Clément-Desormes experiment

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, its pressure rises to $p_3$, with $p_a<p_3<p_1$ in the ideal procedure. The rebound reveals cooling during release. ^[ref]

The bottle loses molecules while the valve is open: it is not a closed system whose geometrical volume 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 gives $p_3/p_a=T_0/T_2$ by the ideal gas law. Eliminating the unmeasured $T_2$ yieldsThe 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 $p_3\simeq 100567.39\mathrm{Pa}$. 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. ^[ref]