Given an adiabatic cylindrical chamber with a frictionless movable piston has been placed on smooth horizontal surface. One mole of an ideal monoatomic gas is enclosed inside the Chamber. Take mass of

Question

Given an adiabatic cylindrical chamber with a frictionless movable piston has been placed on smooth horizontal surface. One mole of an ideal monoatomic gas is enclosed inside the Chamber. Take mass of piston be m and remaining mass of Chamber as 4m. The gas is at atmospheric pressure and temperature. Consider a particle of mass m moving horizontally with speed v strikes the piston elastically. The change in temperature of gas when compression of gas is maximum is found to be $\frac{nmv ^{2}}{15 R}$ , find value of n.

TestHub · Accepted Answer

Immediately after collision, the particle comes to rest; momentum conservation: $5 mV_{0} = mv ∴ V_{0} = \frac{v}{5}$

Energy conservation: $nC_{V} Δ T = \frac{1}{2} mv^{2} - \frac{1}{2} 5 m (\frac{v}{5})^{2}$

$1 \times \frac{3}{2} R Δ T = \frac{2}{5} mv^{2} (C_{v} = \frac{3 R}{2}) \Rightarrow Δ T = \frac{4 mv ^{2}}{15 R} \Rightarrow n = 4$

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