.075~ \mathrm{Nm}^{-1}$ and density. Tags and topics: JEE MAIN,JEE Advanced,JEE Mains,KCET,Mechanical Properties of Fluids,Pressure,Surface Tension,NCERT,NEET, JEE Main Physics,Class 11 Physics,Class 12 Physics.">
.075~ \mathrm{Nm}^{-1}$ and density. Tags and topics: JEE MAIN,JEE Advanced,JEE Mains,KCET,Mechanical Properties of Fluids,Pressure,Surface Tension,NCERT,NEET, JEE Main Physics,Class 11 Physics,Class 12 Physics.">
.075~ \mathrm{Nm}^{-1}$ and density. Tags and topics: JEE MAIN,JEE Advanced,JEE Mains,KCET,Mechanical Properties of Fluids,Pressure,Surface Tension,NCERT,NEET, JEE Main Physics,Class 11 Physics,Class 12 Physics.">
There is an air bubble of radius $1.0 \mathrm{~mm}$ in a liquid of surface tension $0.075~ \mathrm{Nm}^{-1}$ and density $1000 \mathrm{~kg} \mathrm{~m}^{-3}$ at a depth of $10 \mathrm{~cm}$ below the free surface. The amount by which the pressure inside the bubble is greater than the atmospheric pressure is _________ $\mathrm{Pa}\left(\mathrm{g}=10 \mathrm{~ms}^{-2}\right)$
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Step-by-Step Solution
Step 1: Identify the Known Quantities
• Radius of the air bubble, R = 1.0 \times 10^{-3} \text{ m}
• Surface tension of the liquid, T = 0.075 \text{ Nm}^{-1}
• Density of the liquid, \rho = 1000 \text{ kg m}^{-3}
• Depth of the bubble below the surface, h = 0.10 \text{ m}
• Acceleration due to gravity, g = 10 \text{ m s}^{-2}
Step 2: Compute the Excess Pressure Due to Surface Tension
For an air bubble inside a liquid, the excess pressure due to surface tension is given by the Young–Laplace equation:
\Delta P_{\text{surface}} = 2 \frac{T}{R}.
Substitute the values:
\Delta P_{\text{surface}} = 2 \times \frac{0.075}{1.0 \times 10^{-3}} = 2 \times 75 = 150 \text{ Pa}.
Step 3: Compute the Hydrostatic Pressure Contribution
Because the bubble lies h meters below the surface, the hydrostatic pressure is
P_{\text{hydrostatic}} = \rho g h.
Substitute the values:
P_{\text{hydrostatic}} = 1000 \times 10 \times 0.10 = 1000 \text{ Pa}.
Step 4: Calculate the Total Excess Pressure
The total excess pressure inside the bubble above the atmospheric pressure is the sum of the surface tension contribution and the hydrostatic pressure:
\Delta P_{\text{total}} = \Delta P_{\text{surface}} + P_{\text{hydrostatic}} = 150 + 1000 = 1150 \text{ Pa}.
Final Answer
Hence, the pressure inside the bubble is greater than the atmospheric pressure by 1150 \text{ Pa} .