Question

A small spherical monoatomic ideal gas bubble $\left(\gamma =\frac{5}{3}\right)$ is trapped inside a liquid of density ${\rho }_{\ell }$ (see figure). 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 bubble is at the bottom is $T_0$, the height of the liquid is $H$ and the atmospheric pressure is $P_0$ (Neglect surface tension).

The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)

• A. ${\rho }_{\ell }nRg{T}_0\frac{{\left(P_0+{\rho }_{\ell }gH\right)}^{2/5}}{{\left(P_0+{\rho }_{\ell }gy\right)}^{7/5}}$
• B. $\frac{{\rho }_{\ell }nRg{T}_0}{{\left(P_0+{\rho }_{\ell }gH\right)}^{2/5}{\left[P_0+{\rho }_{\ell }g(H-y)\right]}^{3/5}}$
• C. ${\rho }_{\ell }nRg{T}_0\frac{{\left(P_0+{\rho }_{\ell }gH\right)}^{3/5}}{{\left(P_0+{\rho }_{\ell }gy\right)}^{8/5}}$
• D. $\frac{{\rho }_{\ell }nRg{T}_0}{{\left(P_0+{\rho }_{\ell }gH\right)}^{3/5}{\left[P_0+{\rho }_{\ell }g(H-y)\right]}^{2/5}}$

Answer 1

Answer: (B)

${\rho }_{\ell }Vg=$ Buoyancy force $={\rho }_{\ell }g\frac{nR{T}_2}{P_2}$
$T_2=T_0{\left[\frac{P_0+{\rho }_{\ell }g(H-y)}{P+{\rho }_{\ell }gH}\right]}^{2/5}$
$P_2=P_0+{\rho }_{\ell }g(H-y)$