First slide

Surface tension

Question

A number of water droplets each of radius coalesce to form a droplet of radius fl. The rise in temperature d $θ$ is

Difficult

A

$(2 T / rJ)$
B

$(3 T / rJ)$
C

$\frac{3 T}{J} (\frac{1}{r} - \frac{1}{R})$
D

$\frac{3 T}{J} (\frac{1}{r} + \frac{1}{R})$

Solution

Let R be the radius of a single drop. Then
$\frac{4}{3} {πR}^{3} = n \times \frac{4}{3} {πr}^{3} or R^{3} = {nr}^{3} Decrease in surface area = (4 {πr}^{2} n - 4 {πR}^{2}) = 4 π [{nr}^{2} - R^{2}] = 4 π [\frac{{nr}^{3}}{r} - R^{2}] = 4 π [\frac{R^{3}}{r} - R^{2}] = 4 {πR}^{3} [\frac{1}{r} - \frac{1}{R}] Heat produced = \frac{4 {πR}^{3} T}{J} [\frac{1}{r} - \frac{1}{R}] Rise in temperature d 0 is given by \frac{4}{3} {πR}^{3} \times 1 \times dθ = \frac{4 {πR}^{3} T}{J} [\frac{1}{r} - \frac{1}{R}] dθ = \frac{3 T}{J} [\frac{1}{r} - \frac{1}{R}]$

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