## Grade 11 Physics Note

# Surface tension

**Surface tension:**

The property of liquid by virtue of which it tries to occupy minimum surface area within the given volume is called surface tension.

**Surface energy;**

When a certain force is applied on a liquid through, then the work done by this force will be stored in the form of energy. Such energy is called free surface energy or elastic potential energy.

**It is better to wash clothes in hot soap solution:**

We know that, the property of liquid by virtue of which, it tries to occupy minimum surface area within a given volume is called surface tension and a liquid having more surface tension occupies less area. We also know that the surface tension of a liquid is inversely proportional to the temperature.

I.e. ${\rm{T}} \propto \frac{1}{\theta }$

Where, T is the surface tension and $\theta $ is the temperature.

A soap solution having more temperature has fewer surfaces and a soap solution having less temperature spreads in a wide area. Due to this reason, hot water spreads in large area than the cold. So it is better to wash clothes in hot soap solution.

Surface tension of liquid is independent to the area of the liquid surface:

Surface tension of liquid is independent to the area of the liquid surface because surface tension of liquid is given by

${\rm{T}} = \frac{{{\rm{rh}}\rho {\rm{g}}}}{{2{\rm{Cos}}\theta }}$

This shows it is independent of area.

**Angle of contact:**

The angle${\rm{\: }}\theta $ which the tangent to the liquid surface at the point of contact makes with the solid surfaces inside the liquid is called angle of contact. It is also defined as the angle made by tangent with glass inside liquid and it is denoted by$\theta $. It is acute for water-glass surface obtuse for mercury and glass and $0\infty $ for pure water and clean water. Its value is about $10\infty $ for top water.

**The angle of contact of mercury with glass is obtuse but that of water with glass is acute:**

The angle made by tangent with glass inside liquid is called angle if contact and it is denoted by $\theta $. It is acute for water-glass surface obtuse for mercury and glass and $0\infty $ for pure water and clean water. It's value is about $10\infty $ for top water.

**Expression for the ascent of a liquid in a capillary tube;**

A tube of very fine hole or bore is called capillary. The rise or fall of liquid through a capillary is called capillarity or capillary action. Due to capillarity water rises through the root of the plant.

Let's take a vessel containing a liquid of density $\rho $ in which a capillary tube of radius r is dipped in it, in such a way that the height of liquid raised above the free surface of liquid is 'h'. The liquid wets the glass and hence the meniscus is concave. Let 'E' be the surface tension of liquid and '$\theta $' be the angle of contact then, the surface tension 'T' has two horizontally and 'Tcos$\theta $' acts vertically in upward direction. If we consider the diameter AB, then the component 'Tsin$\theta $' being equal act in opposite directions and hence they cancel in pair. The remaining component is only the 'Tcos$\theta $’ acting on the meniscus or circumference of the liquid. This component gives the upward force due to surface tension of the liquid. If F_{1} be the upward force due to surface tension then,

F_{1} = surface tension ${\rm{*\: }}$stretched length

or ${{\rm{F}}_1} = {\rm{TCos}}\theta {\rm{*}}2{\rm{\pi r}}$ -------(i)

Due to the force F_{1} the liquid rises up to certain height 'h'. But the liquid does not rise up to infinite height because the weight of liquid below the meniscus gives downward force.

If F_{2} be the downward force due to weight of liquid below the meniscus then,

F_{2} = weight of liquid below the meniscus

= volume of liquid below the meniscus${\rm{\: *}}\rho {\rm{*g}}$

= $\left( {{\rm{volume\: of\: ABFG}} - {\rm{volume\: of\: hemisphere}}} \right){\rm{*}}\rho {\rm{*g}}$

= $\left[ {{\rm{\pi }}{{\rm{r}}^2}\left( {{\rm{h}} + {\rm{r}}} \right) - \frac{1}{2}{\rm{*}}\frac{{4{\rm{\pi }}{{\rm{r}}^3}}}{3}} \right]{\rm{*}}\rho {\rm{*g}}$

= $\left[ {{\rm{\pi }}{{\rm{r}}^2}{\rm{h}} + {\rm{\pi }}{{\rm{r}}^3} - \frac{{2{\rm{\pi }}{{\rm{r}}^3}}}{3}} \right]{\rm{*}}\rho {\rm{g}}$

= ${\rm{\pi }}{{\rm{r}}^2}\left( {{\rm{h}} + {\rm{r}} - \frac{{2{\rm{r}}}}{3}} \right)\rho {\rm{g}}$

= ${\rm{\pi }}{{\rm{r}}^2}\left( {{\rm{h}} + \frac{{\rm{r}}}{3}} \right)\rho {\rm{g}}$ -------------- (ii)

'F_{2}' is the downward force due to weight of liquid. When, the upward force F_{1 }becomes equal to downward force F_{2} then, the liquid remains stationary. so for, stationary state of liquid, we have

F_{1} = F_{2 }

or, ${\rm{Tcos}}\theta {\rm{*}}2{\rm{\pi r}} = {\rm{\pi }}{{\rm{r}}^2}\left( {{\rm{n}} + \frac{{\rm{\pi }}}{3}} \right)\rho {\rm{g}}$

or, ${\rm{Tcos}}\theta {\rm{*}}2 = {\rm{r}}\left( {{\rm{h}} + \frac{{\rm{r}}}{3}} \right)\rho {\rm{g}}$

or, $\frac{{2{\rm{Tcos}}\theta }}{{{\rm{r}}\rho {\rm{g}}}} = {\rm{h}} + \frac{{\rm{r}}}{3}$

or, ${\rm{h}} = {\rm{\: }}\frac{{2{\rm{Tcos}}\theta }}{{{\rm{r}}\rho {\rm{g}}}} - \frac{{\rm{r}}}{3}$ -----------(iii)

Equation (iii) gives the expression for rise of liquid through the capillary tube.

If the radius of the tube is very small as compared to h, then we can neglect $\frac{{\rm{\pi }}}{3}$ under this assumption equation (iii) becomes,

${\rm{h}} = \frac{{2{\rm{Tcos}}\theta }}{{{\rm{r}}\rho {\rm{g}}}}$ ---------------- (iv)

Equation (iv) gives the expression for rise of liquid through a capillary tube of very small radius. Again, from equation (iv), we can write

$2{\rm{TCos}}\theta = {\rm{rh}}\rho {\rm{g}}$

or, ${\rm{T}} = \frac{{{\rm{h}}\rho {\rm{g}}}}{{2{\rm{Cos}}\theta }}$ -----------(v)

Equation (v) gives the expression for surface tension of a liquid which wets the glass. If the liquid is pure water and the glass is clean, then $\theta = 0\infty $

so, ${\rm{T}} = \frac{{{\rm{rh}}\rho {\rm{g}}}}{{2{\rm{Cos}}\theta }}$

or, ${\rm{T}} = \frac{{{\rm{rh}}\rho {\rm{g}}}}{2}$ -------------(vi)

**Rise of liquid in a tube of inefficient length:**

Suppose a capillary tube of r is held vertically in a liquid which has concave meniscus. If $\rho $ and T are the density and surface tension of the liquid respectively, Then capillary rise h is given by

${\rm{h}} = \frac{{2{\rm{Tcos}}\theta }}{{{\rm{r}}\rho {\rm{g}}

${\rm{cos}}\theta = \frac{{{\rm{AC}}}}{{{\rm{AO}}}} = \frac{{\rm{r}}}{{\rm{R}}}$

${\rm{r}} = {\rm{Rcos}}\theta $

${\rm{putting\: the\: value\: in\: equn}}.\left( {\rm{i}} \right)$

${\rm{\: \: h}} = \frac{{2{\rm{Tcos}}\theta }}{{{\rm{rcos}}\theta \rho {\rm{g}}}}$

= $\frac{{2{\rm{T}}}}{{{\rm{R\: }}\rho {\rm{g}}}}$

${\rm{hR}} = \frac{{2{\rm{T}}}}{{\rho {\rm{g}}}} = {\rm{constant\: For\: given\: liquid}}$

According to figure,

Since h >h’

Therefore R’ >R

So there is an increase in the radius of curvature of the liquid meniscus. But the angle of contact remains constant.

**Surface tension is numerically equal to the surface energy:**

The property of liquid by virtue of which it tries to occupy minimum surface area within the given volume is called surface tension. The minimum area of a liquid is round or spherical in shape.

The force per unit stretched length gives the measurement of surface tension. If 'f' be the force acting on the stretched length 'l', then the surface tension is

${\rm{T}} = \frac{{\rm{F}}}{{\rm{L}}}$

In which, the S.I. unit of surface tension is N/m and its dimension is $\left[ {{{\rm{M}}^1}{{\rm{L}}^0}{{\rm{T}}^{ - 2}}} \right]$

When a certain force is applied on a liquid through, then the work done by this force will be stored in the form of energy. Such energy is called free surface energy or elastic potential energy.

The work done per unit stretched area is called surface energy and it is denoted by '$\sigma $'. so, ${\rm{surface\: energy}}\sigma = \frac{{{\rm{work\: done\: }}}}{{{\rm{increase\: in\: area}}}}$

$ = \frac{{\Delta {\rm{W}}}}{{\Delta {\rm{A}}}}$ ---------- (i)

The unit of surface energy is 1/m^{2}. This surface energy is also called the surface energy density.

We also know that, the force per unit stretched length is called surface tension and it is given by,

${\rm{T}} = \frac{{{\rm{Force}}}}{{{\rm{Stretched\: length}}}}$

$ = \frac{{\rm{F}}}{{{{\rm{l}}^1}}}$ ------------- (ii)

**An expression of excess pressure inside a liquid drop:**

To find the magnitude of excess pressure in a liquid, we consider a liquid drop of finite radius. If P_{1} and P_{0} be the inside and outside pressure in a liquid drop then, the excess pressure is P

P = P_{1} - P_{0} in which pressure is always more than

Let's take a liquid through of center O and radius R in which the inside pressure is P and outside pressure is P_{0}. Due to the difference in pressure or excess pressure, the size of liquid trough increases through the distance OR then, the work done in including the area or distance by DR is

${\rm{dw}} = {\rm{force*distance}}$

$ = {\rm{P*}}4{\rm{\pi }}{{\rm{R}}^2}{\rm{*dR}}$ ------- (i)

But we also know that the surface energy is,

$\sigma = \frac{{\Delta {\rm{W}}}}{{\Delta {\rm{A}}}}$

or, $\Delta {\rm{W}} = \sigma {\rm{*}}\Delta {\rm{A}}$

$ = {\rm{T*}}\Delta {\rm{A}}$ --------- (iii)

Where, $\sigma = {\rm{T}}$

The work done in equation (ii) and (iii) is same, so

${\rm{T*}}\Delta {\rm{A}} = {\rm{P*}}4{\rm{\pi }}{{\rm{R}}^2}{\rm{*dR}}$ ------------ (iv)

But,

$\Delta {\rm{A}} = {{\rm{A}}_2} - {{\rm{A}}_1}$

$ = 4{\rm{\pi }}{\left( {{\rm{R}} + {\rm{dR}}} \right)^2} - 4{\rm{\pi }}{{\rm{R}}^2}$

$ = 4{\rm{\pi }}\left[ {{{\rm{R}}^2} + 2{\rm{RdR}} + {{\left( {{\rm{dR}}} \right)}^2} - {{\rm{R}}^2}} \right]$

$ = 4{\rm{\pi }}\left[ {2{\rm{RdR}} + {{\left( {{\rm{dR}}} \right)}^2}} \right]$

Since ${\rm{dR}}$ is small, ${\left( {{\rm{dR}}} \right)^2}$ will be very small and we can neglect it. So, neglecting ${\left( {{\rm{dR}}} \right)^2}$ we get

$\Delta {\rm{A}} = 4{\rm{\pi *}}2{\rm{RdR}}$ ---------- (v)

Putting the value of $\Delta {\rm{A}}$ in equation (iv) we get

${\rm{T*}}4{\rm{\pi *}}2{\rm{RdR}} = {\rm{P*}}4{\rm{\pi }}{{\rm{R}}^2}{\rm{*dR}}$

Or, $2{\rm{T}} = {\rm{P*R}}$

or, ${\rm{P}} = \frac{{2{\rm{T}}}}{{\rm{R}}}$

or, ${{\rm{P}}_2} - {{\rm{P}}_6} = \frac{{2{\rm{T}}}}{{\rm{R}}}$ ----------(vi)

Equation (vi) gives the expression for excess pressure in a liquid trough in which the angle of contact is$0\infty $.

**Molecular theory of surface tension:**

The property of liquid by virtue of which it tries to occupy minimum surface area within the given volume is called surface tension. The minimum area of a liquid is round or spherical in shape.

The force per unit stretched length gives the measurement of surface tension. If 'f' be the force acting on the stretched length 'l', then the surface tension is

${\rm{T}} = \frac{{\rm{F}}}{{\rm{L}}}$

In which, the S.I. unit of surface tension is N/m and its dimension is $\left[ {{{\rm{M}}^1}{{\rm{L}}^0}{{\rm{T}}^{ - 2}}} \right]$

Surface tension is explained on the basis of the molecular theory of matter .A liquid consist of a very large number of molecules .These molecules attract each other by cohesive force. But these forces have only a short range .The maximum distance up to which the force is called the range of molecular attraction. A sphere drawn with a molecule as centre and the range of molecular attraction as radius is called the sphere of influence of that molecule.

Angle of contact:

The angle${\rm{\: }}$θwhich the tangent to the liquid surface at the point of contact makes with the solid surfaces inside the liquid is called angle of contact. It is also defined as the angle made by tangent with glass inside liquid and it is denoted by$\theta $.

The angle of contact depends on the following factors;

1. It depends on the nature of the liquid and solid in contact.

2. It depends upon the medium that exists above the free surface of the liquid.

It is acute for water-glass surface obtuse for mercury and glass and $0\infty $ for pure water and clean water. Its value is about $10\infty $ for top water.