A spherical bubble inside water has radius R. | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For an adiabatic process,$PV^{\gamma} = 	ext{constant}$.Differentiating,$V^{\gamma} dP + \gamma P V^{\gamma-1} dV = 0$,which gives $dP = -\gamma

Vedclass · Accepted Answer

$2.05$

Vedclass · Answer

(C) For an adiabatic process,$PV^{\gamma} = 	ext{constant}$.Differentiating,$V^{\gamma} dP + \gamma P V^{\gamma-1} dV = 0$,which gives $dP = -\gamma P \frac{dV}{V}$.The change in volume $dV$ for a small change in radius $a$ is $dV = 4 \pi R^2 a$.The pressure change is $\Delta P = |dP| = \gamma P_0 \frac{4 \pi R^2 a}{\frac{4}{3} \pi R^3} = \frac{3 \gamma P_0 a}{R}$.The work done $W$ is approximately the average pressure change multiplied by the change in volume:$W = \frac{1}{2} (\Delta P) (dV) = \frac{1}{2} \left( \frac{3 \gamma P_0 a}{R} ight) (4 \pi R^2 a) = 6 \pi \gamma P_0 R a^2$.Given $W = (4 \pi P_0 R a^2) X$,we equate:$4 \pi P_0 R a^2 X = 6 \pi \gamma P_0 R a^2 \Rightarrow X = \frac{6 \gamma}{4} = 1.5 \gamma$.Substituting $\gamma = 41/30$:$X = 1.5 	imes \frac{41}{30} = \frac{3}{2} 	imes \frac{41}{30} = \frac{41}{20} = 2.05$.

Similar Questions

$5.6 \ L$ of Helium gas at $STP$ is compressed adiabatically to $0.7 \ L$. If the initial temperature is $T_1$,then the work done during the process is...?

By what factor should the volume of an ideal gas $(\gamma = 1.5)$ be increased through an adiabatic expansion so that its $rms$ speed becomes half?

You feel refreshed by taking a shower in summer but not in winter. Why?

Two moles of an ideal monoatomic gas at $27^{\circ}C$ occupies a volume of $V$. If the gas is expanded adiabatically to the volume $2V$,then the work done by the gas will be ....... $J$ $[\gamma = 5/3, R = 8.31 \text{ J/mol K}]$

The pressure and density of a diatomic gas $(\gamma = 7/5)$ change adiabatically from $(P, d)$ to $(P', d')$. If $\frac{d'}{d} = 32$,then $\frac{P'}{P}$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module