Guest post by Robert G. Brown
Duke University Physics Department
The Problem
In 2003 a paper was published in Energy & Environment by Hans Jelbring that asserted that a gravitationally bound, adiabatically isolated shell of ideal gas would exhibit a thermodynamically stable adiabatic lapse rate. No plausible explanation was offered for this state being thermodynamically stable – indeed, the explanation involved a moving air parcel:
An adiabatically moving air parcel has no energy loss or gain to the surroundings. For example, when an air parcel ascends the temperature has to decrease because of internal energy exchange due to the work against the gravity field.
This argument was not unique to Jelbring (in spite of his assertion otherwise):
The theoretically deducible influence of gravity on GE has rarely been acknowledged by climate change scientists for unknown reasons.
The adiabatic lapse rate was and is a standard feature in nearly every textbook on physical climatology. It is equally well known there that it is a dynamical consequence of the atmosphere being an open system. Those same textbooks carefully demonstrate that there is no lapse rate in an ideal gas in a gravitational field in thermal equilibrium because, as is well known, thermal equilibrium is an isothermal state; nothing as simple as gravity can function like a “Maxwell’s Demon” to cause the spontaneous stable equilibrium separation of gas molecules into hotter and colder reservoirs.
Spontaneous separation of a reservoir of gas into stable sub-reservoirs at different temperatures violates the second law of thermodynamics. It is a direct, literal violation of the refrigerator statement of the second law of thermodynamics as it causes and maintains such a separation without the input of external work. As is usually the case, violation of the refrigeration statement allows heat engines to be constructed that do nothing but convert heat into work – violating the “no perfectly efficient heat engine” statement as well.
The proposed adiabatic thermal lapse rate in EEJ is:
where g is the gravitational acceleration (presumed approximately constant throughout the spherical shell) and cp is the heat capacity per kilogram of the particular “ideal” gas at constant pressure. The details of the arguments for an adiabatic lapse rate in open systems is unimportant, nor does it matter what cp is as long as it is not zero or infinity.
What matters is that EEJ asserts that 
in stable thermodynamic equilibrium.
The purpose of this short paper is to demonstrate that such a system is not, in fact, in thermal equilibrium and that the correct static equilibrium distribution of gas in the system is the usual isothermal distribution.
The Failure of Equilibrium
In figure 1 above, an adiabatically isolated column of an ideal gas is illustrated. According to EEJ, this gas spontaneously equilibrates into a state where the temperature at the bottom of the column Tb is strictly greater than the temperature Tt at the top of the column. The magnitude of the difference, and the mechanism proposed for this separation are irrelevant, save to note that the internal conductivity of the ideal gas is completely neglected. It is assumed that the only mechanism for achieving equilibrium is physical (adiabatic) mixing of the air, mixing that in some fundamental sense does not allow for the fact that even an ideal gas conducts heat.
Note well the implication of stability. If additional heat is added to or removed from this container, it will always distribute itself in such a way as to maintain the lapse rate, which is a constant independent of absolute temperature. If the distribution of energy in the container is changed, then gravity will cause a flow of heat that will return the distribution of energy to one with Tb > Tt . For an ideal gas in an adiabatic container in a gravitational field, one will always observe the gas in this state once equilibrium is established, and while the time required to achieve equilibrium is not given in EEJ, it is presumably commensurate with convective mixing times of ordinary gases within the container and hence not terribly long.
Now imagine that the bottom of the container and top of the container are connected with a solid conductive material, e.g. a silver wire (adiabatically insulated except where it is in good thermal contact with the gas at the top and bottom of the container) of length L . Such a wire admits the thermally driven conduction of heat according to Fourier’s Law:
where λ is the thermal conductivity of silver, A is the cross-sectional area of the wire, and ΔT=Tb–Tt . This is an empirical law, and in no way depends on whether or not the wire is oriented horizontally or vertically (although there is a small correction for the bends in the wire above if one actually solves the heat equation for the particular geometry – this correction is completely irrelevant to the argument, however).
As one can see in figure 2, there can be no question that heat will flow in this silver wire. Its two ends are maintained at different temperatures. It will therefore systematically transfer heat energy from the bottom of the air column to the top via thermal conduction through the silver as long as the temperature difference is maintained.
One now has a choice:
- If EEJ is correct, the heat added to the top will redistribute itself to maintain the adiabatic lapse rate. How rapidly it does so compared to the rate of heat flow through the silver is irrelevant. The inescapable point is that in order to do so, there has to be net heat transfer from the top of the gas column to the bottom whenever the temperature of the top and bottom deviate from the adiabatic lapse rate if it is indeed a thermal equilibrium state.
- Otherwise, heat will flow from the bottom to the top until they are at the same temperature. At this point the top and the bottom are indeed in thermal equilibrium.
It is hopefully clear that the first of these statements is impossible. Heat will flow in this system forever; it will never reach thermal equilibrium. Thermal equilibrium for the silver no longer means the same thing as thermal equilibrium for the gas – heat only fails to flow in the silver when it is isothermal, but heat only fails to flow in the gas when it exhibits an adiabatic lapse in temperature that leaves it explicitly not isothermal. The combined system can literally never reach thermal equilibrium.
Of course this is nonsense. Any such system would quickly reach thermal equilibrium – one where the top and bottom of the gas are at an equal temperature. Nor does one require a silver wire to accomplish this. The gas is perfectly capable of conducting heat from the bottom of the container to the top all by itself!
One is then left with an uncomfortable picture of the gas moving constantly – heat must be adiabatically convected downward to the bottom of the container in figure 1 in ongoing opposition to the upward directed flow of heat due to the fact that Fourier’s Law applies to the ideal gas in such a way that equilibrium is never reached!
Of course, this will not happen. The gas in the container will quickly reach equilibrium. What will that equilibrium look like? The answer is contained in almost any introductory physics textbook. Take an ideal gas in thermal equilibrium:
where N is the number of molecules in the volume V, k is Boltzmann’s constant, and T is the temperature in degrees Kelvin. n is the number of moles of gas in question and R is the ideal gas constant. If we assume a constant temperature in the adiabatically isolated container, one gets the following formula for the density of an ideal gas:
where M is the molar mass, the number of kilograms of the gas per mole.
The formula for that describes the static equilibrium of a fluid is unchanged by the compressibility (or lack thereof) of the fluid – for the fluid to be in force balance the variation of the pressure must be:
(so that the pressure decreases with height, assuming a non-negative density). If we multiply both sides by dz and integrate, now we get:
Exponentiating both sides of this expression, we get the usual exponential isothermal lapse in the pressure, and by extension the density:
where P0 is the pressure at z=0 (the bottom of the container).
This describes a gas that is manifestly:
- In static force equilibrium. There is no bulk transport of the gas as buoyancy and gravity are in perfect balance throughout.
- In thermal equilibrium. There is no thermal gradient in the gas to drive the conduction of heat.
If this system is perturbed away from equilibrium, it will quickly return to this combination of static and thermal equilibrium, as both are stable. Even in the case of a gas with an adiabatic lapse rate (e.g. the atmosphere) remarkably small deviations are observed from the predicted P(z) one gets treating the atmosphere as an ideal gas. An adiabatically isolated gas initially prepared in a state with an adiabatic lapse rate will thermally equilibrate due to the internal conduction of heat within the gas by all mechanisms and relax to precisely this state.
Conclusion
As we can see, it is an introductory physics textbook exercise to demonstrate that an adiabatically isolated column of gas in a gravitational field cannot have a thermal gradient maintained by gravity. The same can readily be demonstrated by correctly using thermodynamics at a higher level or by using statistical mechanics, but it is not really necessary. The elementary argument already suffices to show violation of both the zeroth and second laws of thermodynamics by the assertion itself.
In nature, the dry adiabatic lapse rate of air in the atmosphere is maintained because the system is differentially heated from below causing parcels of air to constantly move up and down. Reverse that to a cooling, like those observed during the winter in the air above Antarctica, and the lapse rate readily inverts. Follow the air column up above the troposphere and the lapse rate fails to be observed in the stratosphere, precisely where vertical convection stops dominating heat transport. The EEJ assertion, that the dry adiabatic lapse rate alone explains the bulk of so-called “greenhouse warming” of the atmosphere as a stable feature of a bulk equilibrium gas, is incorrect.
Joe Born says:
January 25, 2012 at 6:04 pm
“It’s true that Velasco et al. is a statistical-mechanical analysis. But it’s not true that “arrives at the same conclusion.” Quite the contrary. In connection with their Equation 8, what Velasco et al. say is, “i.e., for a finite adiabatically enclosed ideal gas in a gravitational field the average molecular kinetic energy decreases with height.””
They do come to the same conclusion; they quite clearly show – and state – that, for any macroscopic system, canonic or microcanonic – the sort of system we are actually interested in here – the result is isothermal, as it should be. You keep refusing to accept – or even adequately acknowledge – that their Equation 8 doesn’t mean what you want it to mean, because temperature itself doesn’t mean what you think it means in that isolated small-number regime.
Joe: ” … if … you believe that temperature is mean molecular translational kinetic energy … ”
But that’s the point. For small isolated systems it isn’t. The limiting case of the single isolated particle, which has no thermodynamic temperature, proves this.
Velasco et al do not calculate a thermodynamic lapse rate in the small-number microcanonic regime at all. They do not prove it to be isothermal; nor do they prove the contrary. What they do show is that their statistical mechanics confirms the thermodynamic conclusion that – for ensembles large enough to apply said thermodynamics to – the answer is indeed isothermal.
Joe: “On the other hand, you can accept Paul Birch’s analysis, which in my view is nothing more than so redefining lapse rate as to exclude anything exhibited by a maximum-entropy configuration.”
Please stop misquoting me. I said nothing whatsoever about “maximum entropy”. I didn’t even mention entropy. I didn’t need to. My “redefinition” (not actually a redefinition at all, but the utterly bog-standard original definition) is simply the statement that things in thermal equilibrium are at the same temperature; that’s what “same temperature” means.
Doug Cotton said @ur momisugly January 26, 2012 at 1:15 am
No. It lacks a defined temperature because temperature does depend on density of molecules. Consider a universe in which there is exactly one molecule. What temperature is it? Consider a universe in which there are two molecules and we know the exact distance between them. What is their temperature? Rinse and repeat…
Pompous said:
“Consider a universe in which there are two molecules and we know the exact distance between them. What is their temperature? Rinse and repeat…”
Come on, you sound more intelligent than that. You have just described an instant in time when you know the distance between two particles. You have not indicated mass, velocity, acceleration, vectors…
Rinse and repeat…
robr says:
January 24, 2012 at 9:32 pm
“Does the atmospheric pressure, no GH gases, play a role in the dry adiabatic lapse rate?”
Willis Eschenbach says:
January 25, 2012 at 7:40 pm
“The DALR is defined as g / Cp, where g is gravity and Cp is the specific heat of the atmosphere at constant pressure. Since I see no pressure term in there, I’m gonna say no. ”
Hmmm. Interesting.
What is Cp, or “specific heat”, really?
http://en.wikipedia.org/wiki/Heat_capacity
I am just trying to repeat all this again (25 years since my school-days);
If Cp had been “molar heat capacity” in that formula, then pressure wouldnt matter IMO.
But, since Cp is “Geat Capacity per unit mass” of a material….that means that if you increase the mass…then Cp increase. Right? Or am I misunderstanding this sentence?
So, how can you increase the mass? Well, increased pressure is really just that the number of molcules per volume unit has increased. Right? Hence, the mass has increased. And therefore Cp increase.
So, In my opinion , yes, increased pressure => increased mass => increased Cp.
Robert Brown’s refutation is contrived and irrelevant.
Any column of any substance from the top to the bottom of the cylinder is subject to the same gravity field with gravitational constant g as the gas in the cylinder. Or, what is the same according to the general theory of relativity, the cylinder with gas as well as the conductor (whatever material it is made of) are subject to the same constant acceleration (vector) of g m/s2. Therefore, the same temperature gradient will appear in the conductor and or along the walls of the cylinder as exists in the gas. The bottom of the cylinder pushes against the gas mass and any other top-to-bottom material like an adiabatic piston, raising the temperature at the bottom, whereas the top of the cylinder pulls like a similar piston, decreasing the temperature at the top.
As long as the acceleration is maintained, the temperature gradient in the direction of the acceleration vector will be maintained. If heat is added to the system by radiation from outside, the average temperature will increase, but as long as the acceleration continues, a temperature gradient between top and bottom of the system will be maintained.
Doug, with regard to your selection that particular article, as the knight says in the film Indiana Jones and the Last Crusade “You have chosen … poorly.” To chose better, let’s recall the Freeman Dyson quote that several recent WUWT posters have quoted (I think Willis Eschenbach was the first):
Here on WUWT, it should be clear to everyone that the “clear physical picture” approach may have worked well for Fermi (who was one of the greatest physicists of his generation), but it does not work wll in on-line forums, for the simple reason that that various posters advocate physical pictures that are so diverse, and so contradictory in their predictions, that most of these physical pictures must be wrong, even though some them may be right. So unless one is similarly talented to Fermi, physical pictures are not all that trustworthy.
That brings us to Fermi’s second option: “have a precise and self-consistent mathematical formalism”. Regrettably, here too the conceptual challenges are severe. In recent years the trend is toward geometric theories of thermodynamics, and a recommended starting point is Quevedo and Vazquez, The geometry of thermodynamics, whose geometric language is explained in John Lee’s highly regarded Introduction to Smooth Manifolds (known to students as “Smooth Introduction to Manifolds”). Here “geometry” is an older name for what nowadays is called “mathematical naturality.”
The increasing and unifying dominance of mathematically natural geometric frameworks in describing classical, quantum, and thermodynamical physics is a relatively recent development (mainly since the 1980s), and these frameworks constitute the nearest approach that we have to Fermi’s “precise and self-consistent mathematical formalism”.
The third option is a brute-force fine-grained numerical simulation of atoms bouncing off one-another. Here a good reference is Frenkel and Smit’s Understanding Molecular Simulation: from Algorithms to Applications.
So when should one have confidence? Whenever all three theoretical approaches (physical, geometric, and computational) lead to the same conclusion, and moreover agree with experiments and observation, then confidence is justified.
And so I want to emphasize that, in the present case, all three paths (physical, geometric, and computational) lead to the same conclusion, which is entirely consistent with experiments and observation: Gravito-thermal theories are just plain wrong.
Joules Verne says at 1/25 8:57pm:
Interesting find. Thanks for posting; on my reading list for later today. Means this thread is open since 1866!
One point to make:
“Later, he derived the Maxwell-Boltzmann distribution law that made the temperature the same throughout the column.”
This wording could incorrectly be interpreted that M-B derived the column isothermal. Actually, the M-B theory starts & ends with the assumption that there are no external forces (no electrostatic, no gravity, etc.) acting on the ideal particles in the column to come up with the distribution of speeds.
That quoted statement above makes more sense if the word “made” is replaced with “assumed” to get the proper meaning. Can only correctly apply the M-B distribution to column without gravity which is then isothermal since the particle kinetic energy velocities are isotropic.
One could speculate the ideal particle speed distribution is deterministic in a gravity field also and M-B could be extended to include gravity – get a non-isothermal column in hydrostatic equilibrium due to non-isotropic particle velocities.
Trick says
“Actually, the M-B theory starts & ends with the assumption that there are no external forces (no electrostatic, no gravity, etc.) acting on the ideal particles in the column to come up with the distribution of speeds.”
Yes and when some use the MB statistics to prove the isothermal distribution in a gravitational field they are indulging in a fine bit of circular reasoning.
Albert Stienstra said @ur momisugly January 26, 2012 at 4:44 am
While The Git’s brains are subject to the force of gravity, they do not plummet through his arse. They are prevented from accelerating through there by the opposing force of electrostatic attraction between the molecules in his skull. Another way of saying this is that The Git’s brains, skull and arse are all in the same inertial reference frame. The molecules of gas in the cylinder are in the same inertial reference frame and are therefore not accelerating. That’s if I’m remembering Einsteinian relativity correctly. If I’m wrong, then my brains may well plummet through my arse 🙁
Doug Cotton says:
Temperature has nothing to do with the density of molecules. If it did, consider a near vacuum such as between the walls of a vacuum flask – does that have a temperature of near absolute zero? Hardly!
————————————————————–
Poor analogy, Doug. The space in between a vacuum flask recieves a lot of radiative energy, which just confuses the matter. And yes, a stationary molecule in a vacuum in deep space, with no interractions with other molecules which recieves little or no radiative energy, will aproximate to near absolute zero.
Please find another analogy at explains your idea better.
.
.
@Joe Kirklin Born
Let’s look at this another way. There are three particles adiabatically isolated in a box. You tell me their temperature is 100K. I say, no it’s 300K. How could we tell? What measurement could we perform that would not contradict the very conditions of the experiment? So then you tell me, oh yes, it’s 300K at the bottom, but only 100K at the top. Again, how could we tell? How could we measure this supposed lapse rate? You can stick a thermometer or a thermocouple in a real gas. You can’t in a microcanonic ensemble comprising just a few particles in all. And if we can’t measure this temperature or lapse rate, what do those concepts even mean? You’re into the land of metaphysics, not hard practical thermodynamics.
Robert Brown:
The connection of a ‘shorting’ thermal conductor to your isolated column in Fig.2 could be viewed in a similar light to a shorting wire across an electrical capacitor.
When a capacitor is shorted, the voltage across its terminals drops to zero very quickly, but, due to dielectric absorption, when the shorting link is removed, the capacitor ‘self charges’, some times up to 10% of its originally charged value (a real problem in high voltage engineering).
The dielectric has ‘remembered’ a portion of the previously applied voltage.
Many theories exist about what could cause the dielectric absorption effect, but it would appear to be related to distortion of molecules etc
A capacitor dielectric is an insulator, just like gases in the atmosphere.
Could Gravity have a similar effect on molecules, distorting orbits etc?
Your isolated column without shorting conductor would stabilise with a slightly higher temperature at the lower end and be thermally settled (ie no further change).
If the thermal short was connected, as per Fig.2, I believe that (similar to the capacitor example) the effective slight temperature difference between the top of the container and the bottom would be neutralised.
The question is, after removing the short, would the temperature gradient reinstate itself, in whole, or in part?
There is no loss or gain in energy in this system. The temperature in the column has a gradient due to gravity.
The addition and removal of the short allows for energy distribution, no work is done.
Similar to two columns filled with gas, connected by two pipe, one top, one bottom.
Inject some heat energy in the left column, the gas within the columns would circulate and the temperature would equalise, no work done, no energy lost or gained.
Some papers that record the influence that gravity has upon physical properties of matter:
“Effect of gravity on critical opalescence: The turbidity” found that gravity does effect scattering.
“Gravity influence on thermal relaxation near the critical point” specific heat is influenced by gravity.
regards
S.
Ralph says:
January 25, 2012 at 9:22 am
Stephen Goddard was expounding this ‘temperature is due to pressure’ theory some time ago on WUWT. Where is he, to defend his theory??
http://wattsupwiththat.com/2010/05/06/hyperventilating-on-venus/
Come on Stephen, what say you now?
Pretty much banned by Anthony and now has his own excellent web site at
http://www.real-science.com/
A physicist says:
January 25, 2012 at 12:29 pm Are you sure, it actually says.
“Dirac’s equation treats the centrifuge in a way that is independent of the internal operation of the machine. It is useful in that it correctly guides the experimentalist to seek foremost high peripheral velocities (v a ), second low temperatures (T),”
Further to my previous post,
Magnetism observed in a gas for the first time in:
http://web.mit.edu/newsoffice/2009/magnetic-gas-0918.html
Shows that we are just getting to understand non-obvious properties of gas.
If gas (currently under extreme conditions) can demonstrate magnetism, which is usually considered related to the alignment of molecules etc, we should not be surprised if other new physical properties emerge.
Steve Richards says:
January 26, 2012 at 6:42 am
A gas of lithium atoms at 0.00000015 K. Yes that’s relevant.
thepompousgit says:
January 26, 2012 at 3:47 am
“Gosh! I’ve been wasting money on putting greenhouse film on my greenhouse when I didn’t really need any. Whoda thunkit? People who use greenhouse film to raise the temperature around their crops are all idiots wasting their money? Don’t think so Bill…”
First I said greenhousing the collectors increases the efficiency of the system. So your argument is without merit. The system still works albiet less efficiently without a greenhouse.
Sure you can create a thought experiment that denies it works but you cannot ignore the fact that non-greenhoused passive solar water heating solutions are physically demonstrated to work each and every day.
The difference is conduction.
If you walk barefoot on a hot asphalt street during mid-day you start burning your feet. When you pick up your foot it feels cooler but mid day the temperature is not much cooler 3 inches off the ground. The difference is you removed conduction. This effect occurs regardless of whether you are inside or outside a greenhouse.
Thus the water system works despite radiation. On a sunny day when its 70 degrees at the beach according to the weather service the sand can be 120 degrees with no greenhouse. This 50 degree differential can be bled off into a water heating system via conduction and convection. Add some insulation and the fact that convection is self regulating where it only runs when the water in the storage is colder than the water in the collectors and you get without any doubt whatsoever, average water temperatures in excess of the local ambient average temperature.
There is absolutely no question this works. An argument that radiation control by a greenhouse boosts the performance is acknowledged and was acknowledged. You need to go back and read the original post.
People can complain their system does not deliver much hot water but systems have to be properly sized so as to allow for people washing dishes and taking showers and replacing the water with underground piped water which is at the temperature of the daily average ambient temperature.
We have a meaured above surface atmosphere temperature measurement going on. It is a poor proxy for surface temperature as it also should be measuring too warm. At night convection does not run so the atmosphere will be warmer than the ground by a considerable margin. During the day the surface is warmer than where its measured. But its difference is less because convection is moving the heat from the surface to the collectors at a much faster rate.
Opps I misspoke. Above I said: “An argument that radiation control by a greenhouse boosts the performance is acknowledged and was acknowledged.” was incorrect. It should read “An argument that heat loss control (ostensibly primarily by control of ambient convection outside of the collectors) by a greenhouse is acknowledged and was acknowledged.”
Yes. To flesh out the reasoning:
(1) The UF6 gas at the outer rim of a gas centrifuge is at many atmospheres of pressure and room temperature (or optionally a little warmer, as supplied by a heating coil), per these pictures of an operating centrifuge cascade.
(2) The UF6 gas at the inner axis is a very low pressure: a tiny fraction of an atmosphere (far lower than the pressure in the anvil cloud of thunderstorm).
(3) “Gravito-thermal” theory predicts that the on-axis gas temperature will be far below room temperature, in consequence of the adiabatic lapse rate combined with a large pressure gradient.
(4) Yet as it turns out, UF6 freezes via a gas->solid transition at lower-than-room temperatures (just as packets of moist air, lifted high into a thunderstorm’s anvil cloud, and thus cooled adiabatically by the pressure drop, condense their H20 vapor as hail and snow).
(5) It is observed that gas centrifuges do not fill with frozen UF6 on-axis; thus the “gravito-thermal” prediction of a stable adiabatic temperature gradient is disconfirmed.
A physicist,
“(5) It is observed that gas centrifuges do not fill with frozen UF6 on-axis; thus the “gravito-thermal” prediction of a stable adiabatic temperature gradient is disconfirmed.”
I am sorry, but, is this centrifuge operating in space away from the earth’s gravitational field?? If not, it proves NOTHING!!!
I have a question / thought experiment. Please keep in mind that I am only a student here in this discussion.
Assume that we take the non-GHG, as Dr Brown has used in his thought experiment, and placed it (1 cu. ft.) at 1 atmosphere pressure (absolute) in a baloon and placed that inside 1 cu. ft. insulated rigid container at say 70 F while aboard the space station at zero gravity. Please note the absolute pressure is 1 atmosphere so the baloon had to have a potential volume of 1 cu. ft. empty.
Next we quickly fly down and land in Florida near/at mean sea level (1 atmosphere pressure) where the temperature is 70 F. Now we have the gas still in the container at the same pressure as the ambient air pressure and the volume hasn’t changed. The only difference is that now the gas is under the influence of gravity.
First question: Did gravity heat (temp increase) the gas in the container?
Second question: If we opened the container and pulled out the baloon, would it immediately shrink or expand or stay at 1 cu. ft.? Would there be a temperature change of the gas?
Remember, I’m only a student here so your comments are appreciated.
Folks, Maxwell, Lagrange, Hamilton and the rest figured this out before there were even molecules and atoms ;-).
Thermodynamics is only a generalization. It is an absolutely correct generalization, but it is still only a generalization. It was developed before anyone knew what the bits and pieces of the world are. It is still useful and most used when someone analyzing a system doesn’t want to, or can’t, find and account for all of the real pieces of the system, atoms, photons and gravitons.
This thread shows the great difficultly of applying the “laws of thermodynamics” correctly to an open system. “Heat” is not real, it is only a useful concept. Just as “thermal equilibrium”, “ideal gas law”, “action”, etc are not real, but are only useful concepts when applied correctly. *** Molecules and elastic collisions and conservation of momentum and conservation of energy are REAL and are ABSOLUTELY TRUE. *** One only needs to look at a single molecule in motion, subject to a linear force (like gravity) to develop a “atmospheric temperature lapse rate”. The Lagrangian of the molecule is an absolutely correct description of the state of the molecule; momentum and energy are absolutely conserved. Emmy Noether is still correct today 😉
I finally plucked up the courage to read Hans Jelbring’s 2003 paper for myself. Not as painful as I expected.
My main aim in reading it was to see (a) whether his theory would apply to liquids as well as gases, and (b) whether he attributed the temperature gradient to compression/rarefaction or to some other mechanism.
On (a), his model is explicitly confined to ‘ideal gases’, but I see nothing in his reasoning that would not apply equally to liquids.
On (b), he does not offer any mechanism at all to explain the temperature gradient, which he deduces from general energetic considerations. Of course, if his proof of the temperature gradient is sound, it must stand whether or not we can find a detailed mechanism to explain it, but the lack of a mechanism is unfortunate.
But is the proof sound? It hinges on the proposition that “Nature will redistribute the contained atmospheric energy (using both convective and radiative processes) until each molecule, in an
average sense, will have the same total energy.” Since the total energy of a molecule for present purposes consists mainly of kinetic energy and gravitational potential energy (chemical, electrical, etc, energy not being very relevant), it follows from this proposition (if true) that those molecules which have high gravitational potential energy must have low kinetic energy and vice versa. Since molecules higher in the atmosphere have greater gravitational potential energy, they must have lower kinetic energy, hence the temperature higher in the atmosphere is lower, etc.
But is it true that “Nature will redistribute the contained atmospheric energy (using both convective and radiative processes) until each molecule, in an average sense, will have the same total energy”? Leaving aside the apparent neglect of conduction, and the vagueness of the phrase “in an average sense”, the proposition seems to me fallacious, at least for the point at issue. The movement of molecules in a gas is a quasi-random process. The kinetic energy and direction of motion of a molecule fluctuate from moment to moment, depending on its collisions with other molecules. Those molecules which are ‘high’ in the atmosphere (in Jelbring’s simplified world, which has no external heat source) are those which happen to have had a history in which there is a surplus of impulses in the direction ‘up’. In general there is no reason to think that they have ‘swapped’ their own kinetic energy for potential energy. Over the long term, no doubt, we would expect all molecules “in an average sense”, to have equal kinetic energy, equal potential energy, and equal total energy. But Jelbring’s proposition is not concerned with ‘the long term’, but with the actual distribution of energy at any given moment, and for this purpose I don’t think it can carry the weight he puts on it.
That’s my opinion anyway, but I’m a mere amateur, and I’m reluctant to believe that a professional scientist would commit a blatant fallacy. What do others think?
Robert Birch: “Please stop misquoting me. I said nothing whatsoever about “maximum entropy”. I didn’t even mention entropy. I didn’t need to. My “redefinition” (not actually a redefinition at all, but the utterly bog-standard original definition) is simply the statement that things in thermal equilibrium are at the same temperature; that’s what “same temperature” means.”
You’re absolutely right: you did not explicitly mention entropy. I think of a system of things in thermal equilibrium as being in a maximum-entropy state, and I inadvertently made that substitution. I apologize; the last thing I want to do is antagonize you, since you and DeWitt Payne are the only participants here who engaged substantively, at my request, on the Velasco et al. paper, and I do sincerely appreciate it.
Actually, with the exception of whether in the posed hypothetical the gas is allowed to exchange heat with the walls that confine it–you think so, I don’t–I believe we agree on the ultimate facts. It’s just that to me the most basic definition of temperature is indeed mean molecular translational kinetic energy; it avoids imprecise qualifications like “for ensembles large enough to apply said thermodynamics to.”
In fact, I think in essence I agree with what you mean by “What [Velasco et al.] do show is that their statistical mechanics confirms the thermodynamic conclusion that – for ensembles large enough to apply said thermodynamics to – the answer is indeed isothermal.” At least I agree if by that you mean that according to Velasco et al. the rate at which per-molecule average translational-kinetic-energy in a microcanonical ensemble changes with altitude approaches zero–i.e., the system approaches isothermality–as the total system energy approaches infinity. (Rather than total system energy, I’ve been speaking of the number of molecules, but there was an assumption–unstated on my part, mea culpa–that the average energy per molecule remained the same. As perusal of Velasco’s Equation 8 reveals, it’s more precise to speak in terms of total system energy.) That is, for any delta > 0 there exists an energy value E_d such that rate of change is less than delta (“is indeed isothermal”) whenever the total system energy is greater than E_d (the ensemble is “large enough to apply said thermodynamics to”). I agree with that. In fact, I emphasized it when I was discussing Loschmidt’s and Jelbring’s theories over at Tallbloke’s Talkshop.
On this thread I instead emphasized that the rate of mean kinetic-energy change with altitude–which I call lapse rate, as you don’t consider proper–remains non-zero for any finite number of molecules (more precisely, for any finite total system energy). The reason I did so was to emphasize that Dr. Brown’s refutation actually begged the question; it tacitly assumed that any non-zero lapse-rate-exhibiting system (such as the gas-wire system) would not be in a maximum-entropy state. What I think he should have based it on is a showing that a system exhibiting the specific lapse rate Jelbring argues for would not be in a maximum-entropy state.
In contrast, even if their mean molecular translational kinetic energies differed in accordance with Velasco et al.’s Equation 8, your view as I understand it is that the top and the bottom of a gas column would have no temperature difference, because heat flows from hot to cold, and no heat flows there. No temperature difference, so no lapse rate, and Dr. Brown’s refutation is adequate as is.
Incidentally, DeWitt Payne largely agrees with you, saying that “temperature is only strictly proportional to the kinetic energy in the canonical limit. . . . So you can’t directly convert kinetic energy to temperature for a microcanonical ensemble,” i.e., for a thermally isolated column of air, such as the one in Dr. Brown’s thought experiment. I don’t find this view satisfying, since it is in a microcanonical ensemble that the thought experiment in which mean kinetic energy is derived from temperature in accordance with PV = NkT is carried out –with, incidentally, no requirement that it be “large enough to apply said thermodynamics to.”
But there is one thing I’ll take back, and that is my attempt to picture the (minuscule) temperature lapse rate as resulting from offsetting diffusion and drift fluxes. That probably works as an alternate view of the pressure gradient, but for lapse rate it problematically implies that gravity drives heat flow.
Anyway, I again thank you for participating.
“”””” Jim Z says:
January 26, 2012 at 9:18 am
Folks, Maxwell, Lagrange, Hamilton and the rest figured this out before there were even molecules and atoms ;-).
Thermodynamics is only a generalization. It is an absolutely correct generalization, but it is still only a generalization “””””
Right on Jim Z, Thermodynamic properties of a system and thermodynamic state variables are MACROSCOPIC PROPERTIES, and generally relate to the statistical properties of a large number of “particles”, or if you will a large number of “degrees of freedom” to which the equipartition principle can be applied. In a “gas” in thermal equilibrium, the individual particles are not stationary, with no motion, but at any time the vector sum of the momenta of all the particles is zero, so the center of mass of the “particles” is going nowhere, or is in free flight relative to some frame of reference that is external to the system, and per Einstein’s relativity, is somewhat irrelevant as far as the internal state of the system itself.And thermodynamic equilibrium requires that the system IS isothermal, meaning it has a common Temperature everywhere. But remember that Temperature is itself a macroscopic property, since it involves the statistical distribution of the kinetic energies of a very large number of “particles.” That distribution is the Maxwell-Boltzmann distribution, that can be found in any elementary Physics text book.
The concept of the Temperature of a single particle, is meaningless in a thermodynamic sense.
However I have argued, that we can actually talk about the Temperature of individual particles.
BUT, that also is a statistical situation. At any time, we can argue that a system in thermal equilibrium, let’s say 22.4 litres of STP gas, contains 6.023E+23 particles, and everyone of those particles occupies one of the 6.023E+23 “spots” on the Maxwell-Boltzmann graph. Next time we look, all the cells are still occupied, but everyone played musical chairs, so generally nobody is in the same place on the chart. Over time it is reasonable to presume that some particular “particle” will eventually occupy each one of those cells at some time. In my whimsical phantasy world, my “Mother Gaia”, who is simply the mother of all Maxwell’s demons, can read the serial number on each and everyone of those particles from # 1 up to # 6.023E+23, so she sees them flitting around; but we can’t. So MG sees that eventually any particular particle can and will occupy every single lattice cell point on the chart, bearing in mind that each momentum value has a certain number of occupants as given by the M-B curve.
So if MG audits the movements of particle # 1066 on the M-B chart, over time, she will find that it goes everywhere on the chart, so SHE can argue that the statistical kinetic energy value of #1066 over time, is exactly the same curve, as the flash picture of the present locations of the whole assemblage of particles.
So Mother Gaia knows that over time, the statistics for any one particle is the same as the instantaneous statistics of the whole gas so she is entitled to claim that the TEMPERATURE of any one particle is identical to the Temperature of the whole system. But we can’t know that because we can’t read the serial numbers on all those particles; they are indistinguishable to us.
So that is why I argue, that Mother Gaia (MY Maxwell’s demon) has a thermometer on EVERY particle; molecule or atom or whatever, which is why in her world the weather and climate is exactly what it is supposed to be, because she has experimentally observed the Temperature of everything, and calculated what their average really is; not that the state of the climate is ever determined only by some Temperature.
We humans on the other hand do a really lousy job of sampling the Temperatures compared to MG; and also sadly compared to what the Nyquist criterion tells us that we MUST do, in order for us to get the true average Temperature correct.
So single molecules do not have any instantaneous Temperature, because Temperature relates to molecular collisions, which don’t happen instantaneously. Temperature is a macroscopic state variable, just like everything in thermodynamics.
“”””” DeWitt Payne says:
January 26, 2012 at 7:27 am
Steve Richards says:
January 26, 2012 at 6:42 am
Further to my previous post,
Magnetism observed in a gas for the first time
A gas of lithium atoms at 0.00000015 K. Yes that’s relevant. “””””
But are we talking Ferromagnetism, or is it either diamagnetism or paramagnetism, which are far less rare ?
Paul Birch: “How could we measure this supposed lapse rate? You can stick a thermometer or a thermocouple in a real gas. You can’t in a microcanonic ensemble comprising just a few particles in all. And if we can’t measure this temperature or lapse rate, what do those concepts even mean? You’re into the land of metaphysics, not hard practical thermodynamics.”
Paul, I understand your point of view, but I personally think there is a sound when a tree falls in the forest even if no one’s around to hear it.
Bill Hunter said @ur momisugly January 26, 2012 at 8:24 am
No. You said:
No numbers; just armwaving. You say “Greenhousing… only adds a few degrees to the system”. I say, if I don’t open the vents on my greenhouse early enough in the day, the crop will rapidly die. That’s not just “a few degrees”. But you remind me that I don’t have numbers. Time for some experiments so I can put some numbers on the greenhouse effect that is not The Greenhouse Effect (so to speak).