Question

An adiabatic cylindrical chamber with a frictionless movable piston has been placed on a smooth horizontal surface as shown. One mole of an ideal monotonic gas is enclosed inside the chamber. Mass of the piston is $M$ and mass of the remaining chamber including the gas is $4M$. The gas is at atmospheric pressure and temperature. A particle of mass $M$ moving horizontally with speed $v$, strikes the piston elastically. Find the change in temperature of the gas when the compression is maximum. [$R$ is the ideal gas constant]

• A. $\Delta T=\frac{2M{V}^2}{3R}$
• B. $\Delta T=\frac{2M{V}^2}{15R}$
• C. $\Delta T=\frac{3M{V}^2}{7R}$
• D. $\Delta T=\frac{4M{V}^2}{15R}$

Answer 1

The correct option is D $\Delta T=\frac{4M{V}^2}{15R}$

The first figure shows the situation immediately after collision and the second figure shows situation at maximum compression.


Immediately after collision, the particle comes to rest and the piston starts moving with velocity $V$, because the collision is elastic and the two colliding objects have equal mass.
The compression is maximum when the piston stops moving relative to the cylinder.
Let $V_0=$ velocity of entire system at the point of maximum compression

Momentum conservation gives -
$(M+4M)V_0=Mv\Rightarrow V_0=\frac{V}{5}$
Energy conservation gives -
$n{C}_v\Delta T=\frac{1}{2}M{V}^2-\frac{1}{2}\left(5M\right){\left(\frac{V}{5}\right)}^2$
$\Rightarrow 1\times \frac{3}{2}R\Delta T=\frac{2}{5}M{V}^2$ $[\text{for monoatomic gas},C_v=\frac{3}{2}R]$
$\Rightarrow \Delta T=\frac{4M{V}^2}{15R}$