Daily Physicist

André Lichnerowicz image from George M. Bergman under the Creative Commons Attribution-Share Alike 4.0 International License.

Today is the birthday of

• André Lichnerowicz (1915-1998).
  
• Daniel Chalonge (1895-1977).
  

Today’s Problem

Show that the internal energy of an ideal gas can be written in the following form

\[U=\frac{PV}{\gamma -1}\]

where \(\gamma =C_p/C_v\) and \(C_p\), \(C_v\) are the molar specific heats at constant pressure and volume.

Answer

If there are \(N\) molecules and each has \(f\) degrees of freedom then by the equipartition theorem for an ideal gas we have \[U=\frac{f}{2}NkT\] From the equation of state for an ideal gas \(NkT=PV\) so that \[U=\frac{f}{2}PV\] and we just need to show that \[\frac{f}{2}=\frac{1}{\gamma -1}\] For an ideal gas \[C_v=\frac{f}{2}R\] \[C_p=\frac{f+2}{2}R\] so that \[\gamma=\frac{C_p}{C_v}=\frac{f+2}{f}\] which can be rearranged to get \[\frac{f}{2}=\frac{1}{\gamma -1}\]