A mercury drop of radius 1 cm is sprayed into 10^6 droplets of equal size, Calculate the energy

step 1 so let's solve this question when a drop of radius r is separated into end droplets of equal rad

A mercury drop of radius 1 cm is sprayed into 10^6 droplets...

A mercury drop of radius $1 \mathrm{~cm}$ is sprayed into $10^{6}$ droplets of equal size, Calculate the energy expended if surface tension of mercury is $35 \times 10^{-3} \mathrm{~N} / \mathrm{m}$.

Key Concepts

Geometric Scaling of Surface Area

The geometric relationship between volume and surface area in spheres plays a critical role when a drop is divided into smaller droplets. While the volume remains constant, breaking a large sphere into many smaller spheres increases the total surface area because surface area scales with the square of the radius, whereas volume scales with the cube. This scaling law explains why a greater amount of energy is required to create the additional surfaces when splitting a drop into many tiny droplets.

Conservation of Energy in Surface Modification

In processes that alter the surface of a liquid, such as breaking up a drop into smaller droplets, the principle of energy conservation is applied by equating the work done (or energy expended) to the increase in the surface energy of the system. This approach provides a framework for determining the energy required to overcome intermolecular forces and expand the surface area.

Surface Tension

Surface tension is a physical property of liquids that represents the cohesive forces acting at the interface between the liquid and another medium (often air). It is defined as the force per unit length and is responsible for minimizing the surface area of the liquid. In many problems, the work done to increase the surface area of the liquid is directly related to the surface tension, making it a key parameter when calculating energies associated with changes in surface configurations.

Surface Energy

Surface energy is the energy required to create a new surface area in a liquid, and it is directly proportional to the surface tension. When a liquid is split into droplets, additional surface area is created, and the associated increase in energy is given by the product of surface tension and the newly formed area. This concept is crucial in understanding how energy is expended when a single drop is divided into many smaller droplets.

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