Experimental test of Loschmidt effect

Does  a lapse rate exist in an isolated gas under a gravitational field, or is it only a bulk convective effect? This essentially was the dispute between Lochsmidt, Maxwell and Boltzman at the end of the 19th century. Maxwell and Boltzman argued that such an isolated atmosphere would reach an isothermal temperature.  Nearly every physicist since then would agree with Maxwell since a persistent temperature gradient appears to violate the second law of thermodynamics. A modern analysis of this problem has recently been given by Christian Fronsal, a theoretical physicist at the University of California [1]. He proposes using a centrifuge to resolve this issue one way or the other. Renewed interest  has also been triggered by a seemingly positive temperature gradient observed in an isolated water column under gravity [2]. Such an  experimental test using a gas centrifuge is shown in figure 1, and described  below.

Fig 1: Experiment to decide whether Loschmidt could have been right.

Gas centrifuges such as those used for Uranium isotope separation can spin at well over 1000 revs/sec simulating huge gravitational fields. For a gas chamber with radius R and an  inner bore of radius ‘a’ spinning at an angular velocity \Omega, the effective gravitatonal acceleration at a distance z inwards from the rim is g =\Omega^2(R-z)

For a gas with specific heat Cp the lapse rate is

\frac{dT}{dz} = -\frac{\Omega^2(R-z)}{C_{p}}

T(z) = -\frac{\Omega^2}{C_{p}} \int_0^z (R-z)\mathrm{d}z = -\frac{\Omega^2}{C_{p}}[_R^zRz -\frac{z^2}{2}+C]

putting  z=0, -\frac{\Omega^2}{C_{p}}C = T_{0}

T(z) = T_{0}+\frac{\Omega^2}{C_{p}}(\frac{z^2}{2}-Rz)

Assuming a centrifuge with an outer radius of 10cm and an inner radius of 1 cm spinning at 100 rev/sec, the predicted lapse rate temperature gradient  is shown in figure 2. It is assumed that the centrifuge is surrounded by a heat bath held at 15C and that the inner bore is a vacuum. Using air  in the centrifuge the temperature at the inner radius is then predicted to be 2 degreec C. lower than the outer radius.  Such a large effect would be easy to measure with thermocouples. Now substituting CO2 instead of air increases slightly the effect by a further 0.3C, while replacing it with argon produces the largest effect with a drop of 3.7 degrees C. Lochsmidt originally calculated a lapse rate  g/Cv  instead of g/Cp. This corresponds to zero work being done (PdV) ( see  understanding the lapse rate ). The results assume in figure 2 assume g/Cp.

Fig 2: Predicted lapse rates for a Loschmidt effect in a gas cyclotron.

This experiment would be of moderate cost and well within the means of a university physics department. My personal opinion is that initially adiabatic compression  will create a temperature gradient which then rapidly  conducts through collisions to a single uniform temperature.  Despite this, such an experiment is still worthwhile because it would resolve this dispute once and for all. Perhaps it has even already been done !

References

1.  Heat and Gravitation. I. The Action Principle, Christian Fronsal, Univ. Califormia, 2011.

2.  Graeff, R.W.  see http://firstgravitymachine.com/descript_B372_V_5.pdf   2007.

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