The R\"uchardt experiment revisited: using simple theory, accurate measurement and python based data analysis

This project uses the R\"uchardt experiment to determine the ratio of specific heats and hence the number of degrees of freedom $f$ of different gases by measuring the frequency of damped simple harmonic motion where the gas provides the Hooke's law like spring of a cylinder-piston system. This project links mechanics, electromagnetism, thermodynamics, statistical mechanics and quantum mechanics making it an excellent synoptic experiment for a mid year undergraduate student. We present simple derivations of the main relationships that govern the experiment, a detailed data analysis of the physics of the apparatus and of the experimental data. We find $f(\textrm{He}) = 3.48 \pm 0.14$, $f(\textrm{N}_2) = 4.92 \pm 0.24$, $f(\textrm{Air}) = 4.96 \pm 0.25$, $f(\textrm{CO}_2) = 6.46 \pm 0.39$ at room temperature and atmospheric pressure. The results for CO$_2$ requires a statistical analysis of its vibrational modes. These results show that the expected results can be measured using fairly simple apparatus, coupled with careful analysis of large data sets.