Temperature of Rising Gas Bubble in Liquid Column

To find the temperature of the gas bubble as a function of height $y$ from the bottom of the container, we analyze the thermodynamics of the process:

1. Pressure of the Bubble:
   Neglecting surface tension, the pressure inside the bubble is equal to the hydrostatic pressure of the liquid at that depth plus the atmospheric pressure $P_0$.

   • At the bottom of the liquid column (depth $= H$), the pressure is:
     $P_{\text{bottom}} = P_0 + \rho_l g H$
     The temperature of the gas at this position is given as $T_0$.
   • At a height $y$ from the bottom, the depth of the bubble below the free liquid surface is $(H - y)$. Hence, the pressure inside the bubble is:
     $P(y) = P_0 + \rho_l g (H - y)$

2. Thermodynamic Process:
   Since the bubble does not exchange any heat with the surrounding liquid, the expansion of the gas as it rises is an adiabatic process.

   For an adiabatic process involving an ideal gas:
   $P^{1-\gamma} T^\gamma = \text{constant} \implies T \propto P^{\frac{\gamma - 1}{\gamma}}$

3. Calculation of Temperature:
   For a monoatomic ideal gas, the adiabatic index is $\gamma = \frac{5}{3}$.
   The exponent is:
   $\frac{\gamma - 1}{\gamma} = \frac{\frac{5}{3} - 1}{\frac{5}{3}} = \frac{2/3}{5/3} = \frac{2}{5}$

   Applying the relation between the initial state (at the bottom) and the state at height $y$:
   $\frac{T(y)}{T_0} = \left(\frac{P(y)}{P_{\text{bottom}}}\right)^{\frac{\gamma - 1}{\gamma}}$

   Substituting the pressures:
   $T(y) = T_0 \left(\frac{P_0 + \rho_l g (H - y)}{P_0 + \rho_l g H}\right)^{2/5}$

Therefore, the correct option is B.