Question 103: Thermodynamics & KTG

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 ______.

Accepted Answer

Case - 1
$$\begin{aligned}
& P-P_0=\Delta P=\frac{4 T}{R} \
& P=\left(P_0+\frac{4 T}{R}ight)
\end{aligned}$$
Case-2
$$\begin{aligned}
& \mathrm{P}_1-\frac{8 \mathrm{P}_0}{27}=\Delta \mathrm{P}_1=\frac{4 \mathrm{~T}}{\mathrm{R}_1} \
& \mathrm{P}_1=\frac{4 \mathrm{~T}}{\mathrm{R}_1}+\frac{8 \mathrm{P}_0}{27}
\end{aligned}$$
Constant temperature process
$$\begin{aligned}
& \mathrm{PV}=\mathrm{P}_1 \mathrm{~V}_1 \
& \left(\mathrm{P}_0+\frac{4 \mathrm{~T}}{\mathrm{R}}ight) \frac{4}{3} \pi \mathrm{R}^3=\left(\frac{4 \mathrm{~T}}{\mathrm{R}_1}+\frac{8 \mathrm{P}_0}{27}ight) \frac{4}{3} \pi \mathrm{R}_1^3 ;\left(\frac{4 \mathrm{~T}}{\mathrm{R}}ight),\left(\frac{4 \mathrm{~T}}{\mathrm{R}_1}ight) ightarrow 	ext { (Neglected) } \
& \mathrm{R}=\frac{2}{3} \mathrm{R}_1 \Rightarrow \mathrm{R}_1=\frac{3}{2} \mathrm{R} \
& \Delta \mathrm{P}_1=\frac{4 \mathrm{~T}}{\mathrm{R}_1}=\frac{4 \mathrm{~T}}{3 \mathrm{R}} 	imes 2=\frac{2}{3} 	imes(144)=96 \mathrm{~Pa}
\end{aligned}$$

Thermodynamics & KTG

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