One mole of the ideal gas goes through the process $p=p_0\left[1-\alpha\left(\frac{V}{V_0}\right)^3\right]$, where $p$ and $V$ are pressure and volume, $p_0, V_0$ and $\alpha$ are constants. If the maximum attainable temperature of the gas is $\left(\frac{3}{4}\right) \frac{p_0 V_0}{R}$, then the value of $\alpha$ is

Two containers of equal volume contain the same gas at pressures $P_1$ and $P_2$ and absolute temperatures $T_1$ and $T_2$ respectively. On joining the vessels,the gas reaches a common pressure $P$ and common temperature $T$. The ratio $P/T$ is equal to

An insulated system contains $4$ moles of an ideal diatomic gas at temperature $T$. When a heat $Q$ is supplied to the gas,$2$ moles of the gas is dissociated into atoms and the temperature remains constant. Then the relation between $Q$ and $T$ is ($R=$ universal gas constant.)

Which graph represents the variation of $PV$ with $V$ for a fixed mass of an ideal gas at constant temperature?

The density of water is $1000 \; kg \; m^{-3}$. The density of water vapour at $100 \; ^{\circ}C$ and $1 \; atm$ pressure is $0.6 \; kg \; m^{-3}$. The volume of a molecule multiplied by the total number gives what is called molecular volume. Estimate the ratio (or fraction) of the molecular volume to the total volume occupied by the water vapour under the above conditions of temperature and pressure.

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: