An engine takes in $5$ moles of air at $20\,^{\circ}C$ and $1\,atm$,and compresses it adiabatically to $1/10^{\text{th}}$ of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules,the change in its internal energy during this process is $X\,kJ$. The value of $X$ to the nearest integer is

$A$ 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 $3.$

An ideal gas at $27^{\circ}\text{C}$ is compressed adiabatically to $(8/27)$ of its original volume. If $\gamma = 5/3$, then the rise in temperature of a gas is (in $\text{ K}$)

One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $100 K$ and the universal gas constant $R = 8.0 J mol^{-1} K^{-1}$,the decrease in its internal energy,in Joule,is. . . . .

$A$ diatomic gas of volume $2 \ m^3$ at pressure $2 \times 10^5 \ N \ m^{-2}$ is compressed adiabatically to a volume $0.5 \ m^3$. The work done in this process is, $[$Use $4^{1.4} = 6.96]$

An ideal gas at a pressure of $1 \text{ atm}$ and temperature of $27^{\circ}C$ is compressed adiabatically until its pressure becomes $8$ times the initial pressure. The final temperature is ..... $^{\circ}C$ (given $\gamma = 3/2$).