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Mechanical Properties of Fluids — Physics STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsMechanical Properties of Fluids4 Marks
Question
Obtain the relation between surface energy and surface tension.
Calculate the work done in blowing a soap bubble to a radius of $1 \ cm$. The surface tension of a soap solution is $2.5 \times 10^{-2} \ N / m$.
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Answer

The relation between surface tension and surface energy:
$i.$ Let $\text{ABCD}$ be a rectangular frame of wire, fitted with a movable arm $\text{PQ.}$
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$ii.$ The frame held in a horizontal position is dipped into a soap solution and taken out so that a soap film $\text{APQB}$ is formed. Due to the surface tension of the soap solution, a force $‘F \ ’$ will act on each arm of the frame. Under the action of this force, the movable arm $ \text{PQ}$ moves towards $\text{AB.}$
$iii$.The magnitude of force due to surface tension is, $F = 2Tl .....(\because T = F/l)$
$($A factor of $2$ appears because soap film has two surfaces that are in contact with a wire.$)$
$iv.$ Let the wire $PQ$ be pulled outwards through a small distance $‘dx \ ’$ to the position $P \ ′Q \ ′,$ by applying an external force $F \ ′$ isothermally, which is equal and opposite to $F.$ Work done by this force, $dW = F \ ′dx = 2Tldx.$
$v.$ But, $2ldx = dA =$ increase in the area of two surfaces of the film.
$\therefore dW = T dA$
$vi.$This work done in stretching the film is stored in the area $dA$ in the form of potential energy $($surface energy$).$
$\therefore$ Surface energy, $E = T dA$
$\therefore \frac{E}{d A}=T$
Hence, surface tension $=$ surface energy per unit area.
$vii.$Thus, surface tension is equal to the mechanical work done per unit surface area of the liquid, which is also called surface energy.
Given:
$T = 2.5 \times 10^{-2} N/m, r_1 = 0 m, r_2 = 1 \ cm = 10^{-2}m, W =$ ?
Formula:
$W =2 T dA =2 T \times 4 \pi\left(r_2^2-r_1^2\right)$
$\therefore W=2 \times 2.5 \times 10^{-2} \times 4 \pi \times 10^{-4}$
$\therefore W=6.283 \times 10^{-5} J$

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