At $27^\circ C$,a gas is suddenly compressed such that its pressure becomes $1/8$th of the original pressure. The temperature of the gas will be $(\gamma = 5/3)$.

$A$ 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).
$1.$ 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 $y$ from the bottom, its temperature is:
$(A)$ $T_0\left(\frac{P_0+\rho_{\ell} gH}{P_0+\rho_{\ell} gy}\right)^{2 / 5}$
$(B)$ $T_0\left(\frac{P_0+\rho_{\ell} g(H-y)}{P_0+\rho_{\ell} g H}\right)^{2 / 5}$
$(C)$ $T_0\left(\frac{P_0+\rho_{\ell} gH}{P_0+\rho_{\ell} gy}\right)^{3 / 5}$
$(D)$ $T_0\left(\frac{P_0+\rho_{\ell} g(H-y)}{P_0+\rho_{\ell} g H}\right)^{3 / 5}$
$3.$ The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant):
$(A)$ $\rho_{\ell} nRgT_0 \frac{\left(P_0+\rho_{\ell} gH\right)^{2 / 5}}{\left(P_0+\rho_{\ell} gy\right)^{7 / 5}}$
$(B)$ $\frac{\rho_{\ell} nRgT_0}{\left(P_0+\rho_{\ell} gH\right)^{2 / 5}\left[P_0+\rho_{\ell} g(H-y)\right]^{3 / 5}}$
$(C)$ $\rho_{\ell} nRgT_0 \frac{\left(P_0+\rho_{\ell} g H\right)^{3 / 5}}{\left(P_0+\rho_{\ell} g(H-y)\right)^{8 / 5}}$
$(D)$ $\frac{\rho_{\ell} nRgT_0}{\left(P_0+\rho_{\ell} gH\right)^{3 / 5}\left[P_0+\rho_{\ell} g(H-y)\right]^{2 / 5}}$
Give the answer for questions $1, 2,$ and $3.$

The volume of a gas is reduced adiabatically to $(1/4)^{th}$ of its initial volume. If the initial temperature is $27\,^{\circ}C$ and $\gamma = 1.4$, the new temperature is:

$A$ monoatomic gas at pressure $P$ and volume $V$ is suddenly compressed to one-eighth of its original volume. The final pressure at constant entropy will be $.....P$.

The process in which no heat enters or leaves the system is termed as

An ideal gas has a specific heat capacity at constant pressure of $\frac{11}{10} R$. If one mole of this ideal gas at $125^{\circ} C$ does $83 \,J$ of work adiabatically, then the final temperature of the gas would be (Universal gas constant, $R=8.3 \,J \,K^{-1} \,mol^{-1}$ ). (in $^{\circ} C$)