May 1, 2002

Temperature Differences in the Gravitational Force Field

In THE DANCING MOLECULES I described that gas molecules, arranged in a vertical column should show a temperature difference between the temperature at the top T1 and the one at the bottom T2. I call this difference T1-T2 = T(Gr), the temperature difference due to gravity.

In trying to measure it, it is important to know the course of it and its size. Once we know its course we can plan meaningful test setups, once we know its size we can judge if the measurement of this difference, if it exists, appears possible or not. These questions are covered in the following paper:

March 10, 2009

Temperature of the walls of a container filled with a gas

Summary:

In an isolated system the temperature of the walls depend on the speed of the impinging molecules. It can be shown that this results in a higher temperature of the lower and a lower temperature of the upper wall in a container filled with a real gas.

The result refutes the argument of Maxwell for equal temperatures over height and limits that of Boltzmann to ideal gases.

Introduction:

Late in the 19th century J. Loschmidt believed that a vertical column of gas or a solid in an isolated system would show a temperature gradient under the influence of gravity, being cold at the top and warm at the bottom. L. Boltzmann and J. C. Maxwell disagreed. Their theories and understanding of the Second Law supported an equal temperature over height. This historical discussion between J. Loschmidt, L. Boltzmann and J. C. Maxwell is covered in [1], [2], and [3]. A. Trupp gives a good summary in [4].

The historical dispute between J. Loschmidt, L. Boltzmann and J. C. Maxwell

In trying to formulate and understand the Second Law, Boltzmann calculated in 1868 based on the Maxellian speed distribution and statistical evaluations that a column of gas should have the same temperature at the top and at the bottom. But his calculations were limited to ideal gases.

Loschmidt disagreed with some of his conclusions and assumptions. He thought that gravity would create a temperature gradient, cold at the top and warm at the bottom, especially in solids. He felt that this would not contradict the Second Law .

Maxwell expected equal temperatures at the top and bottom and in his book "Theory of heat", published in London in 1877 because only this result would be congruent with the Second Law. He writes (p. 320):

"... if two vertical columns of different substances stand on the same perfectly conducting horizontal plate, the temperature of the bottom of each column will be the same; and if each column is in thermal equilibrium of itself, the temperatures at all equal heights must be the same. In fact, if the temperatures of the tops of the two columns were different, we might drive an engine with this difference of temperature, and the refuse heat would pass down the colder column, through the conducting plate, and up the warmer column; and this would go on till all the heat was converted into work, contrary to the second law of thermodynamics. But we know that if one of the columns is gaseous, its temperature is uniform. Hence that of the other must be uniform, whatever its material."

Trupp [ ] disagrees with certain arguments of Boltzmann. Using similar statistical calculations and accepting some aspects of real gases he finds lower temperatures at the top and higher ones at the bottom.

The author uses similar arguments as Loschmidt for gases and liquids and finds lower temperatures at the top and higher ones at the bottom. He proposes a method for calculating the size of this temperature difference.

1. Why is the temperature of molecules in an isolated system colder at the top than at the bottom.

1.1. Knudson gases:

A gas in a container is considered to be a Knudson gas when the free path length of the molecules is great in comparison to the distance of the container walls. This means that a molecule moves between the container walls without hitting another molecule,

Top wall

Bottom wall

When a molecule hits a wall it will, on the average, leave the wall with a speed corresponding to the temperature of the wall. This means that it will transfer energy to the wall when it arrives with a higher speed or it will receive energy whenever it arrives with a lower speed.

Once equilibrium is reached, which means no temperature changes over time, a molecule will leave a wall on the average with the same speed and direction with which another one just arrived.

Let's assume that initially the temperatures of all walls are identical. A molecule leaving the upper wall on its downward pass will be accelerated by the influence of gravity. It will reach the lower wall with an increased speed which is a measure of a higher temperature, initially higher than the wall temperature. When it hits the lower wall it will leave this wall with a reduced speed. As stated above, in consequence of this it will have transferred energy to the wall, increasing its temperature.

On the way up the opposite takes place. The speed of the molecule gets reduced by gravity and when hitting the upper wall the wall will transfer energy to the molecule with the result of lowering the upper wall's temperature.

So the molecules traveling between the upper and lower wall continuously transfer energy from the upper to the lower wall, lowering the upper temperature T1 and increasing the lower wall temperature T2. This will continue until the speed of the molecule hitting a wall will correspond to a temperature identical to the temperature of the wall. The speed which has to be considered for this is the vertical component of the molecule's speed.

Very rarely but once in a while even in a Knudsen gas a molecule will hit another molecule. This will not affect the amount of energy transferred from wall to wall as in the average the vertical speed component will be transferred between the molecules when hitting each other.

All the molecules leaving the upper wall will hit the lower wall. Contrary to this there will be a small number of molecules leaving the lower wall but having not enough energy to reach the upper wall. On the way up gravity will reduce the vertical speed component to zero and they will fall back until they hit the lower wall with the same speed with which they left this wall. This means that they don't effect any temperature change of the wall.

1.2. Dense gases:

Let us arrange a horizontal wall in the middle of our container. This wall will be hit by molecules arriving from the upper and lower walls.(Fehlt noch im Bild)

Top wall

Bottom wall

If we assume that these molecules came from a surface where in the average the energy was distributed equally between all degrees of freedom then on their way to the horizontal wall gravity will have affected only the vertical speed component. That would mean that all molecules hitting the horizontal partition arrive with their energy not being equally distributed between their degrees of freedom. This means that we do not have a Maxwellian distribution of speed as this is based on an equal partitioning of energies between all degrees of freedom.

The reasoning described under 1.1 is equally valid for our horizontal wall. This means that the temperature of this wall will at equilibrium be lower than the temperature of the upper wall and higher than that of the lower wall.

As a next step we now can imagine our container being divided not by one but by a multitude of horizontal walls. Again, this would not affect our reasoning that the lower wall will be warmer that the upper wall.

In a next step we now assume that the horizontal walls are exchanged against the molecules of a dense gas. Without discussing it in detail, we now can assume that in a dense gas a horizontal layer of molecules has the same effect as a horizontal wall.

2. How can we calculate the temperature difference described under 1.1 and 1.2?

No published treatise is known to the author for calculating the vertical temperature gradient TGr in solids or liquids under the influence of gravity. Instead of using Maxwellian speed distribution and statistical methods it is possible to calculate TGr by looking at the energy transport through the molecules between the top and the bottom walls under the influence of gravity. Following this, the value of TGr can be calculated by equating the potential energy of the molecules to the increase of their speed on their downward path. Their speed is related to their temperature. In their downward movement their potential energy is converted to an increase of the vertical component of their lateral speed.. When bouncing off the bottom wall their kinetic energy is zero at the moment of impact. A heat transfer takes place between molecules and the upper and the lower walls of the tube, until the wall temperatures are equal to the "temperature" of the impinging molecules and equilibrium has been reached.

The potential energy is

(1)

with M = mass; g = constant of gravity; H = height difference

(negative, because g and H are measured in opposite directions)

We equate this potential energy Ep with the amount of energy available for a temperature increase of this mass

(2)

with cGr = effective specific heat; T = Temperature difference

We now can equate (1) with (2) or

(3)

or

(4)

cGr is not the normal specific heat of the liquid in question, because the acceleration through g affects only the vertical speed component of the molecule. The potential energy is converted only into an increase of their speed in their lateral downward direction. No energy is added to the other degrees of freedom, like the remaining two lateral directions (left to right and front to back) or to the rotational energy in molecules consisting of more than one atom. Therefore,

(5)

with c = specific heat; n = number of degrees of freedom

We therefore get

(6)

With this formula for a height of 1 meter and taking the number of degrees of freedom for water as 18 and for air as 5 , we obtain

for water TGr = -0,04 K/m

for air TGr = -0,07 K/m

These values correspond well with measured values published by the author in [5] and [8].

Conclusion:

In an isolated system the temperature of the walls depend on the speed of the impinging molecules. The average of their speed is lower at the top than at the bottom as each molecule gets accelerated on its way downwards and decelerated upwards. Through this energy is transported from the upper to the lower wall until equilibrium is reached.

This results in a temperature difference between the upper and lower wall, with the upper wall having a lower and the lower wall having a higher temperature .

Roderich Graeff

REFERENCES

1. Boltzmann, L., Wissenschaftliche Abhandlungen, Vol.2, edited by F.Hasenoehrl, Leipzig, 1909
2. Maxwell, J.C., The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 35, p. 215 (1868)
3. Loschmidt, L., über den Zustand des Wärmegleichgewichts eines Systems von Körpern mit Rücksicht auf die Schwerkraft, Sitzungsberichte der Mathematisch-Naturwissenschaftl. Klasse der Kaiserlichen Akademie der Wissenschaften 73.2 (1876), p. 135
4. Trupp, A., Physics Essays, 12, No. 4, 1999
5. Graeff, R.W., Gravity Machine, Web Site
6. Bohren, C.F. Atmospheric Thermodynamics, pp. 167 ff., Oxford University Press, New York, 1998
7. Graeff, R.W., Gravity Machine, my Search for Peaceful Energy, to be published autumn 2009
8. Graeff, R.W., Measuring the temperature distribution in gas columns, CP 643, Quantum Limits to the Second Law, New York 2002, American Institute of Physics, AIP Conference Proceedings, Vol. 643
9. Graeff, R.W.,Produktion von nutzbarer Energie aus einem Wärmebad, NET-Journal, ISSN 1420-9292, Jupiter-Verlag, Zurich, February 2003
10. Capek,V., and Sheehan, D.P., Challenges to the Second Law of Thermodynamics, pp 203 ff., Springer, 2005
11. Clausius, R., Abhandlungen ueber die mechanische Waermetheorie, Vol. 1, (F.Vieweg, Braunschweig, 1864); Vol. 2, (1867).