You feel refreshed by taking a shower in summer but not in winter. Why?

$A$ monoatomic ideal gas, initially at temperature $T_1$, is enclosed in a cylinder fitted with a massless, frictionless piston. By releasing the piston suddenly, the gas is allowed to expand adiabatically to a temperature $T_2$. If $L_1$ and $L_2$ are the lengths of the gas columns before and after expansion respectively, then $(T_2 / T_1)$ is given by

$A$ sample of gas at temperature $T$ is adiabatically expanded to double its volume. The work done by the gas in the process is (given, $\gamma = 3/2$):

$A$ diatomic gas $\left(\gamma = \frac{7}{5}\right)$ is compressed adiabatically to volume $\frac{V_0}{32}$,where $V_0$ is its initial volume. The initial temperature of the gas is $T_i$ in Kelvin and the final temperature is $xT_i$ in Kelvin. The value of $x$ 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 is expanded adiabatically. How many times has the gas to be expanded to reduce the r.m.s. speed of molecules $3$ times (in $\times$)? $(\gamma = 1.5)$