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Question

The gas inside a spherical bubble expands uniformly and slowly so that its radius increases from R to 2R. Let the atmospheric pressure be P0 and surface tension be S. The work done by the gas in the process is

A
28πP0R33+24πSR2
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B
25πP0R33+24πSR2
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C
28πP0R33+23πSR22
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D
none of these
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Solution

The correct option is A 28πP0R33+24πSR2
Since the bubble is expanding slowly it is a quasi static process.
Let at any intermidiate time the radius of the bubble is r. S is the surface tension.
Excess pressure at any given time(for two boundaries) is 2×2Sr
Pgas=Po+4Sr
Work done for a quasi static process, dW=PdV
W=dW=PdV

The gas is expanding againt the surface tension pressure and atmospheric pressure.P=Po+Psurface


Consider a small surface on the sphere, the surface tension acts tangentially.
The net force acting on this small cross sectional area.
The component of force along the tangent of the surface cancels out and the perpendicular component gets added. This happens to all the elements. The component of force along tangent cancels at every point and the force along the radial component gets added.
Psurface=ΔFrΔA=4Sr
dV=4πr2dr and P=Po+4Sr
The work done is
W=2RRPdV
=2RR(Po+4Sr)4πr2dr
After simplyfying we get
W=28πPoR33+24πSR2

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