Question

An insulated container containing monoatomic gas of molar mass m is moving with a velocity v_o. If the container is suddenly stopped, find the change in temperature.

Answer 1

Since the container is suddenly stopped which is initially moving with velocity v₀, there is no time for the exchange of heat in the process.

The total KE of the container is transferred to gas molecules in the form of translational KE, thereby increasing the absolute temperature.

Let n be the no. of moles of the monoatomic gas in the container. Since the molar mass of the gas is m.

The total mass of the container, M = mn

KE of molecules due to velocity v₀,

KE = `1/2 (mn) v_0^2`   ......(i)

Final KE of gas = 0

Change in kinetic energy, `ΔK = 1/2 (nm)v^2` 

If ΔT = Change in absolute temperature,

Then the internal energy of the gas

`ΔU = nC_vΔT = n(3/2 R)ΔT`  .....(ii)

According to the conservation of mechanical energy, we get

ΔK = ΔU

By equations (i) and (ii), we get

⇒ `1/2 (mn)v_0^2 = n 3/2 R(ΔT)`

`(mn)v_0^2 = n3R(ΔT)`

⇒ ΔT = `((mn)v_0^2)/(3nR) = (mv_0^2)/(3R)`