A cylinder with a movable piston contains air under a pressure p_… - Vidyadip

MCQ

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)

✓
$\left[ {8{p_1} + \frac{{24\sigma }}{r}} \right]$
B
$\left[ {4{p_1} + \frac{{24\sigma }}{r}} \right]$
C
$\left[ {2{p_1} + \frac{{24\sigma }}{r}} \right]$
D
$\left[ {2{p_1} + \frac{{12\sigma }}{r}} \right]$

✓

Answer

Correct option: A.

$\left[ {8{p_1} + \frac{{24\sigma }}{r}} \right]$

a
Lets say, initially, the pressure due to air inside the bubble is $\mathrm{P}_{\mathrm{air}}$

$\Rightarrow P_{\operatorname{air}}-P_{1}=\frac{4 \sigma}{r}$ $...(i)$

Finally, the radius becomes half; so volume becomes $\frac{1}{8}$ th and hence pressure becomes $8 \mathrm{P}_{\text {air: }}$

So $8 \mathrm{P}_{\mathrm{air}}-\mathrm{P}_{2}=\frac{4 \sigma}{\mathrm{r} / 2}$ $...(ii)$

Solving $(i)$ and $(ii)$

$\operatorname{get} \mathrm{P}_{2}=8 \mathrm{P}_{1}+\frac{24 \sigma}{\mathrm{r}}$

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