Question

A soap bubble has radius R and thickness \[d( < < R)\] as shown in the figure. It collapses into a spherical drop. The ratio of excess pressure in the spherical drop to the excess pressure inside the bubble is \[{\left( {\dfrac{R}{{xD}}} \right)^{\dfrac{1}{3}}}\]. Find the value of x.

Answer 1

Hint: For a soap bubble, the pressurized bubble of air is contained within a thin, elastic surface of the liquid having a large volume and surface area. When the bubble bursts, it will form a number of spherical drops with lesser volume and surface area, the difference in pressure causes an audible sound.

Formula Used:
The equation of volume of a soap bubble is given by,
\[4\pi d{R^2} = \dfrac{4}{3}\pi {r^3}\]…………. (1)
Where, \[R\] is the radius of the soap bubble, \[d\] is the diameter of the soap bubble and \[r\] is the radius of the spherical drop.

Complete step by step solution:
To find the value of x we need to find the ratio of excess pressure.
The formula to find the ratio of excess pressure is given by,
\[{P_1} = \dfrac{{4S}}{R}\]..........(Excess pressure of the soap bubble initially)
\[{P_2} = \dfrac{{2S}}{r}\]..........(Excess pressure of the drop finally)

Now, \[\dfrac{{{P_2}}}{{{P_1}}} = \left( {\dfrac{{\dfrac{{4S}}{R}}}{{\dfrac{{2S}}{r}}}} \right)\]
\[\dfrac{{{P_2}}}{{{P_1}}} = \dfrac{R}{{2r}}\]
By rearranging the equation (1), the value of r will be written as,
\[r = {\left( {3{R^2}d} \right)^{\dfrac{1}{3}}}\]
Put the value of r in equation (2) then we get,
\[\dfrac{{{P_2}}}{{{P_1}}} = \dfrac{R}{{2r}}\]


\[\Rightarrow \dfrac{{{P_2}}}{{{P_1}}} = \dfrac{R}{{2{{\left( {3{R^2}d} \right)}^{\dfrac{1}{3}}}}}\\\]
 \[\therefore \dfrac{{{P_2}}}{{{P_1}}} = \dfrac{R}{{{{\left( {24d} \right)}^{\dfrac{1}{3}}}}}\]

Therefore, the value of x is 24.

Note:Surface tension is defined as the property of any liquid by virtue of which it tries to minimize its surface area. The surface tension of water provides the necessary wall tension for the formation of the water bubbles. This tendency to minimize the tension on the walls pulls the bubbles into spherical shapes.