A spherical soap bubble inside an air chamber at pressure P0 = 10^5 Pa has a certain radius so that

Question

A spherical soap bubble inside an air chamber at pressure \(P_0=10^5 \mathrm{~Pa}\) has a certain radius so that the excess pressure inside the bubble is \(\Delta P=144 \mathrm{~Pa}\). Now, the chamber pressure is reduced to \(8 P_0 / 27\) so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure \(\Delta P\) in both the cases to be much smaller than the chamber pressure. The new excess pressure \(\Delta P\) in \(\mathrm{Pa}\) is _____.

Answer 1

Correct answer: 96

Case-1

\(\mathrm{P}-\mathrm{P}_0=\Delta \mathrm{P}=\frac{4 \mathrm{~T}}{\mathrm{R}}\)

\(\mathrm{P}=\left(\mathrm{P}_0+\frac{4 \mathrm{~T}}{\mathrm{R}}\right)\)

Case-2

\(\mathrm{P}_1-\frac{8 \mathrm{P}_0}{27}=\Delta \mathrm{P}_1=\frac{4 \mathrm{~T}}{\mathrm{R}_1}\)

\( \mathrm{P}_1=\frac{4 \mathrm{~T}}{\mathrm{R}_1}+\frac{8 \mathrm{P}_0}{27}\)

Constant temperature process

\(\mathrm{PV}=\mathrm{P}_1 \mathrm{~V}_1 \)

\(\left(\mathrm{P}_0+\frac{4 \mathrm{~T}}{\mathrm{R}}\right) \frac{4}{3} \pi \mathrm{R}^3=\left(\frac{4 \mathrm{~T}}{\mathrm{R}_1}+\frac{8 \mathrm{P}_0}{27}\right) \frac{4}{3} \pi \mathrm{R}_1^3 ;\)

\(\left(\frac{4 \mathrm{~T}}{\mathrm{R}}\right)\left(\frac{4 \mathrm{~T}}{\mathrm{R}_1}\right) \rightarrow \text { (Neglected) }\)

\(\mathrm{R}=\frac{2}{3} \mathrm{R}_1 \Rightarrow \mathrm{R}_1=\frac{3}{2} \mathrm{R} \)

\( \Delta \mathrm{P}_1=\frac{4 \mathrm{~T}}{\mathrm{R}_1}=\frac{4 \mathrm{~T}}{3 \mathrm{R}} \times 2=\frac{2}{3} \times(144)=96 \mathrm{~Pa}\)

Comments

what is the term 4T/R in the place of excess pressure in case 1. please explain how that term was derived