An ideal gas is filled in an insulated cylinder fitted with a movable piston connected with an ideal spring (of force constant $\mathrm{K}$) as shown in figure. Initially spring is relaxed and gas in the piston has volume ${\mathrm{V}}_0$ and temperature ${\mathrm{T}}_0$. Now the gas in the cylinder is heated slowly so that the piston slide towards right to a distance $\mathrm{x}$. Find final pressure and temperature of the gas. (Given atmospheric pressure $={\mathrm{P}}_0$ and area of cross-section of the cylinder $=\mathrm{A}$ ).

Two identical vessel $\mathrm{A}$ and $\mathrm{B}$ each of volume ${\mathrm{V}}_0$ joined together with a small tube (as shown in the figure) initially containing the same gas at pressure ${\mathrm{P}}_0$ and absolute temperature ${\mathrm{T}}_0$ The vessel $\mathrm{A}$ is now heated to maintain its temperature at $2{\mathrm{T}}_0$ while the vessel $\mathrm{B}$ is maintained at the same temperature ${\mathrm{T}}_0$.

Find the number of moles of the gas in each vessel.

Two identical vessel $\mathrm{A}$ and $\mathrm{B}$ each of volume ${\mathrm{V}}_0$ joined together with a small tube (as shown in the figure) initially containing the same gas at pressure ${\mathrm{P}}_0$ and absolute temperature ${\mathrm{T}}_0$ The vessel $\mathrm{A}$ is now heated to maintain its temperature at $2{\mathrm{T}}_0$ while the vessel $\mathrm{B}$ is maintained at the same temperature ${\mathrm{T}}_{\mathrm{n}}$.

Find the common pressure of the gas.

A thermally insulated vessel, with gaseous nitrogen, at a temperature $T= 27 °\mathrm{C}$, moves with velocity $v=100 \mathrm{m} {\mathrm{s}}^{-1}$. How much (in per cent) and in what way will the gas pressure change, on a sudden stoppage of the vessel?