A big drop is formed by coalescing 1000 small identical drops of water. If be the total surface energy of 1000 small drops of water and be the surface energy of single big drop of water, the is where x=

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Solution

Q.
A big drop is formed by coalescing 1000 small identical drops of water. If $E_{1}$ be the total surface energy of 1000 small drops of water and $E_{2}$ be the surface energy of single big drop of water, the $E_{1} : E_{0}$ is $x : 1$ where $x =$ __________ . [2024]

Ans.
(10)   Let radius of small drops = r

  Let radius of big drops = R

  Here volume will remain conserve (volume conservation)

   $∴ (1000) V_{s} = (1) V_{b} \to volume of big drop$

   $Volume of small drop \Rightarrow 10^{3} (\frac{4}{3} π r^{3}) = \frac{4}{3} π R^{3}$

   $R = 10 r$

   $We know, surface tension = \frac{Surface energy}{Area}$

   $∴ Surface energy = (Surface tension) (Area)$

   $\Rightarrow ε_{1} = T \times 4 π r^{2} \times 10^{3}$

   $ε_{2} = T \times 4 π R^{2}$

   $∴ \frac{ε_{1}}{ε_{2}} = \frac{(10^{3}) (r^{2})}{R^{2}} = \frac{(10^{3}) r^{2}}{{(10 r)}^{2}} = \frac{10}{1}$

$\frac{ε_{1}}{ε_{2}} = \frac{10}{1} = \frac{X}{1},$

   $∴ x = 10$

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