Which of the following statements is true?

For two different gases $X$ and $Y$, having degrees of freedom $f_1$ and $f_2$ and molar heat capacities at constant volume $C_{V1}$ and $C_{V2}$ respectively, the $\ln P$ versus $\ln V$ graph is plotted for an adiabatic process, as shown.

If the potential energy of a gas molecule is $U = \frac{M}{r^6} - \frac{N}{r^{12}}$,where $M$ and $N$ are positive constants,then the potential energy at equilibrium must be

The heat energy that must be supplied to $14 \text{ g}$ of nitrogen at room temperature to raise its temperature by $48^{\circ}C$ at constant pressure is ($R$ = gas constant, Molecular weight of nitrogen $(N_2) = 28$) (in $R$)

$(a)$ 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 contains $14 \, g$ ($7$ moles) of hydrogen and $96 \, g$ ($3$ moles) of oxygen at $STP$. $A$ chemical reaction is induced by passing an electric spark in the vessel until one of the gases is consumed. The temperature is brought back to its starting value of $273 \, K$. The pressure in the vessel is ...... $atm$.