Mechanical Properties of Fluids

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Question

What is the excess pressure inside a bubble of soap solution of radius 5.00 mm, given that the surface tension of soap solution at the temperature (20 °C) is 2.50 × 10^–2 N m^–1? If an air bubble of the same dimension were formed at depth of 40.0 cm inside a container containing the soap solution (of relative density 1.20), what would be the pressure inside the bubble? (1 atmospheric pressure is 1.01 × 10⁵ Pa).

Solution

Excess space pressure space inside space the space soap space bubble space is space 20 space Pa Pressure space inside space the space air space bubble space is space 1.06 space cross times space 10 to the power of 5 space Pa Soap space bubble space is space of space radius comma space straight r space equals space 5.00 space mm space equals space 5 space cross times space 10 to the power of – 3 end exponent space straight m Surface space tension space of space the space soap space solution comma space straight S space equals space 2.50 space cross times space 10 to the power of – 2 space end exponent Nm to the power of – 1 end exponent Relative space density space of space the space soap space solution space equals space 1.20 Therefore comma space Density space of space the space soap space solution comma space straight rho space equals space 1.2 space cross times space 10 to the power of 3 space end exponent kg divided by straight m cubed Air space bubble space formed space at space straight a space depth comma space straight h space equals space 40 space cm space equals space 0.4 space straight m Radius space of space the space air space bubble comma space straight r space equals space 5 space mm space equals space 5 space cross times space 10 to the power of – 3 end exponent space straight m 1 space atmospheric space pressure space equals space 1.01 space cross times space 10 to the power of 5 space Pa Acceleration space due space to space gravity comma space straight g space equals space 9.8 space straight m divided by straight s squared So comma space excess space pressure space inside space the space soap space bubble space is comma space straight P thin space equals space fraction numerator 4 straight S over denominator straight r end fraction space equals space fraction numerator 4 cross times 2.5 cross times 10 to the power of negative 2 end exponent over denominator 5 space cross times space 10 to the power of negative 3 end exponent end fraction space equals space 20 space Pa space Now comma space Excess space pressure space inside space the space air space bubble space is space given space by comma space straight P apostrophe equals fraction numerator 2 straight S over denominator straight r end fraction space equals space fraction numerator 2 cross times 2.5 cross times 10 to the power of negative 2 end exponent over denominator 5 space cross times space 10 to the power of negative 3 end exponent end fraction space equals space 10 space Pa Total space pressure space inside space the space air space bubble comma space at space straight a space depth space of space 0.4 space straight m space is comma space equals Atmospheric space pressure space plus space hρg space plus space straight P ’ equals 1.01 space cross times space 10 to the power of 5 space end exponent plus space 0.4 space cross times space 1.2 space cross times space 10 cubed space cross times space 9.8 space plus space 10 equals 1.06 space cross times space 10 to the power of 5 space Pa Therefore comma space Pressure space inside space the space air space bubble space equals space 1.06 space cross times space 10 to the power of 5 space Pa

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Mechanical Properties Of Fluids

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