A monoatomic gas of mass 4.0 u is kept in an insulated container. The container is moving with velocity 30 m s$$^{-1}$$. If the container is suddenly stopped then a change in temperature of the gas ($$R$$ = gas constant) is $$\frac{x}{3R}$$. Value of $$x$$ is,

A monoatomic gas (molar mass $$M = 4.0\,\text{g/mol}$$, i.e., Helium) is kept in an insulated container moving with velocity $$v = 30\,\text{m/s}$$. When the container is suddenly stopped, the kinetic energy of the bulk motion is entirely converted into internal (thermal) energy of the gas.

For $$n$$ moles of gas, the kinetic energy of bulk motion is $$\frac{1}{2}(nM)v^2$$, where $$M$$ is the molar mass. For a monoatomic ideal gas, the change in internal energy is $$\Delta U = n \cdot \frac{3}{2}R\Delta T$$.

Setting these equal: $$\frac{1}{2}nMv^2 = \frac{3}{2}nR\Delta T$$.

Solving for $$\Delta T$$: $$\Delta T = \frac{Mv^2}{3R}$$.

Substituting $$M = 4$$ and $$v = 30\,\text{m/s}$$: $$\Delta T = \frac{4 \times (30)^2}{3R} = \frac{4 \times 900}{3R} = \frac{3600}{3R}$$.

Comparing with the given expression $$\Delta T = \frac{x}{3R}$$, we get $$x = 3600$$.