Question

A spherical bubble inside water has radius $R.$ Take the pressure inside the bubble and the water pressure to be $p_{0}.$ The bubble now gets compressed radially in an adiabatic manner so that its radius becomes $\left(\right.R-a\left.\right).$ For $a < < R$ the magnitude of the work done in the process is given by $\left(4 \pi p_{0} \left(Ra\right)^{2}\right)X,$ , where $X$ is a constant and $\gamma =\frac{C_{P}}{C_{v}}=\frac{41}{30}.$ The value of $X$ is _______ correct to nearest integer.

Answer 1

$W =\left(\Delta P \times 4 \pi R ^{2} a =\left|\frac{ dP }{2} \cdot 4 \pi R ^{2} a \right|\right.$
{for small change $(\Delta P < P >$ Arithmetic mean]
$\Rightarrow PV^{\gamma }=c\Rightarrow dP=-\gamma \frac{P}{V}dV=-\frac{\gamma P_{0}}{ V}4\pi R^{2}a$
$=\frac{\gamma P_{0}}{2 V}\times 4\pi R^{2}a\times 4\pi R^{2}a$
$=\frac{\gamma P_{0}}{2 \times 4 \pi R^{3}}4\pi R^{2}a\times 4\pi R^{2}a$
$=\left(4 \pi RP \times a^{2}\right)\frac{3 \gamma }{2}$
$\therefore X \simeq 2.05\approx2$