A spherical drop of liquid splits into 1000 identical spherical drops. If $$u_i$$ is the surface energy of the original drop and $$u_f$$ is the total surface energy of the resulting drops, the (ignoring evaporation), $$\frac{u_f}{u_i} = (\frac{10}{x})$$. Then value of $$x$$ is _____:

Surface energy is directly proportional to surface area.

Let radius of original drop be R.

Surface energy of original drop :

$$u_i \propto 4\pi R^2$$

The drop splits into 1000 identical drops.

If radius of each small drop is r, then by conservation of volume :

$$\frac{4}{3}\pi R^3 = 1000 \times \frac{4}{3}\pi r^3$$

$$R^3 = 1000r^3$$

$$R = 10r$$

$$r = \frac{R}{10}$$

Total surface area of 1000 drops :

$$= 1000 \times 4\pi r^2$$

$$= 1000 \times 4\pi \left(\frac{R}{10}\right)^2$$

$$= 10 \times 4\pi R^2$$

Hence,

$$\frac{u_f}{u_i} = 10$$

Given,

$$\frac{u_f}{u_i} = \left(\frac{1}{x}\right)^{-1} = x$$

Therefore,

$$x = 10$$

Final Answer :

$$10$$