A spherical soap bubble inside an air chamber at pressure has a certain radius so that the excess pressure inside the bubble is . Now, the chamber pressure is reduced to so that the bubble radius and its excess pressure change. [2024]

Question

A spherical soap bubble inside an air chamber at pressure $P_{0} = 10^{5} Pa$ has a certain radius so that the excess pressure inside the bubble is $Δ P = 144 Pa$ . Now, the chamber pressure is reduced to $\frac{8 P_{0}}{27}$ 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 $Δ P$ in both the cases to be much smaller than the chamber pressure. The new excess pressure $Δ P$ in $P a$ is __________. [2024]

Smart Achievers · Accepted Answer

(96)

$Since temperature remains unchanged i.e., T = constant$

$So P_{1} V_{1} = P_{2} V_{2}$

$Δ P$

$Excess pressure, P_{1} = P_{0} + Δ P_{1} and Δ P_{1} = \frac{4 T}{R_{1}}$

$P_{2} = \frac{8 P_{0}}{27} + Δ P_{2} and Δ P_{2} = \frac{4 T}{R_{2}}$

$∴$ $(P_{0} + Δ P_{1}) \times \frac{4}{3} π R_{1}^{3} = (\frac{8 P_{0}}{27} + Δ P_{2}) \times \frac{4}{3} π R_{2}^{3}$

$or, (P_{0} + Δ P_{1}) {(\frac{4 T}{Δ P_{1}})}^{3} = (\frac{8 P_{0}}{27} + Δ P_{2}) {(\frac{4 T}{Δ P_{2}})}^{3}$ $[∵ Δ P_{1} ≪ P_{0}]$

$\frac{P_{0}}{{(Δ P_{1})}^{3}} \approx \frac{8 P_{0}}{27} \times \frac{1}{{(Δ P_{2})}^{3}}$

$or, Δ P_{2} = \frac{2}{3} Δ P_{1} = \frac{2}{3} \times (144 Pa)$

$∴$ $Δ P_{2} = 96 Pa$

Solution

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