An ideal gas has $N$ molecules in a closed box at temperature $T_1$ and pressure $P_1$. If the number of molecules in the box is doubled and the total kinetic energy is kept the same,what will be the new pressure $P_2$ and temperature $T_2$?

What is/are the same for $O_2$ and $NH_3$ in gaseous state?

The internal energy of $10 \, g$ of nitrogen at $S.T.P.$ is about ......... $J$.

$A$ horizontal uniform glass tube of $100 \ cm$ length, sealed at both ends, contains a $10 \ cm$ mercury column in the middle. The temperature and pressure of air on either side of the mercury column are $31^{\circ} C$ and $76 \ cm$ of mercury, respectively. If the air column at one end is kept at $0^{\circ} C$ and the other end at $273^{\circ} C$, then the pressure of air which is at $0^{\circ} C$ is (in $cm$ of $Hg$):

$(a)$ When a molecule (or an elastic ball) hits a massive wall,it rebounds with the same speed. When a ball hits a massive bat held firmly,the same thing happens. However,when the bat is moving towards the ball,the ball rebounds with a different speed. Does the ball move faster or slower?
$(b)$ When gas in a cylinder is compressed by pushing in a piston,its temperature rises. Guess at an explanation of this in terms of kinetic theory using $(a)$ above.
$(c)$ What happens when a compressed gas pushes a piston out and expands? What would you observe?
$(d)$ Sachin Tendulkar used a heavy cricket bat while playing. Did it help him in any way?

$A$ 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 $\,g$)