Inside pressure in a hollow soap bubble in the air is : (where d is the diameter of the bubble)

ANS:D -

The expression 8σ/d​ represents the inside pressure in a hollow soap bubble in the air, where σ is the surface tension of the soap solution and d is the diameter of the bubble. The formula is derived from the Young-Laplace equation, which describes the relationship between the pressure inside a curved liquid interface (such as a soap bubble) and the surface tension of the liquid and the curvature of the interface. The Young-Laplace equation states: P = 2σ/r​ Where:

• P is the pressure difference across the interface,
• σ is the surface tension of the soap solution,
• r is the radius of curvature of the bubble.

For a soap bubble, the curvature is inversely proportional to the diameter. So, if d is the diameter of the bubble, r=d/2​. Substituting r=d/2​ into the Young-Laplace equation, we get: P= 2σ​ /(d/2)= 4/dσ​ However, this expression considers the pressure difference across one interface of the bubble. Since a soap bubble has two interfaces (one on the inside and one on the outside), we need to double the pressure to get the total pressure difference. Therefore, the total pressure difference P inside the soap bubble is: P=2 × 4σ/d ​= 8σ/d​ Hence, 8σ/d​ represents the inside pressure in a soap bubble. As the diameter of the bubble decreases, the pressure inside the bubble increases due to the surface tension of the soap solution.