In a cylinder-piston arrangement, air is under a pressure P1. A soap bubble of radius r lies inside the cylinder, soap bubble has surface tension T. The radius of soap bubble is to be reduced to half, The new pressure P2 to which air should be compressed isothermally. (Assume r is very small as compared to height of cylinder)

Engineering

Physics

Degree of Freedom Internal Energy of Gas and Specific Heat Capacity of Gases

Thermodynamics

Surface Tension and Surface Energy

Question

In a cylinder-piston arrangement, air is under a pressure P₁. A soap bubble of radius r lies inside the cylinder, soap bubble has surface tension T. The radius of soap bubble is to be reduced to half, The new pressure P₂ to which air should be compressed isothermally. (Assume r is very small as compared to height of cylinder)

$4 P_{1} + \frac{12 T}{r}$

$8 P_{1} + \frac{24 T}{r}$

$P_{1} + \frac{8 T}{r}$

$P_{1} + \frac{4 T}{r}$

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$P = P_{1} + \frac{4 s}{r}$

$(P_{1} + \frac{4 S}{r}) \times (\frac{4}{3} {πr}^{3}) = (P_{2} + \frac{8 S}{r}) (\frac{4}{3} π \times {(\frac{r}{2})}^{3})$

$P_{2} + \frac{8 S}{r} = 8 P_{1} + \frac{32 S}{r}$

$P_{2} = 8 P_{1} + \frac{24 S}{r}$