0$.

The liquid level falls below the outside level. This phenomenon is called capillary depression.

Factors Affecting Capillary Rise

1. Radius of Tube

$h\propto\frac{1}{a}$

Smaller radius → Greater capillary rise

2. Surface Tension

$h\propto S$

Greater surface tension → Greater capillary rise

3. Density of Liquid

$h\propto\frac{1}{\rho}$

Lower density → Greater capillary rise

4. Angle of Contact

$h\propto\cos\theta$

  • $0^\circ<\theta<90^\circ$ → Rise
  • $\theta=90^\circ$ → No rise
  • $\theta>90^\circ$ → Depression

Applications of Capillarity

  • Rise of kerosene through lamp wicks.
  • Absorption of ink by blotting paper.
  • Movement of water through soil.
  • Transport of water in plants.
  • Absorption of water by towels and sponges.
  • Working of fountain pens.

Key Differences Between Water and Mercury

Property Water Mercury
Meniscus Concave Convex
Angle of Contact Less than 90° Greater than 90°
cosθ Positive Negative
Effect Rise Depression

Frequently Asked Questions (FAQs)

1. What is capillary rise?

Capillary rise is the upward movement of a liquid in a narrow tube due to surface tension.

2. What is capillarity?

Capillarity is the phenomenon of rise or depression of a liquid in a narrow tube.

3. Why does water rise in a capillary tube?

Water rises because the adhesive force between water and glass is greater than the cohesive force between water molecules.

4. Why does mercury show capillary depression?

Mercury forms a convex meniscus and has an angle of contact greater than 90°, causing the liquid level to fall.

5. What is the formula for capillary rise?

$h=\frac{2S\cos\theta}{\rho ga}$

6. What is a meniscus?

The curved surface of a liquid near the walls of a container is called a meniscus.

7. What is the SI unit of surface tension?

The SI unit of surface tension is N m⁻¹.

MCQs

1. Capillary rise occurs due to:

A. Viscosity
B. Surface tension
C. Gravity
D. Buoyancy

Answer: B. Surface tension

2. Water in a glass capillary tube forms:

A. Convex meniscus
B. Plane surface
C. Concave meniscus
D. Irregular surface

Answer: C. Concave meniscus

3. The SI unit of surface tension is:

A. N
B. J
C. N m⁻¹
D. Pa

Answer: C. N m⁻¹

4. Capillary rise is inversely proportional to:

A. Surface tension
B. Density
C. Radius of tube
D. Gravity

Answer: C. Radius of tube

5. Mercury in a capillary tube shows:

A. Rise
B. Depression
C. No change
D. Overflow

Answer: B. Depression

True or False

1. Water forms a concave meniscus in a glass tube. True

2. Mercury rises in a glass capillary tube. False

3. Capillary rise depends on surface tension. True

4. Smaller capillary tubes produce greater capillary rise. True

5. Mercury forms a convex meniscus. True

6. Capillary rise is directly proportional to density. False

7. Water wets glass. True

8. Surface tension is responsible for capillarity. True

Fill in the Blanks

1. The curved liquid surface is called a __________.

Answer: Meniscus

2. Water forms a __________ meniscus in glass.

Answer: Concave

3. Mercury forms a __________ meniscus in glass.

Answer: Convex

4. The SI unit of surface tension is __________.

Answer: N m⁻¹

5. Capillary rise is caused by __________.

Answer: Surface tension

6. Water rises in a __________ tube.

Answer: Capillary

7. The height of capillary rise is represented by __________.

Answer: h

Very Short Answer Questions

1. What is capillary rise?

The upward movement of a liquid in a narrow tube due to surface tension.

2. What is capillary depression?

The lowering of a liquid level in a capillary tube.

3. What is a capillary tube?

A tube having a very small internal radius.

4. Which liquid shows capillary depression?

Mercury.

5. What is the SI unit of surface tension?

N m⁻¹.

6. What type of meniscus does water form in glass?

Concave meniscus.

7. What type of meniscus does mercury form in glass?

Convex meniscus.

Short Answer Questions

1. Define capillarity.

Capillarity is the phenomenon of rise or depression of a liquid in a narrow tube due to surface tension.

2. Why does water rise in a capillary tube?

Water rises because the adhesive force between water and glass is greater than the cohesive force between water molecules. This forms a concave meniscus and creates a pressure difference that causes the liquid to rise.

3. Why does mercury show capillary depression?

Mercury forms a convex meniscus and has an angle of contact greater than 90°. Therefore, the liquid level falls below the outside level.

4. State two applications of capillarity.

1. Rise of kerosene in lamp wicks.
2. Transport of water in plants.

5. How does tube radius affect capillary rise?

Capillary rise is inversely proportional to the radius of the tube.

$h\propto\frac{1}{a}$

Long Answer Questions

1. Derive the formula for capillary rise.

For a capillary tube of radius a and angle of contact θ,

$r=a\sec\theta$

The pressure difference across the curved liquid surface is

$(P_i-P_0)=\frac{2S}{r}$

Substituting the value of r,

$(P_i-P_0)=\frac{2S\cos\theta}{a} \qquad ...(1)$

At the same horizontal level,

$P_0+h\rho g=P_i \qquad ...(2)$

Therefore,

$h\rho g=P_i-P_0$

Using Equation (1),

$h\rho g=\frac{2S\cos\theta}{a}$

Hence,

$\boxed{h=\frac{2S\cos\theta}{\rho ga}}$

This is the formula for capillary rise.

2. Discuss the factors affecting capillary rise.

From the formula

$h=\frac{2S\cos\theta}{\rho ga}$

  • Capillary rise is directly proportional to surface tension.
  • Capillary rise is inversely proportional to tube radius.
  • Capillary rise is inversely proportional to density.
  • Capillary rise depends upon the angle of contact.

These factors determine the height to which a liquid rises in a capillary tube.

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