Cauchy–Euler operator

The Rüchardt experiment, 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 due to the fact that 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

A gas is compressed adiabatically, i.e. without the outflow of heat from the system, the temperature rises due to increase of the pressure at more than one isothermal, compression, wherein the resulting compression 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 - known or adiabatic. This 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, Gläserner cylinder with piston and Logger Pro Meters

To implement the experiment, a glass tube (volume $V$), which upwardly has a tubular opening (tube cross-section $A$), are available. In the open pipe, a piston, or in the original experiment, a ball, is released (mass $m$) to fall (gravity $g$), which closes the opening air-tight. Figure 1 below describes the components of the experiment.

By the weight of the piston (or ball), the entrapped gas is first compressed, which leads to an increase in temperature. In the course of the piston falling, a gas cushion is created, and the ball bounces, a harmonic oscillation occurs, which slowly damps. The result is a rapid sequence of expansion and compression of the gas. 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 $p_{0}$ and the weight of the piston (or ball) yielding the equation

$p=p_{0}+{\frac {m.g}{A}}$

When the ball moves beyond the equilibrium by a distance $X$, the pressure changes by $\mathrm {p}$. By this a force $F=A.\mathrm {dp}$ is exerted on the ball applying an acceleration defined by Newton's second law of motion:

$A.\mathrm {dp} =m{\cfrac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}$ (1)

As this process is Adiabatic process#Ideal gas (reversible process), the equation for ideal gas, which is Poissons equation is:

$PV^{\gamma }=\operatorname {constant} \qquad$ (2)

It follows using differentiation from the equation above that:

$V^{\gamma }\mathrm {dp} +p{\gamma }V^{\gamma -1}{\mathrm {d} V}=0$ (3)

${\mathrm {d} p}=-{\frac {\gamma p}{V}}{\mathrm {d} V}$ (4)

The piston (or ball) moves a distance $x$ in the glass tube, which gives a change of volume:

${\mathrm {d} V}=A.x$ (5)

By substituting equation 5 into equation 4 and inserting equation 1 we can rewrite the equation of motion as follows:

${\cfrac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\frac {\gamma p{A}^{2}}{mV}}x=0$

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

$\omega ={\sqrt {\cfrac {\gamma p{A}^{2}}{\mathrm {m} V}}}$

From this the period ${T}$ of harmonic oscillation performed by the ball is:

${T}={\cfrac {2\pi }{\omega }}=2{\pi }{\sqrt {\cfrac {\mathrm {m} V}{\gamma p{A}^{2}}}}$

Measuring the period of oscillation ${T}$, and the relative pressure p in the vessel yields the equation for the adiabatic exponent:

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

References

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