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15.Soap bubbles and liquid drops have excess pressure inside it. a. What is the reason for this excess pressure (½) b. Deduce the expression for excess pressure of a liquid drop. (3)

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

Let's now derive expressions for excess pressure inside a bubble and a liquid drop.

Excess of pressure inside the liquid drop: Consider a liquid drop of radius 'R' and of surface tension T. Let P_i and P_o be the pressures inside and outside the drop respectively, then we have, P_i > P_o

Therefore, net force on the drop is radially outwards and is given by,

       F = (P_i - P_o) 4 ? R²          (1)

If ?R be the increase in radius of the drop due to this net force, then work done by this force to increase the radius,

            W = F ? R
            W = (P_i - P_o) 4 ? R²   ?R        (2)

Also, we know that amount of work done in increasing the surface area of a free surface gets stored in it in the form of potential energy, which can also be calculated as,

                                
                             W = T ?A
                or               W = T [4? (R + ?R)² - 4 ?R²]

?R is very small, therefore, the term of ?R² can be neglected.

?            W = 8 ?RT ?R             (3)

Now from (2) and (3), we have,

Tht is, Excess of pressure inside a liquid drop is directly proportional to its surface tension and inversely proportional to the radius of the drop.

Excess of pressure inside a liquid bubble:
The concept is same as that of liquid drop except for one difference that liquid bubble has two free surfaces, therefore, total change in area is double that of (in the case of ) drop. Thus,

Excess of pressure inside an air bubble:
Air bubble is formed inside a liquid, therefore, it also has only one free surface. In the case of air bubble,