Buoyancy Force Acting on a Gas Bubble

To determine the buoyancy force acting on the gas bubble, we consider the thermodynamics of the rising bubble:

1. Pressure of the Bubble:
   Let $H$ be the depth of the liquid column and $y$ be the vertical height of the bubble measured from the bottom of the container.
   The depth of the bubble below the free surface of the liquid is $h = H - y$.
   The pressure $P$ inside the bubble at height $y$ is equal to the hydrostatic pressure at that depth:
   $P = P_0 + \rho_l g (H - y)$
   At the bottom ($y = 0$), the initial pressure is:
   $P_{\text{bottom}} = P_0 + \rho_l g H$

2. Adiabatic Expansion of the Gas:
   The gas inside the bubble consists of $n$ moles of a monoatomic ideal gas with an adiabatic index $\gamma = \frac{5}{3}$.
   Since the expansion is adiabatic, the temperature $T$ and pressure $P$ are related by:
   $P^{1-\gamma} T^\gamma = \text{constant} \implies T \propto P^{\frac{\gamma - 1}{\gamma}}$
   Substituting $\gamma = \frac{5}{3}$:
   $\frac{\gamma - 1}{\gamma} = \frac{\frac{5}{3} - 1}{\frac{5}{3}} = \frac{2}{5}$
   Given that the temperature of the gas at the bottom is $T_0$, the temperature at height $y$ is:
   $T = T_0 \left( \frac{P}{P_{\text{bottom}}} \right)^{2/5} = T_0 \left[ \frac{P_0 + \rho_l g (H - y)}{P_0 + \rho_l g H} \right]^{2/5}$

3. Volume of the Bubble:
   Using the ideal gas equation $P V = n R T$, the volume $V$ of the bubble at height $y$ is:
   $V = \frac{n R T}{P} = \frac{n R T_0}{P} \left( \frac{P}{P_{\text{bottom}}} \right)^{2/5} = \frac{n R T_0}{P_{\text{bottom}}^{2/5} P^{3/5}}$
   Substituting the expressions for $P_{\text{bottom}}$ and $P$:
   $V = \frac{n R T_0}{(P_0 + \rho_l g H)^{2/5} [P_0 + \rho_l g (H - y)]^{3/5}}$

4. Buoyancy Force:
   According to Archimedes' principle, the buoyant force $F_B$ exerted on the bubble by the liquid of density $\rho_l$ is:
   $F_B = \rho_l V g = \frac{\rho_l n R g T_0}{(P_0 + \rho_l g H)^{2/5} [P_0 + \rho_l g (H - y)]^{3/5}}$

Therefore, the correct option is (B).