## What is cslled capillarity?

*Capillarity is the phenomenon of rising or dipping of liquid surface in a narrow tube when it is dipped in the liquid.*

The surface of liquid in the capillary tube is called **meniscus.**

- If the liquid wets the tube ( i.e. adhesion is predominant ) the meniscus is concave upward and the liquid level rises in the tube.
- If the liquid does not wet the tube ( i.e. cohesion is predominant ) the meniscus is convex upward and the liquid level falls in the tube.

### Angle of contact

*Angle ( \theta ) made by the tangent to the liquid surface at the point of contact and the wall of the tube is called angle of contact.*

The value of angle of contact depends upon the following factors –

- Nature of the solid surface and the liquid in contact.
- Cleanliness of the surface in contact.
- Medium above the free surface of the liquid.
- Temperature of the liquid.

The liquid surface is usually curved when it is in contact with a solid surface. This is called ** meniscus**. The shape which the liquid surface takes depends upon the relative strengths of cohesive and adhesive forces.

- If \text {adhesive force} > \text {Cohesive force} – Liquid wets the solid surface. The angle of contact is acute and the liquid surface has a concave meniscus.
- If \text {adhesive force} < \text {Cohesive force} – Liquid does not wet the solid surface. The angle of contact is obtuse and liquid surface has a convex meniscus.
- If \text {adhesive force} = \text {Cohesive force} – Liquid surface is plane. The angle of contact is a right angle.

- The above phenomenon is known as capillarity.

### Capillary Tube

A narrow glass tube of small diameter which is open at both ends is called a **capillary tube. **

- When a capillary tube is dipped into a liquid like water, we find that the liquid rises in the tube above the general level of the outside liquid as shown in figure ( A ).
- But, if the same glass tube is dipped in a heavy liquid like mercury, the level of the liquid in the tube is lower than the general level of the outside liquid as shown in figure ( B ).

## Ascent formula of capillary rise

Consider about the figure (A) for the capillary rise of a liquid in a tube of radius ( r ) . Let, the rise of the liquid in the tube is ( h ) .

Let, ( \sigma ) is the surface tension of the liquid and ( \theta ) is the angle of contact between the liquid and the glass tube.

Then, weight of the column of liquid of height ( h ) in the tube is balanced by the vertical component of the surface tensile force.

- Vertical component of the
*surface tension force*is –

\sigma \times ( 2 \pi r ) \cos \theta

- Weight of the column of liquid in capillary tube is –

\pi r^2 h \rho g

Therefore, for *equilibrium* of the liquid column in tube –

\sigma \times ( 2 \pi r ) \cos \theta = \pi r^2 h \rho g

Or, \quad h = \left ( \frac {2 \sigma \cos \theta}{\rho g r} \right )

*This is called Ascent formula for rise of liquid in a capillary tube.*

Also, \quad h = \left ( \frac {4 \sigma \cos \theta}{\rho g d} \right )

Where ( d = 2r ) is the diameter of the tube.

- \theta = 25 \degree 32' \text {for water, and} 128 \degree 52' for mercury.

The ascent formula shows that the height ( h ) to which a liquid rises in the capillary tube is –

- – inversely proportional to the radius of the tube.
- – inversely proportional to the
*density*of the liquid. - – directly proportional to the
*surface tension*of the liquid.

*Hence, a liquid rises more in a narrower tube ( having less diameter ) than in wider tube ( having larger diameter).*

### Capillary rise in tube of insufficient height

The height to which a liquid rises in a capillary tube is given by –

h = \left ( \frac {2 \sigma \cos \theta}{\rho g r} \right )

The radius ( r ) of the capillary tube and radius of curvature ( R ) of the liquid meniscus are related by –

r = R \cos \theta

Therefore, \quad h = \left ( \frac {2 \sigma \cos \theta}{\rho g R \cos \theta} \right ) = \left ( \frac {2 \sigma}{R \rho g} \right )

As \sigma, \rho \text {and} g are constants, so \quad h R = \left ( \frac {2 \sigma}{\rho g} \right ) ( It is a constant ).

Therefore, \quad h R = h' R'

Where ( R' ) is the radius of curvature of the new meniscus at a height ( h' ) .

Hence in a capillary tube of insufficient height, the liquid rises to the top and spreads out to a new radius of curvature ( R' ) is given by

R' = \left ( \frac {h R}{h'} \right )

- But the liquid will not overflow from the tube.