First slide

Surface tension

Question

A certain number of spherical drops of a liquid of radius r coalesce to form a single drop of radius R and volume V. If T is the surface tension of the liquid, then

Difficult

A

$energy = 4 V T (\frac{1}{r} - \frac{1}{R}) is released.$
B

$energy = 3 V T (\frac{1}{r} + \frac{1}{R}) is absorbed.$
C

$energy = 3 V T (\frac{1}{r} - \frac{1}{R}) is released.$
D

energy is neither released nor absorbed

Solution

Let n droplets each of radius r coalesce to form a big drop of radius R.
$∴$ Volume of n droplets = Volume of big drop
$n \times \frac{4}{3} π r^{3} = \frac{4}{3} π R^{3}$
$n = \frac{R^{3}}{r^{3}} . . . . (i)$
$Volume of big drop, V = \frac{4}{3} π R^{3} . . . . (i i)$
$Initial surface area of n droplets,$
$\begin{aligned} A_{i} = n \times 4 π r^{2} = \frac{R^{3}}{r^{3}} \times 4 π r^{2} \\ (Using . . . (i)) \end{aligned}$
$= 4 π \frac{R^{3}}{r} = (\frac{4}{3} π R^{3}) \frac{3}{r} = \frac{3 V}{r} (Using . . . . . (ii))$
Final surface area of big drop $A_{f} = 4 π R^{2} = (\frac{4}{3} π R^{3}) \frac{3}{R} = \frac{3 V}{R} (Using . . . . (iii))$
Decrease in surface area
$Δ A = A_{i} - A_{f} = \frac{3 V}{r} - \frac{3 V}{R} = 3 V (\frac{1}{r} - \frac{1}{R})$
$∴ Energy released = Surface tension \times Decrease in surface area$
$\begin{array}{l} = T \times Δ A \\ = 3 V T (\frac{1}{r} - \frac{1}{R}) \end{array}$

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