$A$ thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and it is suddenly brought to rest. Assuming no heat is lost to the surroundings,its temperature increases by

$A$ closed vessel contains a mixture of two diatomic gases $A$ and $B$. Molar mass of $A$ is $16$ times that of $B$ $(M_A = 16 M_B)$ and mass of gas $A$ contained in the vessel is $2$ times that of $B$ $(m_A = 2 m_B)$. Which of the following statements are true?
$(i)$ Average kinetic energy per molecule of $A$ is equal to that of $B$.
$(ii)$ Root mean square value of translational velocity of $B$ is four times that of $A$.
$(iii)$ Pressure exerted by $B$ is eight times that exerted by $A$.
$(iv)$ Number of molecules of $B$ in the cylinder is eight times that of $A$.

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3 = \text{constant}$. The coefficient of volume expansion of the gas is . . . . . . .

$A$ hot air balloon with a payload rises in the air. Assume that the balloon is spherical in shape with a diameter of $11.7 \, m$ and the mass of the balloon and the payload (without the hot air inside) is $210 \, kg$. The temperature and pressure of the outside air are $27^{\circ} C$ and $1 \, atm = 10^5 \, N/m^2$,respectively. The molar mass of dry air is $30 \, g/mol$. The temperature of the hot air inside is close to .......... $^{\circ} C$. [The gas constant,$R = 8.31 \, J K^{-1} mol^{-1}$]

$A$ vessel containing nitrogen gas is supplied a heat of $498 \,J$, so as to raise the temperature of the gas by $40^{\circ} C$ at constant pressure. The mass of nitrogen gas in the vessel is (Molecular mass of nitrogen $= 28 \,g/mol$; Universal gas constant $= 8.3 \,J \,mol^{-1} \,K^{-1}$) (in $\,g$)

For a mass $m$ of an ideal gas at constant pressure $P$,the volume versus temperature graph is represented by the straight line $B$. If the mass is changed to $2m$ and the pressure is changed to $2P$,which straight line will represent the new state?