A 2 mm diameter, spherical air bubble is suspended in an insulating fluid . with

Question

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 spherical bubble the pressure difference between the bubble and surrounding fluid is given by delta P = 2s/r , where s is the interfacial tension and r is the radius of the bubble . assuming the interfacial tension is constant at 0.70 N/m, the initial bubble temperature is 300K, and the surrounding fluid has a pressure of 100 kPa calculate (a) the mass of the bubble , (b) the final temperature of the bubble ,(c) the work done by the gas , and(d) the laser energy required to perform the expansion.
Additionally , is this is a realistic scenario?? Why? What assumption are most likely to fail? Note the volume of the bubble is related to the radius by V=(4/3)pr3

Explanation / Answer

initial pressure =100*1000+ 2*0.7/0.001 = 101400 Pa, n= no of moles = PV/RT =101400*(4/3 *pi*0.001^3)/8.314*300 = 1.703*10^-7

a) therefore mass of air inside bubble = ( 1.703*10^-7 ) * 29 = 4.938*10^-6 g = 4.938 micro gram

b)final radius = 0.001*1.9 = 0.0019 m

final P= 100*1000 + 2*0.7/0.0019 = 100736.842 Pa

therefore temperature = PV/nR = 100736.842 * (4/3 * pi * 0.0019^3) / 1.703*10^-7 * 8.314 = 2044.15 K

c) work done by the gas = P* delta V = (100*1000)(4/3*pi)(0.0019^3 - 0.001^3) = 2.45 mJ

d) as from first law of thermodynamics , change in internal energy = -work done by gas + heat given,

here change in internal energy = m*cp*delta T , here cp = air specific heat capacity (at 1200 K )= 1.173 KJ/kg-K, hence change in internal energy = 4.938*10^-6 * 1.173 *1000*(2044.15 - 300) = 10.1025 J , hence heat given = 10.1025 + 2.45 *10^-3 = 10.105J

d) This doesn't seem to be realistic scenario, because at such a higer temperature ( around 2000 K ) bubble would definitely burst off, here assumption for "surface tension to be constatnt" is making it unrealistic.

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