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Question: The radius of a soap bubble is \(r\). The surface tension of soap solution is \(S\). Keeping tempera...

The radius of a soap bubble is rr. The surface tension of soap solution is SS. Keeping temperature constant, the radius of the soap bubble is doubled, the energy necessary for this will be:
A. 24πr2S24\pi {r^2}S
B. 8πr2S8\pi {r^2}S
C. 12πr2S12\pi {r^2}S
D. 16πr2S16\pi {r^2}S

Explanation

Solution

Here we have to use the concept of surface tension. The inclination of liquid surfaces to shrink into the smallest available surface area is surface tension. Surface friction enables insects to float and slip on a water surface, typically denser than water.The results of intermolecular forces at the interface create surface tension. Surface friction depends on the composition of the liquid, temperature and the ambient environment. Liquids were molecules that would have a high surface tension and strong attractive intermolecular forces.

Complete step by step answer:
The surface tension is decreased so that bubbles can form when a soapy dish detergent is applied to water. This eliminates the pull or attraction that the molecules of water exert on each other, decreasing the solution's surface tension.

Given:
Original soap bubble radius =r = r.
Soap solution surface tension =T = T.
Final soap bubble radius =2r = 2r.
The initial energy required to blow the soap bubble is:
E1=2×4πr2×T ⇒E1=8πr2T  {E_1} = 2 \times 4\pi {r^2} \times T \\\ \Rightarrow{E_1}= 8\pi {r^2}T \\\
The final energy needed for the soap bubble to blow is:
{E_2} = 2 \times 4\pi \times {\left( {4r} \right)^2} \\\ \Rightarrow {E_2} = 32\pi {r^2}T \\\
Therefore the additional energy necessary is provided by
E2−E1=32πr2T−8πr2T ∴E2−E1=24πr2T  {E_2} - {E_1} = 32\pi {r^2}T - 8\pi {r^2}T \\\ \therefore {E_2} - {E_1} = 24\pi {r^2}T \\\
Hence, option A is correct.

Note: Here we have to also take the final radius of the soap bubble as radius should be taken from both sides. But if we take only one radius from both sides in confusion then the answer would be wrong. The soap molecules crowd hydrophobic ends to the surface, seeking to escape water, and stay out of the layer of water molecules. Water molecules differ from each other as a result. A reduction in surface tension is caused by the increased distance between the water molecules, causing bubbles to form.