Question

A soap bubble of radius $r$ is blown up to form a bubble of radius $2r$ under isothermal conditions. If $T$ is the surface tension of soap solution, the energy spent in the blowing is
$\begin{aligned} & A)3\pi T{{r}^{2}} \\\ & B)6\pi T{{r}^{2}} \\\ & C)12\pi T{{r}^{2}} \\\ & D)24\pi T{{r}^{2}} \\\ \end{aligned}$

Answer 1

Energy spent in blowing a soap bubble from a particular radius to another particular radius is equal to the energy required for the soap bubble to increase its size. It is mathematically equal to the product of change in surface area of the soap bubble and the surface tension of the soap bubble. A soap bubble has two surface areas – inner surface area and outer surface area.

Formula used:
$1)A=4\pi {{R}^{2}}$
$2)E=(2\Delta A)T$

Complete answer:
When a soap bubble of radius $r$ is blown up to form a bubble of radius $2r$ under isothermal conditions, the energy spent in the blowing is given by
$E=(2\Delta A)T$
where
$E$ is the energy spent in blowing
$2\Delta A$ is the change in inner surface area and outer surface area of the bubble
$T$ is the surface tension of the soap bubble
Let this be equation 1.
We know that surface area of a soap bubble is given by
$A=4\pi {{R}^{2}}$
where
$A$ is the surface area of a soap bubble (in the form of a sphere)
$R$ is the radius of the soap bubble
Let this be equation 2.
From the question, we are provided that a soap bubble of radius $r$ is blown up to a size, whose radius is equal to $2r$. If the surface area of bubble with radius $r$ is denoted as ${{A}_{1}}$, using equation 1, ${{A}_{1}}$ is given by
${{A}_{1}}=4\pi {{r}^{2}}$
Similarly, if ${{A}_{2}}$ represents the surface area of the blown up size of soap bubble of radius $2r$, then, ${{A}_{2}}$ is given by
${{A}_{2}}=4\pi {{(2r)}^{2}}=16\pi {{r}^{2}}$
Now, if change in surface area is denoted as $\Delta A$, then, we know that $\Delta A$is given by
$\Delta A={{A}_{2}}-{{A}_{1}}=16\pi {{r}^{2}}-4\pi {{r}^{2}}=12\pi {{r}^{2}}$
Let this be equation 3.
Since a soap bubble has both inner surface and outer surface, the change in surface area is multiplied by two, in order to determine the energy spent in blowing as given below.
Substituting equation 3 in equation 1, we have
$E=(2\Delta A)T=\Delta A=(2\times 12\pi {{r}^{2}})\times T=24T\pi {{r}^{2}}$
where
$E$ is the energy spent in blowing a soap bubble to a size double its initial radius
$T$ is the surface tension of the soap solution (as provided in the question)
$r$ is the initial radius of the soap bubble
Therefore, the energy spent in blowing is equal to $24T\pi {{r}^{2}}$.

The correct answer is option $D$.

Note:
Surface tension is the tendency of a fluid to occupy the minimum possible surface area. For example, surface tension is the reason behind the spherical shapes of rain drops as well as spilled mercury from a broken thermometer. Mathematically, surface tension is given by
$T=\dfrac{F}{L}$
where
$F$ is the force acting per unit length
$L$ is the length in which the force acts
$T$ is the surface tension