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 $\mathrm{T}_0$, the height of the liquid is $\mathrm{H}$ and the atmospheric pressure is $\mathrm{P}_0$ (Neglect surface tension).
Figure: $Image$
$1.$ As the bubble moves upwards, besides the buoyancy force the following forces are acting on it
$(A)$ Only the force of gravity
$(B)$ The force due to gravity and the force due to the pressure of the liquid
$(C)$ The force due to gravity, the force due to the pressure of the liquid and the force due to viscosity of the liquid
$(D)$ The force due to gravity and the force due to viscosity of the liquid
$2.$ When the gas bubble is at a height $\mathrm{y}$ from the bottom, its temperature is
$(A)$ $\mathrm{T}_0\left(\frac{\mathrm{P}_0+\rho_0 \mathrm{gH}}{\mathrm{P}_0+\rho_t \mathrm{gy}}\right)^{2 / 5}$
$(B)$ $T_0\left(\frac{P_0+\rho_t g(H-y)}{P_0+\rho_t g H}\right)^{2 / 5}$
$(C)$ $\mathrm{T}_0\left(\frac{\mathrm{P}_0+\rho_t \mathrm{gH}}{\mathrm{P}_0+\rho_t \mathrm{gy}}\right)^{3 / 5}$
$(D)$ $T_0\left(\frac{P_0+\rho_t g(H-y)}{P_0+\rho_t g H}\right)^{3 / 5}$
$3.$ The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)
$(A)$ $\rho_t \mathrm{nRgT}_0 \frac{\left(\mathrm{P}_0+\rho_t \mathrm{gH}\right)^{2 / 5}}{\left(\mathrm{P}_0+\rho_t \mathrm{gy}\right)^{7 / 5}}$
$(B)$ $\frac{\rho_{\ell} \mathrm{nRgT}_0}{\left(\mathrm{P}_0+\rho_{\ell} \mathrm{gH}\right)^{2 / 5}\left[\mathrm{P}_0+\rho_{\ell} \mathrm{g}(\mathrm{H}-\mathrm{y})\right]^{3 / 5}}$
$(C)$ $\rho_t \mathrm{nRgT} \frac{\left(\mathrm{P}_0+\rho_t g \mathrm{H}\right)^{3 / 5}}{\left(\mathrm{P}_0+\rho_t g \mathrm{~g}\right)^{8 / 5}}$
$(D)$ $\frac{\rho_{\ell} \mathrm{nRgT}_0}{\left(\mathrm{P}_0+\rho_{\ell} \mathrm{gH}\right)^{3 / 5}\left[\mathrm{P}_0+\rho_t \mathrm{~g}(\mathrm{H}-\mathrm{y})\right]^{2 / 5}}$
Give the answer question $1,2,$ and $3.$
IIT 2008, Advanced
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$1.$ Buoyant force is resultant of pressure-force of liquid.
$2.$ $ \mathrm{P}_1^{1-\gamma} \mathrm{T}_1^\gamma=\mathrm{P}_2^{l-\gamma} \mathrm{T}_2^\gamma $
$ \mathrm{P}_1=\mathrm{P}_0+\rho_{\ell} \mathrm{gH}, \quad \mathrm{T}_1=\mathrm{T}_0 $
$ \mathrm{P}_2=\mathrm{P}_0+\rho_{\ell} \mathrm{g}(\mathrm{H}-\mathrm{y})$
$3.$ $ \rho_{\ell} \mathrm{Vg}=\text { Buoyancy force }=\rho_{\ell} g \frac{n R T_2}{P_2} $
$ \mathrm{~T}_2=\mathrm{T}_0\left[\frac{\mathrm{P}_0+\rho_t g(\mathrm{H}-\mathrm{y})}{\mathrm{P}+\rho_t \mathrm{gH}}\right]^{2 / 5} $
$ \mathrm{P}_2=\mathrm{P}_0+\rho_{\ell} \mathrm{g}(\mathrm{H}-\mathrm{y})$
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