1. Pressure on the bubble at depth

Let the liquid surface be at height $H$ from the bottom, and the bubble be at height $y$ from the bottom.

So the depth of the bubble below the liquid surface is: $h = H-y$

Hence the pressure of the liquid surrounding the bubble at that point is: $P = P_0 + \rho_l g (H-y)$

At the bottom ($y=0$), the bubble pressure is: $P_b = P_0 + \rho_l g H$

2. Nature of the process

The bubble does not exchange heat with the liquid, so the expansion is adiabatic.

For an adiabatic process of an ideal gas: $PV^\gamma = \text{constant}$

Also, using ideal gas law and adiabatic relations: $TV^{\gamma-1} = \text{constant}$

or equivalently, $T^\gamma P^{1-\gamma} = \text{constant}$

From this, $T \propto P^{\frac{\gamma-1}{\gamma}}$

Therefore, $\frac{T}{T_0} = \left(\frac{P}{P_b}\right)^{\frac{\gamma-1}{\gamma}}$

3. Substitute $\gamma = \frac{5}{3}$

We get: $\frac{\gamma-1}{\gamma} = \frac{\frac{5}{3}-1}{\frac{5}{3}} = \frac{\frac{2}{3}}{\frac{5}{3}} = \frac{2}{5}$

So, $\frac{T}{T_0} = \left(\frac{P}{P_b}\right)^{2/5}$

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

4. Match with options

This is exactly Option B: $T_0{\left( \frac{{P_0 + \rho _l g(H - y)}}{{P_0 + \rho _l gH}} \right)^{2/5}}$

5. Check other options briefly

• A uses $P_0+\rho_l gy$, which is incorrect because pressure depends on depth below the surface, i.e. $H-y$.
• C has wrong exponent $3/5$.
• D also has wrong exponent $3/5$.

Therefore, the correct option is B.