A spherical bubble inside water has radius R. Take the pressure inside the bubble and the water pressure to be p0. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes (R − a). For a << R the magnitude of the work done in the process is given by (4πp0Ra2)X, where X is a constant and γ = Cp/CV = 41/30. The value of X is_______

Engineering

Physics

Thermodynamics

Liquid Pressure

Question

A spherical bubble inside water has radius R. Take the pressure inside the bubble and the water pressure to be p₀. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes (R − a). For a << R the magnitude of the work done in the process is given by (4πp₀Ra²)X, where X is a constant and γ = C_p/C_V = 41/30. The value of X is________.

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W = ( $∆$ P)_avg × 4πR²a

$= | \frac{dP}{2} . 4 {πR}^{2} a |$

{for small change ( $∆$ P)_avg <P> arithmetic mean}

$= {PV}^{γ} = c \Rightarrow dP = - γ \frac{P}{V} dV = - \frac{{γP}_{0}}{V} 4 {πR}^{2} a$

$= \frac{{γP}_{0}}{2 V} \times 4 {πR}^{2} a \times 4 {πR}^{2} a$

$= \frac{{γP}_{0}}{2 \times 4 {πR}^{3}} \times 4 {πR}^{2} a \times 4 {πR}^{2} a$

$= (4 pRP \times a^{2}) \frac{3 γ}{2}$

$∴$ x ; 2.05