The pressure and density of a diatomic gas with $\gamma = 7/5$ change adiabatically from $(P, d)$ to $(P', d').$ If $\frac{d'}{d} = 32,$ then $\frac{P'}{P}$ is

The work done in an adiabatic change in a gas depends only on

In a thermodynamic process,if $\Delta W$ and $\Delta U$ are the work done and the change in the internal energy of a system respectively,then:

Compressed air in the tube of a bicycle wheel at normal temperature suddenly starts coming out from a puncture. The air inside:

The following figure shows an adiabatic cylindrical container of volume $V_0$ divided by an adiabatic smooth piston (area of cross-section = $A$) into two equal parts. An ideal gas $(C_P/C_V = \gamma)$ is at pressure $P_1$ and temperature $T_1$ in the left part,and a gas at pressure $P_2$ and temperature $T_2$ is in the right part. The piston is slowly displaced and released at a position where it can stay in equilibrium. The final pressure of the two parts will be (Suppose $x$ = displacement of the piston):

$A$ 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 $3.$