Problem — Cylinder that falls and stops

Problem

A rigid, insulated cylinder contains $n = 5$ mol of air (diatomic, $f=5$) at $T = 290$ K. The cylinder is dropped from a height of $h = 10$ m and comes to rest on hitting the ground (the cylinder does not deform or exchange heat). By how much does the temperature of the air inside it increase? Use the average molar mass of air $M \approx 29\cdot 10^{-3}$ kg/mol.

Solution

Falling from $h$, the gas gains macroscopic kinetic energy equal to the potential energy lost; on impact, since the cylinder is rigid (no work) and insulated (no heat), this energy converts into internal energy of the gas.

Mass of the air contained: $M_\text{gas} = nM = 5\cdot 29\cdot 10^{-3} = 0{,}145\ \text{kg}$ Potential energy converted into internal energy: $\Delta E_\text{int} = M_\text{gas}\,g\,h = 0{,}145\cdot 9{,}8\cdot 10 \approx 14{,}2\ \text{J}$ For a diatomic gas ($f=5$), $\Delta E_\text{int} = \tfrac{5}{2}nR\,\Delta T$, so: $\Delta T = \frac{2\,\Delta E_\text{int}}{5nR} = \frac{2\cdot 14{,}2}{5\cdot 5\cdot 8{,}31}$ $\ev{\Delta T \approx 0{,}14\ \text{K}}$

Connections

Topics: Teoria cinetica dei gas Concepts: Energia interna · Temperatura Skills: Interpretazione micro-macro