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. $\rho_{\ell} n R g T_{0} \frac{\left(P_{0}+\rho_{\ell} g H\right)^{2 / 5}}{\left(P_{0}+\rho_{\ell} g y\right)^{7 / 5}}$
• 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. $\rho_{\ell} nRgT _{0} \frac{\left( P _{0}+\rho_{\ell} gH \right)^{3 / 5}}{\left( P _{0}+\rho_{\ell} gy \right)^{8 / 5}}$
• D. $\frac{\rho_{\ell} nRgT _{0}}{\left( P _{0}+\rho_{\ell} gH \right)^{3 / 5}\left[ P _{0}+\rho_{\ell} g( H - y )\right]^{2 / 5}}$

Answer 1

Answer: (B)

$\rho_{\ell} Vg=$ Buoyancy force $=\rho_{\ell} g \frac{ nR T _{2}}{ P _{2}}$
$T _{2}= T _{0}\left[\frac{ P _{0}+\rho_{\ell} g ( H - y )}{ P +\rho_{\ell} gH }\right]^{2 / 5}$
$P _{2}= P _{0}+\rho_{\ell} g ( H - y )$