= Clément-Desormes experiment
{c}
The Clément-Desormes experiment infers <heat capacity ratio> from a rapid <pressure> reduction followed by reheating. Use a rigid bottle, pump or bellows, a quick release valve, and a U-tube <pressure> gauge connected to the bottle and atmosphere. First pump to a small excess <pressure>, close the inlet, and wait for <thermal equilibrium> with the room at $T_0$. Measure $p_1>p_a$, where $p_a$ is local absolute atmospheric <pressure>. Briefly open the release valve until the bottle reaches $p_a$, then close it promptly. The remaining <gas> is cooler than the room. As it warms back to $T_0$ at fixed bottle <volume (thermodynamics)>, its <pressure> rises to $p_3$, with $p_a<p_3<p_1$ in the ideal procedure. The rebound reveals cooling during release. https://labs.physik.kit.edu/prakt-nebenfach-manuals/Experiments/Adiabatenexponent_Luft/[KIT Faculty of Physics, Clément-Desormes laboratory protocol: release and pressure rebound]{ref}
The bottle loses <molecules> while the valve is open: it is not a <closed system> whose geometrical <volume (thermodynamics)> expands. In the well-mixed, negligible-<heat transfer> discharge model with constant <heat capacity ratio>, the remaining <gas> obeys $T_2/T_0=(p_a/p_1)^{(\gamma-1)/\gamma}$. Subsequent heating at fixed <amount of substance> and bottle <volume (thermodynamics)> gives $p_3/p_a=T_0/T_2$ by the <ideal gas law>. Eliminating the unmeasured $T_2$ yields
$$
\gamma=\frac{\ln(p_1/p_a)}{\ln(p_1/p_3)}
\simeq\frac{\Delta p_1}{\Delta p_1-\Delta p_3}
=\frac{h_1}{h_1-h_3},\qquad
\Delta p_i=p_i-p_a=\rho_{\rm liquid}gh_i.
$$
The <logarithms> use absolute <pressure> ratios. The approximation requires small <gauge pressures> $\Delta p_i\ll p_a$; $h_i$ are signed height differences between the two liquid surfaces, and the final equality uses the same liquid in both readings with <hydrostatic equilibrium>. It is not an instruction to insert gauge readings into the logarithmic formula. For synthetic states generated with $p_a=100000\,\mathrm{Pa}$, $p_1=102000\,\mathrm{Pa}$ and $\gamma=1.4$, the model gives $p_3\simeq100567.39\,\mathrm{Pa}$. The logarithmic inference recovers $1.4$; the small-<pressure> approximation gives approximately $1.3961$.
Wait for a stable initial <pressure> after pumping and for a stable final <pressure> after closing. Vary the valve-open interval and initial excess <pressure> across repeated trials: closing too early leaves residual excess <pressure>, while leaving the valve open during warming lets more <gas> escape and suppresses the rebound. Test the seal for leaks, and read both liquid levels consistently. In a variant with incomplete release, measure the actual immediate post-release absolute <pressure> $p_2$ and replace the numerator by $\ln(p_1/p_2)$. The Zurich laboratory manual explains that liquid-column <oscillations> can delay this reading until appreciable reheating has occurred; its proposed backward extrapolation assumes an exponential thermal relaxation and must be checked against the recorded readings. These controls separate a thermal signal from valve timing, instrumental lag and leakage. https://www.physik.uzh.ch/~matthias/espace-assistant/manuals/en/anleitung-cd_e.pdf[University of Zurich, PHY112/122 laboratory manual, §1.3 and Appendix 1.4(a): pressure readings and reheating correction]{ref}
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