Rüchardt experiment

The Rüchardt experiment,^{[1] [2][3]} invented by Eduard Rüchardt, is a famous experiment in thermodynamics, which determines the molar heat capacity, i.e. the ratio of $C_p$ (constant pressure) and $C_V$ (heat capacity at constant volume) and is denoted by $\gamma$ (gamma, for ideal gas) or $\kappa$ (kappa, isentropic exponent, for real gas). It is because the temperature of a gas changes as pressure changes. The results of the experiment is the calculation for heat capacity ratio or adiabatic index, which is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. The results are sometimes also known as the isentropic expansion factor.

Background

If a gas is compressed adiabatically, i.e. without outflow of heat from the system, the temperature rises (due to the pressure increase) at a higher rate with respect to isothermal compression, where the performed work is dissipated as heat. The exponent, $\kappa$, with which the expansion of the gas can be calculated by the application of heat is called the isentropic - or adiabatic coefficient . Its value is determined by the Rüchardt experiment.

An adiabatic and reversible running state change is isentropic ( entropy S remains the same temperature as T changes). The technique is usually an adiabatic change of state. For example, a steam turbine is not isentropic, as friction, choke and shock processes produce entropy.

Experiment

Rüchardt experiment, Glass vessel with piston and Logger Pro P,V,T sensors

A typical experiment,^[4] consists of a glass tube of volume V, and of cross-section A, which is open on one of its end. A ball (or sometimes a piston) of mass m with the same cross-section, creating an air-tight seal, is allowed to fall under gravity g. The entrapped gas is first compressed by the weight of the piston, which leads to an increase in temperature. In the course of the piston falling, a gas cushion is created, and the piston bounces. Harmonic oscillation occurs, which slowly damps. The result is a rapid sequence of expansion and compression of the gas. The picture shows a revised version of the original Rüchardt setup: the sphere oscillating inside the tube is here replaced by a "breast-pump" which acts as an oscillating glass-piston; in this new setup three sensors allow to measure in real-time the piston oscillations as well as the pressure and temperature oscillations of the air inside the bottle (more details may be found in: revised version of Rüchardt apparatus .

Conceptual diagram of the experiment.

According to Figure 1, the piston inside the tube is in equilibrium if the pressure P inside the glass bottle is equal to the sum of the atmospheric pressure P₀ and the pressure increase due to the piston weight :

$P = P_0 + \frac{mg}{A}$

(Eq. 1)

When the piston moves beyond the equilibrium by a distance dx, the pressure changes by dp. A force F will be exerted on the piston, equal to

$F = A \mathrm{d}P$

(Eq. 2)

According to Newton's second law of motion, this force will create an acceleration a equal to

$a = \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = \frac{A}{m}\mathrm{d}P$

(Eq. 3)

As this process is adiabatic, the equation for ideal gas (Poisson's equation) is:

$P V^\gamma = \mathrm{constant}$

(Eq. 4)

It follows using differentiation from the equation above that:

$V^\gamma \mathrm{d}P + \gamma P V^{\gamma-1} \mathrm{d}V = 0$

(Eq. 5a)

$\mathrm{d}P = - \frac{\gamma P}{V} \mathrm{d}V$

(Eq. 5b)

If the piston moves by a distance $d x$ in the glass tube, the corresponding change in volume will be

$\mathrm{d}V=A dx$

(Eq. 6)

By substituting equation Eq. 5b into equation Eq. 3, we can rewrite Eq. 3 as follows:

$\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} + \frac{\gamma P A^2}{m V}x = 0$

(Eq. 7)

Solving this equation and rearranging terms yields the differential equation of a harmonic oscillation from which the angular frequency ω can be deduced:

$\omega = \sqrt{\frac{\gamma P A^2}{m V}}$

(Eq. 8)

From this, the period T of harmonic oscillation performed by the ball is:

$T = \frac{2 \pi}{\omega} = 2 \pi \sqrt \frac{m V}{\gamma P {A}^2 }$

(Eq. 9)

Measuring the period of oscillation T and the relative pressure P in the tube yields the equation for the adiabatic exponent:

${\gamma}=\frac{4 \pi^2 m V}{A^2 P T^2}$

(Eq. 10)

List of various versions of the Rüchardt experiment

In 1929 Rinkel ^[5] proposed a different method to calculate $\gamma$ while using the Rüchardt apparatus: he noted that it may be shown that the vertical distance L which the sphere falls before it begin to rise is: $L=\frac {2mgV}{PA^2\gamma}$, so $\gamma$ may be calculated from measured values of L,m,V,P and A.

In 1951 Koehler ^[6] and later, in 1972 Flammersfeld ^[7] introduced a trick in the original Rüchardt setup, to increase the number of oscillations that are limited by the unavoidable friction-damping and gas leak (through the piston-tube seal): they made a thin hole on the tube (at half-heigh) and provided a gas-feeding pump to keep constant the pressure inside the vessel. By properly trimming the gas inlet flux (through a throttling valve) they obtained the following result: during the oscillations the piston is pushed-up by the gas overpressure until it crosses the hole position; then the gas leakage through the hole reduces the pressure, and the piston falls back. The force acting onto the piston varies at a rate that is regulated by the piston oscillation frequency leading to forced oscillation; fine adjustment of the throttle valve allows to achieve maximum amplitude at resonance.

In 1958 Christy and Rieser ^[8] used only a gas-feeding pump to stabilize the gas pressure.

A slightly different solution was found in 1964 by Hafner ^[9] who used a tapered tube (conical: slightly larger at the top).

In 1959 Taylor ^[10] used a column of mercury oscillating inside a “U” shaped tube instead of the Rüchardt sphere.

In 1964 Donnally and Jensen ^[11] used a variable load attached to the Rüchardt sphere in order to allows frequency measurements with different oscillating mass.

In 1967 Lerner ^[12] suggested a modified version of the Taylor method (with mercury replaced by water).

In1979 Smith ^[13] reported a simplified version of the complex Rüchardt-resonance method, originally invented by Clark and Katz,^[14] in which an oscillating magnetic piston is driven into resonance by an external coil.

In 1988 Connolly ^[15] suggested the use of a photogate to measure more precisely the frequency of the Rüchardt sphere.

In 2001 Severn and Steffensen ^[16] used a pressure transducer to monitor the pressure oscillations in the original Rüchardt setup.

In 2001 Torzo, Delfitto, Pecori and Scatturin ^[17] implemented the version of Rüchardt apparatus (shown in the top picture) using three sensors: a sonar that monitors the breast-pump oscillations, and pressure and temperature sensors that monitor the changes in pressure and temperature inside the glass vessel.

References

1. Fuchs, Hans U. The Dynamics of Heat: A Unified Approach to Thermodynamics and Heat Transfer, 2010, P212-214
2. Glasser, L. (1990). "Useful papers on Ruchardt's method". Journal of Chemical Education 67 (8): 720. Bibcode:1990JChEd..67..720G. doi:10.1021/ed067p720.3.
3. Rüchardt, E. (1929). "Eine einfache methode zur bestimmung von Cp /Cv". Physikalische Zeitschrift 30: 58–59.
4. "Kinetic theory of gases" (PDF). ld-didactic.de. Retrieved 5 Nov 2013.
5. Rinkel, R. (1929), "Die bestimmung von Cp /Cv", Physikalische Zeitschrift 30: 895
6. Koehler, W.F. (1951), "A Laboratory Experiment on the Determination of γ for Gases by Self-Sustained Oscillation", American Journal of Physics 19: 113, doi:10.1119/1.1932723
7. Flammersfeld, A. (1972), "Messung von Cp /Cv von Gasen mit ungedämpften Schwingunge", Zeitschrift für Naturforschung 27a: 540, doi:10.1515/zna-1972-0327
8. Christy, R.W.; Rieser, M.L. (1958), "Modification of Rüchardt's Experiment", American Journal of Physics 26: 37, doi:10.1119/1.1934595
9. Hafner, E.M. (1964), "Refined Rüchhardt Method for γ", American Journal of Physics 32: XIII, doi:10.1119/1.1970131
10. Taylor, L.W. (1959), Manual od advanced undergraduate experiments in physics, p. 152
11. Donnally, B.; Jensen, H. (1964), "Another Refinement for Rüchardt's Method for γ", American Journal of Physics 32: XVI, doi:10.1119/1.1970327
12. Lerner, I. (1967), "Determination of Cp/Cv", American Journal of Physics 35: XVI, doi:10.1119/1.1974103
13. Smith, D.G. (1979), "Simple Cp/Cv resonance apparatus for the physics teaching laboratory", American Journal of Physics 47: 593, doi:10.1119/1.11760
14. Clark, A.L.; Katz, L. (1940), "Resonance Method for Measuring the Ratio of the Specific Heats of a Gas, Part I", Canadian Journal of Research A 18: 23–38, doi:10.1139/cjr40a-002
15. Connolly, W. (1988), "Measurement of a thermodynamic constant", The Physics Teacher 26: 235, doi:10.1119/1.2342501
16. Severn, G.D.; Steffensen, T. (2001), "A simple extension of Rüchardt’s method for measuring the ratio of specific heats of air using microcomputer-based laboratory sensors", American Journal of Physics 69: 387, doi:10.1119/1.1317558
17. Torzo, G.; Delfitto, G.; Pecori, B.; Scatturin, P. (2001), "A New Microcomputer-Based Laboratory Version of the Rüchardt Experiment for Measuring the Ratio γ = Cp/Cv in Air", American Journal of Physics 69: 1205, doi:10.1119/1.1405505