Question

A big drop is formed by coalescing 1000 small droplets of water. The surface energy will become :

• A. 100 times
• B. 10 times
• C. $\frac{1}{100}th$
• D. $\frac{1}{10}th$

Answer 1

Answer: (D)

$\frac{1}{10}th$

Answer 2

Given: - A big drop is formed by combining 1000 small droplets.

Step 1: Volume Conservation

Since the droplets coalesce to form one big drop, the total volume remains constant. Let $r$ be the radius of each small droplet and $R$ be the radius of the big drop.

The volume of one small droplet is:

$V_{\text{small}} = \frac{4}{3} \pi r^3$

The total volume of 1000 small droplets is:

$V_{\text{total}} = 1000 \times \frac{4}{3} \pi r^3 = \frac{4000}{3} \pi r^3$

The volume of the big drop is:

$V_{\text{big}} = \frac{4}{3} \pi R^3$

Equating the total volumes:

$\frac{4000}{3} \pi r^3 = \frac{4}{3} \pi R^3$

Simplifying:

$R^3 = 1000r^3$

Taking the cube root on both sides:

$R = 10r$

Step 2: Surface Area Calculation

The surface area of one small droplet is:

$A_{\text{small}} = 4 \pi r^2$

The total surface area of 1000 small droplets is:

$A_{\text{total}} = 1000 \times 4 \pi r^2 = 4000 \pi r^2$

The surface area of the big drop is:

$A_{\text{big}} = 4 \pi R^2 = 4 \pi (10r)^2 = 4 \pi \times 100r^2 = 400 \pi r^2$

Step 3: Surface Energy Comparison

Surface energy is directly proportional to the surface area. Let $E_{\text{small}}$ and $E_{\text{big}}$ be the surface energies of the small droplets and the big drop, respectively. The ratio of the surface energies is:

$\frac{E_{\text{big}}}{E_{\text{total}}} = \frac{A_{\text{big}}}{A_{\text{total}}} = \frac{400 \pi r^2}{4000 \pi r^2} = \frac{1}{10}$

Conclusion:

The surface energy will become $\frac{1}{10}$th of its original value.