13. Two spherical liquid droplets coalesce to form one large spherical droplet. The initial
radius of each droplet is r, the surface tension the liquid is s, and its density is d. The
energy released in this process depends on
a. only s
b. s and r
c. s and d
d. s, r and d

Concept: Surface Energy and Coalescence

The energy released when two equal spherical liquid droplets of radius r coalesce into one larger spherical droplet depends only on the surface tension s and the radius r, not on the density d.
Hence, the correct answer is: (b) s and r.

Surface Energy Formula

For a liquid droplet, surface energy E is given by:

E = s × (surface area) = s × 4πR²

where s is the surface tension and R is the droplet radius.

When Droplets Merge

When two droplets merge, the total volume remains constant but the total surface area decreases.
This reduction in surface area causes a decrease in surface energy, and the difference is released as energy.

For two identical droplets of radius r:

• Total initial volume: 2 × (4/3)πr³
• Final droplet radius (R): R = 2^1/3r
• Initial surface area: 2 × 4πr² = 8πr²
• Final surface area: 4πR² = 4π(2^2/3r²)

The energy released (E) is proportional to the change in surface area (ΔA):

E = s × ΔA

Thus, the energy depends only on s and r.

Why Density Does Not Appear

Density d would matter only if the question related the released energy to kinetic energy or temperature rise in the final droplet.
Here, the question focuses solely on the change in surface energy, which depends on surface area and surface tension, not on density.

Evaluation of Each Option

• (a) only s: Wrong. The surface energy involves area, which depends on radius. Without r, energy cannot be fixed.
• (b) s and r: Correct. The change in surface energy is s × ΔA, and ΔA depends on r for two identical droplets.
• (c) s and d: Wrong. Density is irrelevant to surface energy change. It matters only if we convert that energy into motion or heating.
• (d) s, r, and d: Wrong. It includes unnecessary density; the released energy depends solely on surface tension and geometry (radius).