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**Force of cohesion:**It is force between two molecules of similar nature.**Force of adhesion:**It is the force between two molecules of different nature.**Molecular range:**The maximum distance between two molecules so that the force of attraction between them remains effective is called molecular range.**Sphere of influence:**Sphere of influence of any molecule is the sphere with molecule as its center and having a radius equal to molecular range (=10^{-7}cm).**Surface film:**Surface film of a liquid is defined as the portion of liquid lying on the surface and caught between two parallel planes situated molecular range apart.

**Surface tension:**

Surface tension is the property of a liquid by virtue of which its free surface behaves like a stretched membrane and supports, comparatively heavier objects placed over it. It is measured in terms of force of surface tension.

**Force of surface tension:** It is defined as the amount of force acting per unit length on either side of an imaginary line drawn over the liquid surface.**(a)** T = Force/length = F/l**(b)** T = Surface energy/Surface area = W/A

Units: S.I – Nm^{-1}

C.G.S- dyn cm^{-1}

**Additional force****(a)** For a cylindrical rod:- F = T×2πr (Here r is the radius of cylindrical rod)**(b) **For a rectangular block:- F = T×2(l+d) (Here l is the length and d is the thickness of the rectangular block)**(c)** For a ring:- F = T×2×2πr (Here r is the radius of cylindrical rod)

**Surface energy: **Potential energy per unit area of the surface is called surface energy.**(a) Expansion under isothermal condition: **To do work against forces of surface tension:-

W= T × A (Here A is the total increase in surface area)

To supply energy for maintaining the temperature of the film:-

E = T + H**(b) Expansion under adiabatic conditions:**

E = T

Force of surface tension is numerically equal to the surface energy under adiabatic conditions.**DROPS AND BUBBLES****(a) Drop:** Area of surface film of a spherical drop of radius R is given by, A = 4πR^{2}**(b) Bubble:** The surface area of the surface films of a bubble of radius R is, A = 2×4πR^{2}

**Combination of n drops into one big drop****(a)** R = n^{1/3}r**(b) **E_{i }= n (4πr^{2}T), E_{f} =4πR^{2}T**(c) **E_{f}/ E_{i} = n -1/3**(d)** ΔE/E_{i }= [1-(1/n^{1/3})]**(e)** ΔE = 4πR^{2}T (n^{1/3}-1) = 4πR^{3}T (1/r – 1/R)

**Angle of contact:** Angle of contact, for a pair of solid and liquid, is defined as the angle between tangent to the liquid surface drawn at the point of contact and the solid surface inside the liquid.**(a) When θ < 90º (acute):**

F_{a} >F_{c}/√2**(i) **Force of cohesion between two molecules of liquid is less than the force of adhesion between molecules of solid and liquid.**(ii) **Liquid molecules will stick with the solid, thus making solid wet.**(iii) **Such liquid is put in the solid tube; it will have meniscus concave upwards.**(b) When θ > 90º (obtuse):-F _{a}<F_{c}/√2**

F

The surface of liquid at the point of contact is plane. In this case force of cohesion and adhesion are comparable to each other.

Here, T

**Capillarity**

Capillarity is the phenomenon, by virtue of which the level of liquid in a capillary tube is different from that outside it, is called capillarity.

Weight of liquid, W = Vρg = πr^{2}[h+(r/3)]ρg (Here r is the radius meniscus)

If weight of meniscus is taken into account, the force of surface tension will be,

T = [r(h+(r/3)) ρg]/2 cosθ

For fine capillary, force of surface tension, T = rhρg/2 cosθ

So height, h = 2T cosθ/ rρg

This signifies, height of liquid risen (or depressed) in a capillary tube varies inversely as the radius of tube. Smaller the diameter of capillary tube, greater is the rise of liquid in it.

**Tube of insufficient length**

Rh = 2T/ρg

As, T, ρ and g are all constant, Rh = Constant

Smaller the value of h, greater will be the value of R. But liquid will never flow.

**Effect of temperature affecting surface tension of liquids**

Surface tension of a liquid decreases with an increase in its temperature.

T_{θ }= K (θ_{c}-θ)

Here Tθ is the surface tension at a particular temperature θ while θc is the critical temperature of the liquid and K is constant.

**Effect of density**

Density of liquid also affects its surface tension. Surface tension of a liquid is given by,

T = A (ρ - ρ')^{n}

Here, ρ is the density of liquid, ρ' is the density of saturated vapors of liquid and A is the constant depending on the nature of liquid.**Pressure difference across a liquid surface****(a) Plane surface:** There is no difference of pressure on the two sides of the film.**(b) Convex surface: **Pressure below the surface film must be greater than that just above it.**(c) Concave surface: **Pressure on the upper side is greater than that just below it.**General formula for excess pressure**

P_{excess} =T[1/R_{1} + 1/R_{2}]

**Excess pressure in liquid drop:**P_{excess}= 2T/R, Here R is the radius of liquid drop.**Excess pressure for an air bubble in liquid drop:**P_{excess }= 2T/R**Excess pressure in soap bubble:**P_{excess}= 4T/R, Here R is the radius of soap bubble.**Pressure inside an air bubble at a depth h in a liquid:**P_{in }= P_{atm+}hdg + (2T/R)

**Forces between two plates with thin water film separating them:****(a)** ΔP = T (1/r – 1/R)**(b) **F = ΔT (1/r – 1/R)**(c) **If separation between plate is d, then ΔP = 2T/d and F = 2AT/d

**Radius of curvature of common film:** R_{comon }= rR/R-r

Capillary depression, h = 2T cos (π-θ)/rdg

**Shape of liquid surface****(a) **Plane surface (as for water – silver) if F_{adhesive }> F_{cohesive}/√2**(b) **Concave surface (as for water – glass) if F_{adhesive} > F_{cohesive}/√2**(c) **Convex surface (as for mercury-glass) if F_{adhesive} < F_{cohesive}/√2

**Increase in temperature: **Δθ = 3T/ρs (1/r – 1/R) or Δθ = 3T/ρsJ (1/r – 1/R)

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