A cylinder with a movable piston contains air under a pressure `p_(1)` and a soap bubble of radius `r`. The pressure `p_(2)` to which the air should be compressed by slowly pushing the piston into the cylinder for the soap bubble to reduce its size by half will be : (The surface tension is `sigma`, and the temperature `T` is maintained constant)

Options

A. `[8p_(1) + (24sigma)/(r)]`
B. `[4p_(1) + (24sigma)/(r)]`
C. `[2p_(1) + (24sigma)/(r)]`
D. `[2p_(1) + (12sigma)/(r)]`

Answer

A

Explanation

Let say, initially, the pressure due to air inside the bubble is `P_(air)`.
`rArr P_(air) - P_(1) = (4sigma)/(r) ……(i)`
Finally, the radius becomes half, so volume becomes `(1)/(8)` th and hence pressure becomes `8P_(air)`.
So, `8P_(air) - P_(2) = (4sigma)/(r//2)......(ii)`
Solving `(i)` and `(ii)`
get `P_(2) = 8P_(1) + (24sigma)/(r)`