For a diatomic gas,the change in internal energy at constant pressure and the change in internal energy for a unit temperature change are $U_1$ and $U_2$ respectively. Then $U_1 : U_2$ is equal to:

$(a)$ 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?

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

$A$ 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 ...... $atm$.

An ideal gas $(\gamma = 1.5)$ is expanded adiabatically. How many times must the gas be expanded to reduce the root mean square velocity of the molecules $2.0$ times?

When the gas is heated,in which direction will the piston move?