A 2 mm diameter spherical air bubble is suspended in an insu

A 2 mm diameter, spherical air bubble is suspended in an insulating fluid. With a focused laser, the bubble is heated so that the bubble radius increases by 90%. For a spherical bubble the pressure difference between the bubble and surrounding fluid is given by delta P = 2row/r, where row is the interfacial tension and r is the radius of the bubble. Assuming the interfacial tension is a constant at .070 N/m, the initial bubble temperature is 300K, and the surrounding fluid has a pressure of 100 kPa calculate the mass of the bubble, the final temperature of the bubble, the work done by the gas, and the laser energy required to perform the expansion. Additionally, is this a realistic scenario? Why? What assumptions are most likely to fail?

Solution

Initial radius r = dia/2 = 2/2 = 1 mm = 0.001 m

Final radius R = r + 90/100*r = 1.9*r = 0.0019 m

Initial volume v = 4/3*pi*r^3 = 4/3*3.14*0.001^3 = 4.1866*10^-9 m^3

Final volume V = 4/3*pi*R^3 = 4/3*3.14*0.0019^3 = 2.8716*10^-8 m^3

Initial pressure difference dp = 2*sigma/r = 2*0.07/0.001 = 140 Pa = 0.14 kPa

Initial pressure in bubble p = 100 + 0.14 = 100.14 kPa

Initial Density = p / (RT) = 100.14*10^3 / (287*300) = 1.163 kg/m^3

Final pressure difference dP = 2*sigma/R = 2*0.07/0.0019 = 73.68 Pa = 0.07368 kPa

Final pressure in bubble P = 100 + 0.07368 = 100.07368 kPa

a)

Mass of the bubble = density*volume = 1.163*4.1866*10^-9 = 4.869*10^-9 kg

b)

Final density = mass / volume = 4.869*10^-9 / (2.8716*10^-8) = 0.1696 kg/m^3

P/RT = 0.1696

100.07368*10^3 / (287*T) = 0.1696

T = 2056 K

c)

For adibatic expansion, work done = (P2V2 - P1V1) / (n-1)

= (100.07368*2.8716*10^-8 - 100.14*4.1866*10^-9) / (1.4-1)

= 6.136*10^-6 kJ

d)

Q = m*Cp*(T2 - T1)

= 4.869*10^-9 * 1.005 * (2056 - 300)

= 8.593*10^-6 kJ