= Rüchardt experiment
{c}
The Rüchardt experiment infers <heat capacity ratio> from the <period of an oscillation> of a piston supported by trapped <gas>. Use a vertical precision cylinder with a close-fitting movable piston, known moving <mass> $M$, face <area> $A$, trapped equilibrium <volume (thermodynamics)> $V_0$, and local atmospheric <pressure> $p_{\rm atm}$. With negligible static <friction>, <force balance> gives the absolute gas <pressure> $p_0=p_{\rm atm}+Mg/A$, where $g$ is gravitational acceleration. Displace the piston slightly and release it: compression raises <pressure>, and expansion lowers it, producing a restoring <force>. The <oscillation> is a mechanical probe of thermal response, without separately measuring two <heat> inputs. https://groups.ist.utl.pt/wwwelab/wiki/index.php?title=Determination_of_the_Adiabatic_Constant[Instituto Superior Técnico e-lab, Determination of the Adiabatic Constant: piston apparatus and period model]{ref}
Let $x$ be upward <displacement> from equilibrium, so $V=V_0+Ax$. For constant <heat capacity ratio> and a reversible <adiabatic process>, the net upward <force>, including the ambient load and weight, is
$$
F(x)=A p_0\left[\left(1+\frac{Ax}{V_0}\right)^{-\gamma}-1\right]
\simeq-\frac{\gamma p_0A^2}{V_0}x.
$$
The <linear approximation> requires $|Ax|/V_0\ll1$. Applying <Newton's second law> gives <simple harmonic motion> with
$$
\omega_0^2=\frac{\gamma p_0A^2}{MV_0},\qquad
\gamma=\frac{4\pi^2MV_0}{p_0A^2\mathcal T^2},\qquad
\mathcal T=\frac{2\pi}{\omega_0}.
$$
Here $\omega_0$ is the undamped <angular frequency> and $\mathcal T$ the undamped <oscillation period>, rather than <temperature>. Record repeated cycles, measure $M,A,V_0,p_{\rm atm}$ independently, and include all connected trapped <volume (thermodynamics)>. An illustrative model with $M=0.020\,\mathrm{kg}$, $A=2.0\times10^{-4}\,\mathrm{m^2}$, $V_0=0.010\,\mathrm{m^3}$, $p_{\rm atm}=100000\,\mathrm{Pa}$, $g=9.81\,\mathrm{m\,s^{-2}}$ and $\gamma=1.4$ gives $p_0=100981\,\mathrm{Pa}$ and $\mathcal T\simeq1.182\,\mathrm s$. These stipulated values are not measurements.
\Image[/heat-capacity-ratio-gas-spring.png]
{title=<Rüchardt experiment>: <heat capacity ratio> sets the restoring stiffness of trapped <gas>}
{description=Original schematic and analytical <ideal gas> curves. Upward piston <displacement> increases <volume (thermodynamics)>; the blue curve uses constant $\gamma=1.4$, its dashed tangent gives the small-<amplitude> model, and the orange curve is the <isothermal process> limit. The shaded central interval illustrates a small fractional <volume (thermodynamics)> change, without specifying an experimental acceptance threshold. No measured waveform is shown.}
In Shearwood and Sloan's implementation, piston-mounted magnets induce an <electromotive force> in a stationary coil through <Faraday's law of induction>; an oscilloscope records the signal. They observed faster <oscillations> at smaller trapped <volume (thermodynamics)> values and stronger <damping> at small cylinder <volume (thermodynamics)> values. The signal tracks <velocity> locally, so its <amplitude> is not directly piston <displacement>. Calibration also identified inlet dead space beyond the graduated cylinder volume. https://arxiv.org/html/2209.03323v1[Chris Shearwood and Peter A. Sloan, The Rüchardt experiment revisited, §§3–4: signal, dead volume and observations]{ref}
Repeat at smaller initial <amplitudes> to test the <linear approximation>, and compare several $V_0$ values. <Heat transfer> weakens the restoring <force> toward the <isothermal process> limit, biasing the inference toward $1$. Check leakage through resting-piston drift and before-and-after trapped <volume (thermodynamics)> readings. For viscous <damping> with envelope $e^{-t/\tau}$, the measured <angular frequency> obeys $\omega_0^2=\omega_d^2+\tau^{-2}$; this correction does not remove thermal leakage. Internal equilibration must be faster than an <oscillation period>, with <heat transfer> and leakage slow over a run. Large <amplitude> or sticking <friction> require a richer model. https://arxiv.org/html/2209.03323v1[Shearwood and Sloan, §§2.1, 2.4 and 3.2: damping, heat exchange and leakage limitations]{ref}
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