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# Excess of pressure inside a liquid drop, a soap bubble, and an air bubble

When the free surface of the liquid is curved, there is a difference in pressure between the inner and outer the side of the surface.

Excess of pressure inside a liquid drop, a soap bubble, and an air bubble

As it is discussed earlier, the free surface of a liquid becomes curved when it has contact with a solid. Depending upon the nature of liquid-air or liquid-gas interface, the magnitude of interfacial surface tension varies. In other words, as a consequence of surface tension, the above such interfaces have energy and for a given volume, the surface will have a minimum energy with least area. Due to this reason, the liquid drop becomes spherical (for a smaller radius).

When the free surface of the liquid is curved, there is a difference in pressure between the inner and outer the side of the surface (Figure 7.27).

i) When the liquid surface is plane, the forces due to surface tension (T, T) act tangentially to the liquid surface in opposite directions. Hence, the resultant force on the molecule is zero. Therefore, in the case of a plane liquid surface, the pressure on the liquid side is equal to the pressure on the vapour side. ii) When the liquid surface is curved, every molecule on the liquid surface experiences forces (FT, FT) due to surface tension along the tangent to the surface. Resolving these forces into rectangular components, we find that horizontal components cancel out each other while vertical components get added up. Therefore, the resultant force normal to the surface acts on the curved surface of the liquid. Similarly, for a convex surface, the resultant force is directed inwards towards the centre of curvature, whereas the resultant force is directed outwards from the centre of curvature for a concave surface. Thus, for a curved liquid surface in equilibrium, the pressure on its concave side is greater than the pressure on its convex side.

## Excess of pressure inside a bubble and a liquid drop:

The small bubbles and liquid drops are spherical because of the forces of surface tension. The fact that a bubble or a liquid drop does not collapse due to the combined effect indicates that the pressure inside a bubble or a drop is greater than that outside it.

### 1)  Excess of pressure inside air bubble in a liquid.

Consider an air bubble of radius R inside a liquid having surface tension T as shown in Figure 7.28 (a). Let P1 and P2 be the pressures outside and inside the air bubble, respectively. Now, the excess pressure inside the air bubble is ΔP = P1 − P2. In order to find the excess pressure inside the air bubble, let us consider the forces acting on the air bubble. For the hemispherical portion of the bubble, considering the forces acting on it, we get,

i) The force due to surface tension acting towards right around the rim of length  R is FT = 2πRT

ii) The force due to outside pressure P1 is to the right acting across a cross sectional area of πR2 is FP1= P1πR2

iii) The force due to pressure P2 inside the bubble, acting to the left is FP2= P2πR2.

As the air bubble is in equilibrium under the action of these forces, FP2= FT + FP1

P2πR2 = 2πRT + P1πR2

(P2P1R2 = 2πRT ### 2)  Excess pressure inside a soap bubble

Consider a soap bubble of radius R and the surface tension of the soap bubble be T as shown in Figure 7.28 (b). A soap bubble has two liquid surfaces in contact with air, one inside the bubble and other outside the bubble. Therefore, the force on the soap bubble due to surface tension is 2×2πRT. The various forces acting on the soap bubble are,

i) Force due to surface tension FT=4πRT towards right

ii) Force due to outside pressure, FP1= P1πR2 towards right

iii) Force due to inside pressure, FP2= P2πR2 towards left

As the bubble is in equilibrium, FP2= FT + FP1

P2πR2 = 4πRT + P1πR2

(P2P1R2 = 4πRT  ### 3)  Excess pressure inside the liquid drop

Consider a liquid drop of radius R and the surface tension of the liquid is T as shown in Figure 7.28 (c). The various forces acting on the liquid drop are,

i) Force due to surface tension FT=2πRT towards right

ii) Force due to outside pressure, FP1= P1πR2 towards right

iii) Force due to inside pressure, FP2= P2πR2 towards left

As the drop is in equilibrium,

FP2= FT + FP1

P2πR2 = 2πRT + P1πR2

= > (P2P1R2 = 2πRT ## EXAMPLE 7.11

If excess pressure is balanced by a column of oil (with specific gravity 0.8) 4 mm high, where R = 2.0 cm, find the surface tension of the soap bubble.

### Solution

The excess of pressure inside the soap bubble is T = 15.68 ×10 2 N m1

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11th Physics : Properties of Matter : Excess of pressure inside a liquid drop, a soap bubble, and an air bubble |

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