Daily Physicist

Today is the birthday of

• Lev Davidovich Landau (1908-1968).
  Get a Lev Landau ceramic mug.
  
• Friedrich Georg “Fritz” Houtermans (1903-1966).

Today’s Problem

An ideal gas with pressure \(P_i\) and volume \(V_i\) is contained in an insulated cylinder by a piston of mass \(m_0\). A mass \(m_1\) is dropped onto the piston, compressing the gas. Find the ratio of the final volume to the initial volume, \(V_f/V_i\), when the piston comes to rest.

Answer

Let \(m=m_0+m_1\) be the final mass of the piston, \(h\) be the distance that the piston falls, and \(A\) be the area of the piston. Using the expression derived yesterday for the internal energy of an ideal gas, and the conservation of energy, we have

\[\frac{P_fV_f}{\gamma -1}=\frac{P_iV_i}{\gamma -1}+mgh\]

Substituting \(mg=P_fA\) and \(h=(V_i-V_f)/A\) into this equation we get

\[\frac{P_fV_f}{\gamma -1}=\frac{P_iV_i}{\gamma -1}+P_f(V_i-V_f)\]

This equation can be solved for the compression ratio.

\[\frac{V_f}{V_i}=\frac{\gamma -1}{\gamma}+\frac{P_i}{\gamma P_f}\]

The ratio of pressures is

\[\frac{P_i}{P_f}=\frac{m_0g/A}{mg/A}=\frac{m_0}{m_0+m_1}\]

Reference

Irreversible Adiabatic Compression of an Ideal Gas
Mungan, Carl E.
The Physics Teacher, Volume 41, Issue 8, pp. 450-453 (2003).
November 2003