Question #42190

A thermally insulated rectangular box of length $L$, width $W$, and height $H$ contains $n$ moles of an ideal monoatomic gas of molar mass $M$. The box is placed in a uniform downward gravitational field $g$. Assume the gas is in thermal equilibrium at an initial temperature $T_0$, and the gravitational potential energy is small compared to the thermal energy ($MgH \ll RT_0$, where $R$ is the universal gas constant). The vertical wall at $x=L$ (of area $W \times H$) is flexible. An external agent quasi-statically pushes this wall inwards such that the inward displacement of the wall at a height $y$ from the bottom is given by $\delta(y) = \delta_0 \frac{y}{H}$, where $\delta_0 \ll L$. Find the change in temperature $\Delta T$ of the gas. (Keep terms up to first order in $\frac{MgH}{RT_0}$).