91Ó°ÊÓ

The excess pressure \( \Delta P \) inside a soap bubble can be calculated using the formula:\[ \Delta P = \frac{4T}{r} \]where \( T \) is the surface tension of the soap solution (\( T = 2.50 \times 10^{-2} \mathrm{~N/m} \)) and \( r \) is the radius of the bubble (\( r = 5.00 \times 10^{-3} \mathrm{~m} \)).Substituting these values gives:\[ \Delta P = \frac{4 \times 2.50 \times 10^{-2}}{5.00 \times 10^{-3}} = 20 \times 10^{-2} = 0.20 \mathrm{~N/m^2} \]

When the air bubble is at a depth inside a liquid, the pressure exerted by the liquid should also be considered. This pressure \( P \) is the sum of atmospheric pressure, the excess pressure due to the soap bubble, and the pressure due to the depth in the liquid.Calculate the pressure due to the depth:\[ P_{\text{depth}} = \rho g h \]where \( \rho \) is the relative density of the soap solution (converted to \( 1.20 \times 10^{3} \mathrm{~kg/m^3} \)), \( g \) is gravity \( 9.81 \mathrm{~m/s^2} \), and \( h = 0.40 \mathrm{~m} \).\[ P_{\text{depth}} = 1.20 \times 10^{3} \times 9.81 \times 0.40 = 470.88 \mathrm{~Pa} \]Now, calculate the total pressure inside the bubble:\[ P_{\text{inside}} = P_{\text{atm}} + \Delta P + P_{\text{depth}} \]\[ P_{\text{inside}} = 101 \times 10^{5} + 0.20 + 470.88 \]Simplifying gives:\[ P_{\text{inside}} = 1.005 \times 10^{5} \mathrm{~Pa} = 100500 \mathrm{~Pa} \]Hence, the pressure inside the bubble is approximately \( 1.06 \times 10^{5} \mathrm{~Pa} \).