Question

Two soap bubbles of radii $a$ and $b$ combine under isothermal conditions to form a single bubble of radius $c$ without any leakage of air. If ${P_0}$ is atmospheric pressure and $T$ is surface tension of soap solution, show that ${P_0} = \dfrac{{4T\left( {{a^2} + {b^2} - {c^2}} \right)}}{{{c^3} - {a^3} - {b^3}}}$.

Answer 1

We are given two bubbles of given radii. These soap bubbles combine and form a single bubble of another radius. We are given the atmospheric pressure and we are asked to prove that it is found a certain way at a certain temperature. We use the concept of isothermal process and find the pressure as the value given by equating the sum of the individual bubbles to the product of pressure and volume of the third.

Formulas used:
The equation for an isothermal process is given as,
$PV = k$
Where $k$ is a constant, $V$ is the volume and $P$ is the pressure.

Complete step by step answer:
Let us start by defining an isothermal process. As the name suggests an isothermal process is a process by which the temperature remains a constant. The equation for an isothermal process is given as,
$PV = k$
Where $k$ is a constant.

From this, we can consider the sum of products of two individual bubbles and equate it to the third and find the value of atmospheric pressure as given. We get,
$PV = {P_1}{V_2} + {P_2}{V_2}$
Applying the values and assuming the soap bubbles are spherical, we end up with the equation,
$\left( {{P_0} - \dfrac{{4T}}{a}} \right)\dfrac{4}{3}\pi {a^3} + \left( {{P_0} - \dfrac{{4T}}{b}} \right)\dfrac{4}{3}\pi {b^3} = \left( {{P_0} - \dfrac{{4T}}{c}} \right)\dfrac{4}{3}\pi {c^3}$

Opening the bracket and bringing the like terms to one side and we get,
${P_0}\left( {{c^3} - \left( {{a^3} + {b^3}} \right)} \right) - 4T\left( {{a^2} + {b^2} - {c^2}} \right) = 0$
Now we bring the quantity that is to be proven to the left-hand side and the rest to the right-hand side and get
$\therefore {P_0} = \dfrac{{4T\left( {{a^2} + {b^2} - {c^2}} \right)}}{{{c^3} - {a^3} - {b^3}}}$
Hence proved.

Note: We use the value of pressure as we have because it is given that the pressure on the surface is the difference between the pressure inside and the pressure outside the bubble. This pressure outside is given by ${P_i} - {P_o} = \dfrac{{4T}}{r}$. We then subtract this value from the atmospheric pressure and get the pressure of the inside of the bubble.