Question

A small spherical monoatomic ideal gas bubble $(\gamma =5/3)$ is trapped inside a liquid of density ${\rho }_1$ (see fig). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubbles is at the bottom is $T_0$, the height of the liquid is H and the atmospheric pressure is $P_0$ (neglect surface tension). When the gas bubble is at height $y$ from the bottom, its temperature is:

• A. $T_0{\left(\frac{P_0+{\rho }_{1}gH}{P_0+{\rho }_{1}gy}\right)}^{\frac{2}{5}}$
• B. $T_0{\left(\frac{P_0+{\rho }_{1}g(H-y)}{P_0+{\rho }_{1}gH}\right)}^{\frac{2}{3}}$
• C. $T_0{\left(\frac{P_0+{\rho }_{1}gH}{P_0+{\rho }_{1}gy}\right)}^{\frac{3}{5}}$
• D. $T_0{\left(\frac{P_0+{\rho }_{1}g(H-y)}{P_0+{\rho }_{1}gy}\right)}^{\frac{3}{5}}$

Answer 1

The correct option is B $T_0{\left(\frac{P_0+{\rho }_{1}g(H-y)}{P_0+{\rho }_{1}gH}\right)}^{\frac{2}{3}}$

Since the process is adiabatic,

$P{V}^{\gamma }=constant$ for gas inside bubble.

Thus $P^{1-\gamma }{T}^{\gamma }=constant$

$P_{bottom}^{1-\gamma }{T}_{bottom}^{\gamma }=$ $P_y^{1-\gamma }{T}_y^{\gamma }=constant$

$⟹(P_0+{\rho }_{1}gH{)}^{-2/3}(T_0{)}^{5/3}=(P_0+{\rho }_{1}g(H-y{)}^{-2/3})T^{5/3}$

$⟹T=T_0(\frac{P_0+{\rho }_{1}g(H-y)}{P_0+{\rho }_{1}gH}{)}^{\frac{2}{3}}$