The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting,movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and $1$ moles of an ideal gas,respectively. In the left compartment,the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium,the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions,if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$,then the value of $\alpha$ is:

An ideal gas of molar mass $M$ is contained in a very tall vertical cylindrical column in a uniform gravitational field. Assuming the gas temperature to be $T$,the height at which the centre of gravity of the gas is located is (where $R$ is the universal gas constant).

One kg of a diatomic gas is at a pressure of $8 \times 10^4 \; N/m^2$. The density of the gas is $4 \; kg/m^3$. What is the energy (in $\times 10^4 \; J$) of the gas due to its thermal motion?

$A$ thermally isolated cylindrical closed vessel of height $8 \ m$ is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass $8.3 \ kg$. Thus,the partition is held initially at a distance of $4 \ m$ from the top. Each of the two parts of the vessel contains $0.1 \ mol$ of an ideal gas at temperature $300 \ K$. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached,the distance of the partition from the top (in $m$) will be. . . . . . (take the acceleration due to gravity $g = 10 \ m/s^2$ and the universal gas constant $R = 8.3 \ J \ mol^{-1} \ K^{-1}$).

During an experiment,an ideal gas is found to obey a condition $\frac{P^2}{\rho} = \text{constant}$ [$\rho = \text{density of the gas}$]. The gas is initially at temperature $T$,pressure $P$,and density $\rho$. The gas expands such that density changes to $\rho/2$.

An insulated container contains $4$ moles of an ideal diatomic gas at temperature $T$. Heat $Q$ is supplied to the gas,causing $2$ moles of the gas to dissociate into atoms. If the temperature of the gas remains constant,then: