Phenomenon of surface tension on the basis of molecular theory :

The phenomenon of surface tension arises due to the cohesive forces between the molecules of a liquid. The net cohesive force on the liquid molecules within the surface film differs from that on molecules deep in the interior of the liquid.

Consider three molecules of a liquid : A molecule A well inside the liquid, and molecules B and C lying within the surface film, see Fig.

The figure also shows their spheres of influence of radius R.

• The sphere of influence of molecule A is entirely inside the liquid and the molecule is surrounded by its nearest neighbours on all sides. Hence, molecule A is equally attracted from all sides, so that the resultant cohesive force acting on it is zero. Hence, it is free to move anywhere within the liquid.
• For molecule B, a part of its sphere of influence is outside the liquid surface. This part contains air molecules whose number is negligible compared to the number of molecules in an equal volume of the liquid. Therefore, molecule B experiences a net cohesive force downward.
• For molecule C, the upper half of its sphere of influence is outside the liquid surface. Therefore, the resultant cohesive force on molecule C in the downward direction is maximum.
• Thus, all molecules lying within a surface film or thickness equal to R experience a net cohesive force directed into the liquid.
• The surface area is proportional to the number of molecules on the surface. To increase the surface area, molecules must be brought to the surface from within the liquid. For this, work must be done against the cohesive forces. This work is stored in the liquid surface in the form of potential energy.
• With a tendency to have minimum potential energy, the liquid tries to reduce the number of molecules on the surface so as to have minimum surface area. This is why the surface of a liquid behaves like a stressed elastic membrane.

Relation between the surface tension and surface energy of a liquid:

Suppose a soap film is isothermally stretched over the area enclosed by a U-shaped frame ABCD and a cross-piece RQ that can slide smoothly along the frame, as shown in the figure. Let T be the surface tension of the soap solution and l, the length of wire RQ in contact with the soap film.

The film has two surfaces, both of which are in contact with the wire. The film tends to contract by exerting a force on wire RQ. As each surface exerts a force Tl, the net force on the wire is 2Tl.

Suppose that wire RQ is pulled outward very slowly through a distance dx to the position R’Q' by an external force of magnitude 2Tl. The work done by the external force against the force due to the film is

W = applied force x displacement

∴ W=Fdx = 2Tldx     (… F=2Tl)

This work is stored in the unit surface area in the form of potential energy. This potential energy is called the surface energy.

Due to the displacement dx, the surface area of the film increases. As the film has two surfaces, the increase in its surface area is

A = 2ldx

Thus, the work done per unit surface area is

\(\frac{W}{A}\) =\(\frac{2Tldx}{2ldx}\) = T

Thus, the surface energy per unit area of a Liquid is equal to its surface tension.

Characteristics of angle of contact :

• It depends upon the nature of the liquid and solid in contact, and is constant for a given liquid—solid pair, other factors remaining unchanged.
• It depends upon the medium (gas) above the free surface of the liquid.
• It is independent of the inclination of the solid to the liquid surface.
• It changes with surface tension and, hence, with the temperature and purity of the liquid.

 Characteristics of the angle of contact in :

(i) The angle of contact in the case of a liquid which completely wets.

For a liquid, which completely wets the solid, the angle of contact is zero.

For example, pure water completely wets clean glass. Therefore, the angle of contact at the water glass interface is zero

(ii) The angle of contact in the case of a liquid which partially wets

For a liquid which partially wets the solid, the angle of contact is an acute angle.

For example, kerosine partially wets glass, so that the angle of contact is an acute angle at the kerosine-glass interface

(iii) The angle of contact in the case of a liquid which does not wet the solid.

For a liquid which does not wet the solid, the angle of contact is an obtuse angle.

For example, mercury does not wet glass at all, so that the angle of contact is an obtuse angle at the mercury-glass interface.

(iv) The angle of contact for a given liquid solid pair is constant at a given temperature, provided the liquid is pure and the surface of the solid is clean.

Formation of concave and convex surface of a liquid on the basis of molecular theory.

OR

(Q. why the free surface of some liquids in contact with a solid is not horizontal.?)

For a molecule in the liquid surface which is in contact with a solid, the forces on it are largely the solid-liquid adhesive force F_A = PA and the liquid- liquid cohesive force F_C = PC, F_A is normal to the solid surface and F_C is at 45° with the horizontal See Fig. The free surface of a liquid at rest is always perpendicular to the resultant F_R = PR of these forces

If F_C = \(\sqrt{2}\) F_A, F_R is along the solid surface, the contact angle is 90° and the liquid surface is horizontal at the edge where it meets the solid, as in above Fig. In general this is not so, and the liquid surface is not horizontal at the edge.

For a liquid which completely wets the solid (e.g. pure water in contact with clean glass), F_C << F_A. For a liquid which partially wets the solid (e.g., kerosene or impure water in contact with glass), F_C < \(\sqrt{2}\)F_A

If F_C << F_A or if F_C < \(\sqrt{2}\)F_A, the contact angle is correspondingly zero or acute and the liquid surface curves up and acquires a concave shape until the tangent PT is tangent to F_R ref. below Fig.

If F_C > \(\sqrt{2}\)F_A, the contact angle is obtuse and the liquid surface curves down and acquires a convex shape until the tangent PT is tangent to F_R, see below Fig.

Shape of a liquid drop on solid surface in terms of interfacial tensions.

A liquid surface, in general, is curved where it meets a solid. The angle between the solid surface and the tangent to the liquid surface at the extreme edge of the liquid, as measured through the liquid, is called the angle of contact.

Figure shows the interfacial tensions that act in equilibrium at the common point of the liquid, solid and gas (air + vapour).

T₁ = the liquid-solid interfacial tension

T₂ = the solid-gas interfacial tension

T₃ = the liquid-gas interfacial tension

θ = the angle of contact for the liquid-solid pair is the angle between T₁ and T₃,

The equilibrium force equation (along the solid surface) is

T₃ cos θ + T₁ – T₂ = 0

∴ cos θ =\(\frac{T_2-T_1}{T_3}\) ......... (1)

Case (1) : If T₂ > T₁ , cos θ is positive and contact angle 0 < 90°, so that the liquid wets the surface.

Case (2) : If T₂ < T₁, cos θ is negative and q is obtuse, so that the liquid is non-wetting.

Case (3) : If T₂ — T₁, ≅ T₃, cos θ = 1 and θ ≅ 0°.

Case (4) : If T₂ — T₁, > T₃, cos θ will be greater than 1 which is impossible, so that there will be no equilibrium and the liquid will spread over the solid surface.

Effect of impurity :

(i) The angle of contact or the surface tension of liquid increases with dissolved impurities like common salt. For dissolved impurities, the angle of contact (or surface tension) increases linearly with the concentration of the dissolved materials.

(ii) It decreases with sparingly soluble substances like phenol or alcohol. A detergent is a surfactant whose molecules have hydrophobic and hydrophilic ends, the hydrophobic ends decrease the surface tension of water. With reduced surface tension, the water can penetrate deep into the fibres of a cloth and remove stubborn stains.

(ii) It decreases with insoluble surface impurities like oil, grease or dust. For example, mercury surface contaminated with dust does not form perfect spherical droplets till the dust is removed.

Effect of Temperature :

The surface tension of a liquid decreases with increasing temperature of the liquid. For small temperature differences, the decrease in surface tension is nearly directly proportional to the temperature rise.

If T and T₀ are the surface tensions of a liquid at temperatures q and 0°C, respectively, then

T=T₀(1 − αθ) where α is a constant for a given liquid.

The surface tension of a liquid becomes zero at its critical temperature. The surface tension increases with increasing temperature only in case of molten copper and molten cadmium.

Pressure difference across a curved liquid surface.:

Every molecule lying within the surface film of a static liquid is pulled tangentially by forces due to surface tension. The direction of their resultant,\(\vec f_r\)  on a molecule depends upon the shape of that liquid surface and decides the cohesion pressure at a point just below the liquid surface.

Consider two molecules, A and B, respectively just above and below the free surface of a liquid. So the level difference between them is negligibly small and the atmospheric pressure on both is the same, p_o, Let \(\vec f_A\)  be the downward force on A and B due to the atmospheric pressure.

(i) If the free surface of a liquid is horizontal, the resultant force \(\vec f_r\)  on molecule B is zero, see Fig. Then, the cohesion pressure is negligible and the net force on A and B is \(\vec f_A\) . The pressure difference on the two sides of the liquid surface is zero.

(ii) If the free surface of a liquid is concave, the resultant force \(\vec f_r\)  on molecule B is outwards (away from the liquid), see below Fig. opposite to  \(\vec f_A\)

Then, the net force on B is less than \(\vec f_A\), and the cohesion pressure is decreased. The pressure above the concave liquid surface is greater than that just below the liquid surface.

(iii) If the free surface of a liquid is convex, the resultant force \(\vec f_r\)   on molecule B acts inwards (into the liquid), see below Fig. in the direction of .\(\vec f_A\)

Then, the net force on B is greater than \(\vec f_A\) and the cohesion pressure is increased. The pressure below the convex liquid surface is greater than that just above the liquid surface.

Laplace's law for a spherical membrane. :

(Q. Derive an expression for the excess pressure inside a liquid drop.)

Consider a spherical drop as shown in Fig.

Let p_i be the pressure inside the drop and p₀ be the pressure outside it. As the drop is spherical in shape, the pressure, p_i, inside the drop is greater than p₀, the pressure outside.

Therefore, the excess pressure inside the drop is  p_i - p₀.

Let the radius of the drop increase from r to r + Δr, where Δr is very small,

so that the pressure inside the drop remains almost constant.

Let the initial surface area of the drop be

A₁ = 4πr²,

and the final surface area of the drop be

A₂ = 4π (r+Δr)²

∴ A₂ = 4π (r² + 2rΔr + Δr²)

∴ A₂ = 4π r² + 8πrΔr + 4πΔr²

As Δr is very small, Δr² can be neglected,

∴ A₂ = 4πr² + 8πrΔr

Thus, increase in the surface area of the drop is

dA = A₂ – A₁ = 4πr² + 8πrΔr - 4πr² = 8πrΔr --- (1)

Work done in increasing the surface area by dA is stored as excess surface energy.

∴ dW = TdA= T (8πrΔr) --- (2)

This work done is also equal to the product of the force F which causes increase in the area of the bubble and the displacement Δr which is the increase in the radius of the bubble.

∴ dW = FΔr --- (3)

The excess force is given by,

(Excess pressure) × (Surface area)

∴  F = (p_i – p₀) 4πr² --- (4)

Equating Eq. (2) and Eq. (3), we get,

T(8prΔr) = (p_i – p₀) 4πr² Δr

∴ (p_i – p₀) = \(\frac{2T}{r}\)  --- (5)

This equation gives the excess pressure inside a drop. This is called Laplace’s law of a spherical membrane or Young Laplace Equation in spherical form

Applications of capillarity :

• A blotting paper or a cotton cloth absorbs water, ink by capillary action.
• Oil rises up the wick of an oil lamp and sap rises up xylem tissues of a tree by capillarity.
• Ground water rises to the open surface through the capillaries formed in the soil. In summer, the farmers plough their fields to break these capillaries and prevent excessive evaporation.
• Water rises up the crevices in rocks by capillary action. Expansion and contraction of this water due to daily and seasonal temperature variations cause the rocks to crumble.

[Note : The rise of sap is due to the combined action of capillarity and transpiration. The transpiration pull considered to be the major driving force for water transport throughout a plant.]

Phenomenon of capillarity (Capillary action ):

(1) When a capillary tube is partially immersed a wetting liquid, there is capillary rise and liquid meniscus inside the tube is concave, as shown in Fig. (A).

Consider four points A, B, C, D, of which point is just above the concave meniscus inside the capillary and point B is just below it. Points C and D are just above and below the free liquid surface outside.

Let P_A, P_B, P_C and P_D be the pressures at point A, B, C and D, respectively.

Now, P_A = P_C = atmospheric pressure

The pressure is the same on both sides of the free surface of a liquid, so that

P_C = P_D

∴ P_A = P_D

The pressure on the concave side of a meniscus is always greater than that on the convex side, so that

P_A > P_B

∴ P_D > P_B            (….P_A = P_D)

The excess pressure outside presses the liquid up the capillary until the pressures at B and D (at the same horizontal level) equalize, i.e., P_B becomes equal to P_D. Thus, there is a capillary rise.

(2) For a non-wetting liquid, there is capillary depression and the liquid meniscus in the capillary tube is convex, as shown in Fig. (B).

Consider again four points A, B, C and D when the meniscus in the capillary tube is at the same level as the free surface of the liquid. Points A and B are just above and below the convex meniscus.

Points C and D are just above and below the free liquid surface outside.

The pressure at B (P_B) is greater than that at A(P_A). The pressure at A is the atmospheric pressure H and at D, P_D @ H = P_A. Hence, the hydrostatic pressure at the same levels at B and D are not equal,

P_B > P_D.

Hence, the liquid flows from B to D and the level of the liquid in the capillary falls. This continues till the pressure at B’ is the same as that D’, that is till the pressures at the same level are equal.

An expression for capillary rise for a liquid having a concave meniscus.

Consider a capillary tube of radius r partially immersed into a wetting liquid of density p. Let the capillary rise be h and q be the angle of contact at the edge of contact of the concave meniscus and glass (Fig)..

If R is the radius of curvature of the meniscus then from the figure, r=R cosθ.

Surface tension T is the tangential force per unit length acting along the contact line. It is directed into the liquid making an angle θ with the capillary wall. We ignore the small volume of the liquid in the meniscus. The gauge pressure within the liquid at a depth h, i.e. at the level of the free liquid surface open to the atmosphere, is

P— P_o = ρgh ………. (1)

By Laplace’s law for a spherical membrane, this gauge pressure is

P— P_o = 2T/R  …….(2)

hρg  = 2T/R = \(\frac{2Tcosθ}{r}\)

∴ h =    \(\frac{2Tcosθ}{rρg}\)…….(3)

Thus, narrower the capillary tube, the greater is the capillary rise.

From Eq. (3),

T=   \(\frac{hrρg}{2Tcosθ}\)………(4)

Equations (3) and (4) are also valid for capillary depression h of a non-wetting liquid. In this case, the meniscus is convex and θ is obtuse. Then, cosθ is negative but so is h, indicating a fall or depression of the liquid in the capillary. T is positive in both cases.

[For capillary rise, Eq. (3) is also called the ascent formula. ]