Question

A big water drop is formed by the combination of 'n' small water drops of equal radii. The ratio of the surface energy of 'n 'drops to the surface energy of big drop is ______.

विकल्प

• n² : 1

• n : 1

• ${\displaystyle \sqrt{\text{n}}:1}$

• ${\displaystyle \sqrt[3]{\text{n}}:1}$

Answer 1

A big water drop is formed by the combination of 'n' small water drops of equal radii. The ratio of the surface energy of 'n 'drops to the surface energy of big drop is ${\displaystyle \underline{\sqrt[3]{\text{n}}:1}}$.

Explanation:

Let radius of big drop = R

and of small drop = r

Volume of big drop = n (Volume of small drop)

${\displaystyle \frac{4}{3}\pi {\text{R}}^3=\text{n}.\frac{4}{3}\pi {\text{r}}^3}$ 

R³ = nr³ ⇒ R = n^1/3.r

Surface energy of n drops,

E₂ = n × 4πr² × T 

Surface energy of big drops,

E₁ = 4πR²T

∴ ${\displaystyle \frac{{\text{E}}_2}{{\text{E}}_1}=\frac{{\text{nr}}^2}{{\text{R}}^2}=\frac{{\text{nr}}^2}{{({\text{n}}^{\frac{1}{3}}.\text{r})}^2}}$

= ${\displaystyle \frac{{\text{nr}}^2}{{\text{n}}^{2/3}.{\text{r}}^2}={\text{n}}^{1/3}}$   [∵ R = n^1/3.r]