Question

A bubble forms on the ocean floor far away from the coast. Assume the density of the salt water is 1.05 g/cm', and that the ocean is 0.600 km deep at this point. The pressure in the bubble is equal to the pressure in the surrounding water. a) What is the absolute pressure (in Pa) in the water at this depth? Assume that the air above the ocean is at a pressure of 1.013 x 105 Pa. Be careful with significant figures. b) The bubble initially has a radius of 0.750 cm, and the temperature of air in the bubble is 3.5°C. How many moles of gas are in the bubble? (Possibly useful information: R = 8.31 J/(mol-K); the volume of a sphere = nr3) c) As the bubble rises to the surface, the pressure drops and the temperature of the water increases to 23.5°C. The air in the bubble is in thermal equilibrium with the surrounding water. What is the radius of the bubble (in cm) as it comes to the surface? d) Find the speed of sound (in m/s) in the air in the bubble as it comes to the surface. e) When the bubble pops, the sound it makes has a wavelength equal to twice the diameter of the bubble. What is the frequency (in Hz) of the lowest tone it produces?

Answer 1

(a) pressure P at 0.6 km deep inside sea water that has density 1.05 g/cm3 P = pressure at surface + pressure due to water column = 1.013 X 105 + (Pgh) P = 1.013 X 105 + ( 1050 X 9.8 X 600 ) = 62.753 X 105 Pa (b) for ideal gas, we have, PV = nRT where Vis volume = (4/3)#r3, where r is radius n is number of moles, R is gas constant, Tis temperature in Kelvin n=(PV)/(RT) = ( 62.753 X 105x (4/3)7X (0.75)3x 10-6)/(8.31 X 276.5) n = 4.826 X 10-3 mole (c) Radius at surface we have , PV = nRT V = (n RT)/P= (4.826 X 10-3x 8.31 x 296.5) // 1.013 X 105) = 1.1738 X 10-4 m3 Radius :-(4/3) r3 = 1.1738 X 10-4 radius r = 3.037 cm

(d) velocity of sound v is given by, WRT V = VM where 7 is adiabatic constant of air, R is gas constant and M is molar mass of air for air 7 =1.4 and M = 29 g velocity of sound = [(1.4 X 8.31 X 296.5) / ( 29 X 10-3) 11/ 2 345 m/s (e) diameter of bubble = 2 x 3.037 = 6.074 cm wavelength = 2d = 12.148 cm velocity v = frequency X wavelength frequency = velocity / wavelength = 345 / ( 12.148 X 10-2) 2840 Hz