Question

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)

• A. $[8p_1+\frac{24\sigma }{r}]$
• B. $[4p_1+\frac{24\sigma }{r}]$
• C. $[2p_1+\frac{24\sigma }{r}]$
• D. $[2p_1+\frac{12\sigma }{r}]$

Answer 1

The correct option is A $[8p_1+\frac{24\sigma }{r}]$

Initial pressure $P_1$ for bubble radius $r$ compression pressure $P_2$. Surface Tension $\sigma$.

Pressure gradient will be constant.

$P_1{V}_1=P_2{V}_2$

Apply Bernoulli'sTheorem

$\Rightarrow (P_1+\frac{4\sigma }{r})\left(\frac{4}{3}\pi r^3\right))=(P_2+\frac{4\sigma }{r/2}\left)\right(\frac{4}{3}\pi \frac{r^3}{8})$

$\Rightarrow 8P_1+\frac{32\sigma }{r}=P_2+\frac{8\sigma }{r}$

$\Rightarrow P_2=8P_1+\frac{24\sigma }{r}$