Stephen Wilde: Greenhouse Gases and the Ideal Gas Law

Posted: January 13, 2013 by tallbloke in Analysis, climate, general circulation, Natural Variation, weather

Guest post from Stephen Wilde:

Greenhouse Gases and the Ideal Gas Law
Stephen Wilde – Jan 2013

This is the usual form of the Ideal Gas Law:

where P is the pressure of the gas, V is the volume of the gas, n is the amount of substance of gas (also known as number of moles), T is the temperature of the gas and R is the ideal, or universal, gas constant, equal to the product of Boltzmann’s constant and Avogadro’s constant.

The Ideal Gas Law as set out above is a representation of certain physical relationships and is therefore not about absolute values.

It is widely known how the various terms within that equation respond to changes in any one or more of them,

To summarise:

P and V are inversely proportional to each other so a rise in Pressure results in reduced Volume and vice versa.

Increasing either P or V without reducing the other requires an increase in

n – total atmospheric mass and/or

R – the gas constant which is related to the strength of the gravitational field and/or

T – Temperature.

The product of n, R and T then rises to match the increased product of P and V.

The problem with AGW theory in relation to the Ideal Gas Law.

AGW theory proposes that an increase in GHGs causes an increase in T which then causes an equal increase in V so as to keep the two sides of the equation balanced.

So far so plausible.

However, an increase in V results in a reduction of density (n) throughout an atmosphere which must REDUCE the product of nRT.

Normally a reduction in n would be accompanied by a reduction in P as well because less mass in an atmosphere results in reduced pressure at the surface if the strength of the gravitational field stays the same but there is no reduction in P at the surface from mere expansion even though the density of the entire atmosphere reduces when V increases.

Therefore we cannot look to a reduction in P to correct the imbalance caused by the reduction in density (n).

According to the Ideal Gas Law it is not possible for PV to fail to equal nRT yet that is just what happens if one holds P steady whilst increasing T and V equally but reducing n.

There would only be balance with PV continuing to equal nRT :

if more mass were added to the atmosphere so as to avoid reducing the average density of the atmosphere when expansion occurred.

or, if the strength of the gravitational field increases to pull V back down to restore n to the previous value

Since no extra mass or gravity is being added AGW theory cannot be right because of the residual imbalance.

If a higher T from more GHGs leads to a higher V then the reduction of density (n) results in lower V which must be accompanied by lower T so there is a logical impasse.

That is what I think some sceptics mean when they say that AGW theory is impossible or contrary to the Laws of physics.

In order to resolve the problem we need to look at the same scenario differently.

What is the effect of adding more GHGs ?

We have proof that GHGs expand the region of an atmosphere in which they are situated.

Since they absorb more energy than non GHGs they spread energy more evenly across the whole area that they occupy. The effect is to reduce the rate at which temperature would otherwise decline with height (the lapse rate).

In the Troposphere the dry adiabatic lapse rate without water vapour would be about 10C per km. The presence of water vapour reduces the actual lapse rate to about 6.5C. As a consequence of reducing the lapse rate the distance required for the air to cool between surface and tropopause needs to increase and so the expanded troposphere pushes the tropopause upwards thereby expanding the troposphere beyond the height that it would have achieved without GHGs.

We see the same process in the stratosphere where ozone (a GHG) warmed by the sun actually reverses the lapse rate so that temperature increases with height up to the stratopause. The expansion of the stratosphere can push both up and down because there is no solid surface beneath it and that results in some interesting features of our climate system that are beyond the scope of this article.

So the effect of GHGs is to increase atmospheric height AND reduce the slope of the lapse rate.

As I will now explain, that is important because the combination of expansion and increased height enables the atmosphere to accommodate more GHGs without altering system equilibrium temperature.

How to approach the problem.

Note first that a constant flow of new energy from outside the atmosphere is required to maintain it in gaseous form.

If that supply of energy from external sources were to be cut off the atmosphere would simply collapse and freeze to the surface in solid form.

Those energy sources can be anything such as from a nearby sun, from geothermal energy below the surface, from nearby planetary gas giants large enough to radiate, even the temperature of space being above absolute zero makes some contribution.

Above all it must be constant because energy is also being radiated to space at an equal rate when the atmosphere is at equilibrium.

Note second that a large amount of work is constantly being done in order to keep the gases lifted off the surface against the constant force of gravity. If the work rate drops the atmosphere will contract and if the work rate increases the atmosphere will expand.

The persistence of a gaseous atmosphere despite the efforts of gravity to pull it down to the surface is due to work being done constantly.

Now, recall the problem we had with AGW theory in that we had nothing to counter the reduction in density (n) caused by more GHGs and we needed something to counter it in order to comply with the Ideal Gas Law.

The rise in V on one side of the equation offset the rise in T on the other side of the equation but we couldn’t balance the numbers because n had reduced on one side of the equation but P had been held steady on the other side.

We obviously need another variable but all we have left is the Gas Constant (R)

Let’s look at R more closely and see what can be done.

Dimensions of R

From the general equation PV = nRT we get

R = PV/nT or (pressure × volume) / (amount × temperature).

As P is defined as force per unit area, so we can also write the gas equation as

R = [(force/area) × volume] / (amount × temperature).

Again, area is simply (length)2 and volume is equal to (length)3. Therefore,

R = [force / (length)2] (length)3 / (amount × temperature).

Since force × length = work,

R = (work) / (amount × temperature).

The physical significance of R is work per degree per mole. It may be expressed in any set of units representing work or energy.

It turns out that R, the Gas Constant is not really constant at all except in clearly defined circumstances unique to each planet. R is only a constant for a fixed gravitational field, a fixed amount of atmospheric mass and a fixed height of atmosphere. Change any of those features and the amount of work required will also change and the value of R will rise or fall.

The amount of work per degree per mole will be related to the strength of a gravitational field. A stronger such field will require more work per degree per mole and the value of R will increase.

It will also be related to the amount of mass that is available to be raised off the surface. The more atmospheric mass the higher the value of R will need to be in order to lift it all off the surface.

Crucially, it will also be related to the height of the atmosphere because a higher atmosphere requires more work to raise molecules to the greater height against the continuing force of gravity so for a higher atmosphere the value of R will increase.

The solution.

If the atmosphere expands thereby rising in height the value of R will increase because more work needs to be done in order to raise the molecules higher against the force of gravity.

One can then increase the value of R in the equation nRT which will offset the reduction in n to bring both sides of the equation back into balance.

But there is another step to take.

The final step.

It turns out that it isn’t necessary for GHGs to raise T.

In the Ideal Gas Law Equation T is often taken as simply temperature but it is actually more than that.

T is the amount of energy available from all sources to maintain the constant flow that keeps the atmosphere off the surface.

GHGs do not add to that external energy source. Nor do they detract from it.

So it is wrong to include their thermal characteristics within T.

Instead they go straight to expanding the atmosphere thereby raising V

So how do we balance that on the other side of the equation without raising T?

We have determined that a higher atmosphere requires a higher value of R because more work needs to be done in order to maintain the new higher atmosphere.

So all we have to do is raise R and the product of nRT then balances again with PV at the increased V.

The increase in R is enough to offset both the increase in V on the other side of the equation and the reduction of n on the same side of the equation leading to overall balance without raising T.

What happens to the ‘missing energy’

Since there has been no increase in T the total amount of energy flowing through the atmosphere at equilibrium remains the same as before but the GHGs have absorbed more energy so where is it?

The atmosphere has expanded so the total amount of energy held within the system has obviously increased.

The answer is contained in the fact that the energy held within the system isn’t just kinetic energy which registers as heat. It is also comprised of potential energy which does not register as heat.

The higher the atmosphere is allowed to rise within the gravitational field the more of its energy content takes the form of potential energy.

So the extra energy absorbed by GHGs has all gone to increasing atmospheric height which has converted that initial kinetic energy to potential energy where it will remain until the atmosphere contracts again.

That scenario avoids the problem of imbalance inherent in AGW theory, keeps PV = nRT in balance and explains why any extra energy absorbed by GHGs is no longer available to affect equilibrium temperature.

The higher atmosphere does result in air circulation changes that potentially have a climate impact but that is another story that I have dealt with elsewhere.

1. tallbloke says:

Hi Stephen,
Interesting post. Just to kick things off, could you go a bit further into this definition:

R is the ideal, or universal, gas constant, equal to the product of Boltzmann’s constant and Avogadro’s constant…. which is related to the strength of the gravitational field and/or Temperature

Then I have a few questions about how well ‘ideal gas’ can represent the real atmosphere.

2. oldbrew says:

NASA recorded a drop in the average height of clouds of 1% (= 100ft.) between 2000 and 2010, as explained in the link [1min.28secs]. They say this ‘could function as a negative feedback mechanism’ but more study is needed.

3. tallbloke says:

Thanks oldbrew, good spot. I think that rather than clouds spontaneously lowering altitude the shrinking of the troposphere maybe a result of the cooling of the ocean due to the quiet Sun and the oceans heading into the negative phase of the ~60 year cycle. I have a strong suspicion the ‘adjustments’ to the ARGO data are incorrect.

It clearly shows that natural factors are bigger players than the co2 increase which according to Stephen’s idea here should expand the troposphere. Clearly not enough to resist the opposite tendency due to solar quiet and oceanic oscillation though.

Perhaps Richard Betts can tell us if this lowering of the cloud deck is an emergent property of the new HADgem3 model which he claims better includes natural variation.

Somehow, I doubt it… :)

4. Stephen Wilde says:

Whoah, didn’t expect it to go up yet but no matter.

I was just asking for feedback to make sure there were no glaring scientific errors but since it is up here there can’t have been :)

As regards the query about R I pulled the basic data from here :

http://en.wikipedia.org/wiki/Ideal_gas_law

As regards the issue of how different the atmosphere is from an ideal gas I suspect that it doesn’t matter because expansion, contraction and circulation deal with the thermal effects of any such difference.

The effect to be expected from human emissions of GHGs would be so small as compared to natural solar and oceanic variability that we could never identify it let alone measure it.

5. Stephen Wilde says:

Details on the Gas Constant here:

http://en.wikipedia.org/wiki/Ideal_gas_constant

6. Stephen Wilde says:

The lowering of the tropopause from natural cooling would reconvert more of the PE to KE and act as a negative warming feedback to natural cooling.

7. Joe's World {Progressive Evolution} says:

Stephen,

The assumption part is that all the planet has the same exact gas pressure at the same moment. No matter the land height differences or the planetary velocity differences. The temperature too is assumed by the art of averaging.
The planet is very different from our man-made laws…

8. oldbrew says:

The ‘gas pressure assumption’, aka the Standard Atmosphere, is used every day by the aviation industry.

http://www.digitaldutch.com/atmoscalc/help.htm

9. oldbrew says:

Spot the ‘Skeptical Science’ graphic in that NASA video. No bias there then ;-)

10. Joe's World {Progressive Evolution} says:

oldbrew,

Much of our science and technology is trial and error. Recording the results and ability to project.
BUT that still does NOT look into the science of how it ACTUALLY works. As long as it works, why look at why it works?
Power generation has done that part all along.

11. Stephen Wilde says:

“The planet is very different from our man-made laws”

There is no such thing as man made laws in science.

The real world supplies a law and we simply express it mathematically.

The fact is that for any atmosphere in equilibrium PV = nRT.

If any aspect of the system seeks to diverge from that then there is an immediate and complete negative system response from expansion, contraction and circulation changes.

The only limit as regards immediacy of response is the time it takes for the atmospheric adjustments to have full effect.

In practice, that involves constant ongoing adjustments to a multitude of forcing elements all competing against one another to make the system diverge from PV = nRT.

They always fail but in the adjustment process there are variations around the mean.

12. Joe's World {Progressive Evolution} says:

Stephen,

What is missed in that calculation is that each gas has a different density.

The other inconvenience is planetary tilting in the year…

13. tallbloke says:

Stephen says:
There is no such thing as man made laws in science.
The real world supplies a law and we simply express it mathematically.

In natural philosophy terms, this viewpoint is known as ‘naive realism’.

The real world does not supply laws. The real world changes in accordance with itself. We observe the evident relationships and measure their magnitudes, and draw up hypotheses to explain them. Well validated hypotheses become ‘theory’. Longstanding theory gets its axioms elevated to the status of ‘Laws’.

These are human intellectual constructs, which eventually fall by the wayside when a wider or deeper apprehension of nature invalidates them. Thus science proceeds.

Having said all that, I posted this article without revision because I think you are onto something fundamental and important.

However, the question remains: If the additional greenhouse gases cause the inflation of the atmosphere, then how do they do that if not by raising the temperature? If the expansion of the atmosphere then causes a cooling of it, and the same applies to solar variation, why does the Earth’s surface temperature fluctuate on all timescales? Thermal inertia in the oceans setting up longer term oscillations I would assume plays a large role. If so, why does the ~60 year term seem to predominate?

On the subject of ideal vs real gases. I raised this because James Clark Maxwell performed experiments to find the ratio of Cp to Cv in air. He found it to differ from his theoretical ideal gas calculation by a big percentage. The reason for this discrepancy is connected to Wayne’s investigation of ‘degrees of freedom’ in molecules. Maxwell realised gases don’t behave linearly as pressure increases. There are attractive and repulsive forces at work between molecules. As pressure increases and molecules are forced closer together, the repulsive forces increasingly predominate. Boltzmann worked on this aspect of stat mech.

I’m hopeful that we are homing in on the fundamental variables we need for a better theory of atmospheric thermodynamic physics with which we will be able to improve on the radiative greenhouse theory, which is incrasingly invalidated by observation.

14. Stephen Wilde says:

“The real world changes in accordance with itself”

“Itself’ being comprised of a set of ‘natural laws’ ?

Which we then try to discern and express mathematically.

“If the additional greenhouse gases cause the inflation of the atmosphere, then how do they do that if not by raising the temperature?”

Because the Ideal Gas Law works with ‘energy’ rather than ‘heat’.

That is hidden within the term R.

Additionally the term T refers not to temperature but the amount of available energy from outside the atmosphere.

The relevance of the precise nature of T and R within the Ideal Gas Law for the AGW theory has been overlooked because AGW proponents have been focusing only on radiation.

Everyone has been assuming that temperature needs to rise as you just did.

It doesn’t, because R can be adjusted to take account of the extra work being done to raise the atmosphere higher so that the extra energy from GHGs goes to PE rather than KE.

Thus work can result in either more KE (heat) or more PE (height) and it is gravity which determines the relative proportions.

And work is ever present because it needs a persistent flow of energy to keep the gases floating above the surface against the constant pull of gravity.

Energy is still conserved because energy in equals energy out (my diabatic loop) and as long as the flow of new energy continues the gases can be kept floating by a balanced energy exchange between surface and atmosphere (my adiabatic loop).

Thus all the old arguments about there not being any work done to maintain atmospheric pressure must fall away.

The old objection involving a once and for all compression of gases is inapplicable. There is in fact a constant cycle of decompression and compression with no loss or gain of energy and the work involved in that is what creates both heat (KE) and height (PE) in proportions that can vary so as to frustrate any attempts at destabilisation from anything other than more mass, gravity or incoming energy.

The total energy content is the total of KE plus PE but only KE registers as temperature.

The increase in height converts KE to PE and so no need for a higher temperature, only more total energy within the adiabatic loop.

This feeds back nicely to my earlier article about the adiabatic and diabatic loops in which I explained how shuffling energy between KE and PE can adjust the system to retain equilibrium when anything other than mass, gravity or the amount of incoming energy tries to disrupt that equilibrium.

In effect this post provides the necessary equations to support the thesis.

It turns out that the important point is to realise that R is not a true constant but rather a representation of the effect of gravity on the proportions of KE and PE relative to one another in any given atmosphere.

15. tallbloke says:

“Itself’ being comprised of a set of ‘natural laws’ ?

Last time I checked, nature was comprised of mud, grass, puddles and looking out of the window right now, snow. ;)

‘Laws of nature’ are human intellectual constructs that we fondly believe are followed by nature, until it turns out on closer observation that F doesn’t quite =MA or E doesn’t quite = MC^2

R is not a true constant but rather a representation of the effect of gravity on the proportions of KE and PE relative to one another in any given atmosphere.

This is why Wayne suggested NASA do experiments to measure the speed of sound in other planetary atmospheres. And according to Wiki, it’s the most accurate and practical way to determine R. You and Wayne are converging on the same vital variables independently of each other. Exciting stuff!

But since gravity is a constant for a given planet with a given atmospheric mass, we still have to reckon with the fact that different atmospheric compositions have different ‘degrees of freedom’ and this affects R via the Cv Cp ratio.

16. Stephen Wilde says:

‘Laws of nature’ are human intellectual constructs that we fondly believe are followed by nature.

Well, nature has its own ‘Laws’ but our attempts to formulate them are usually incomplete.

“we still have to reckon with the fact that different atmospheric compositions have different ‘degrees of freedom’ and this affects R via the Cv Cp ratio”

I don’t think we need to revise the Ideal Gas Law to accommodate that.

The issue at hand is whether more GHGs raise KE (heat) or PE (height).

Since the pressure gradient sets the lapse rate and therefore the proportions of KE and PE at any given height and composition changes make the lapse rate less steep or more steep resulting in expansion or contraction then composition changes must cause expansion or contraction rather than a higher or lower temperature so PE it must be.

So all the different compositions and their respective degrees of freedom just determine how much energy is available to cause expansion or contraction but in every case it goes to PE and not KE because gravity sets the pressure gradient.

If composition changes did not change the slope of the lapse rate then there would be no contraction or expansion and it would be KE instead of PE.

That is implicit in my description of how the Ideal Gas Law works out as described above.

The term R converts energy to work to height and that is what stops KE from increasing. The extra height converts all extra energy from anything other than more mass, more gravity or more incoming energy to PE instead of KE.

.

17. Roger Clague says:

The change from gas to liquid of H2O releases latent heat energy of condensation. This energy becomes KE.
Less PE is needed to keep the total energy constant and therefore the lapse rate is less.

The cause of the lapse rate variation and resulting changes in troposphere height is the varying water vapor concentration.

CO2 and methane are IR active but do not change to a liquid in the atmosphere and so do not change the lapse rate.

A discussion of the lapse using the gas laws is usually avoided by CAGW supporters. They say the lapse rate in the troposphere is caused by heating at the surface and cooling at the top.

18. oldbrew says:

Joe’s World says: ‘What is missed in that calculation is that each gas has a different density’

Air is 99% nitrogen+oxygen (densities 0.073+0.083). The other 1% makes little difference, leaving air at 0.075.

But dry air is more dense than moist air.

http://www.engineeringtoolbox.com/density-air-d_680.html

19. pochas says:

Things get a little messy when you try to figure out what happens when a heated, humid parcel of air rises buoyantly in the atmosphere.

20. Stephen Wilde says:

Roger Clague said:

“CO2 and methane are IR active but do not change to a liquid in the atmosphere and so do not change the lapse rate”.

Thank you. I wasn’t sure about the characteristics of each type of molecule.

If those molecules have greater absorption capability then they should affect R.

However, if those molecules are also more radiatively active then their radiation out to space reduces the increase in R that would otherwise have occurred.

So if Roger’s point is correct that CO2 and methane do not change the lapse rate then the only reason can be that their radiative ability keeps them and the surrounding air cool and so there is no extra energy left to cause R and V to rise.

That would be consistent with such molecules providing an additional radiative window to space that is absent for non GHGs.

Indeed there is a case for suggesting that such cooling capability of CO2 and methane would actually contract the atmosphere and reduce R.

It just depends where the balance lies between their absorption capability and their radiative ability.

So, either way my thesis is unaffected.

Note that T in the Ideal Gas Law relates only to energy supplied from outside an atmosphere. For most purposes that is synonymous with temperature but one needs to be more specific when starting to consider energy building up within the atmosphere from other causes.

If additional energy develops within the atmosphere from any cause other than from extra mass, gravity or energy input from outside then it is the value of R that is affected not T and V then responds to comply with the Ideal Gas Law.

As I stated some time ago (I think in the N & Z thread) the Ideal Gas Law operates so as to regulate energy flow through the atmosphere so as to retain top of atmosphere radiative balance.

The radiative flow is a consequence and not a cause.

21. Bill Gilbert says:

Stephen,

You have confused the term “n” in the ideal gas law with mass and density. “n” is the number of “moles” in the system, which is not the same as mass. A “mole” is an Avogadro’s number of atoms or molecules and its mass is dependent on the molecular weight of the atoms or molecules. As V increases, the density does decrease, but “n”, the number of molecules in the system, remains constant. “n” is not density. “n” does not decrease when V increases.

“R” is a universal constant and does not change with height, mass, etc.. R only changes when the gas deviates from “ideal” and demonstrates intermolecular forces. Remember that R is also based on “moles” not mass. The “specific gas constant”, by definition, will vary with molecular weight, but that is not what is used in PV = nRT. R has nothing to do with work lifting mass off the surface of the planet.

The presence of GHG’s does not affect the lapse rate since they are in local radiative thermodynamic equilibrium with their surroundings. The dry adiabatic lapse rate as derived from the first law of thermodynamics can be expressed as dT/dh = -g/Cp. The presence of GHG’s only affects Cp, and then only to a very trivial degree (water vapor is the biggest by far). Water vapor changes the lapse rate due to the release of latent heat upon condensation, not because of its heat capacity or radiative properties.

For a thorough discussion of PE, KE and work in the troposphere see the “Physics” section (pg.265) of my paper here:

http://www.friendsofscience.org/assets/documents/Gilbert-Thermodyn%20surf%20temp%20&%20water%20vapour.pdf

Bill

22. tjfolkerts says:

“n is the amount of substance of gas (also known as number of moles)
and
“However, an increase in V results in a reduction of density (n) throughout an atmosphere which must REDUCE the product of nRT.
You can’t define “n” in two different ways and expect to get a self-consistent set of conclusions!

“It turns out that R, the Gas Constant is not really constant at all except in clearly defined circumstances unique to each planet.”
No — R is a universal constant with a specific, well defined value INDEPENDENT of gravity, temperature, etc.
R = 8.3144621(75) J K−1 mol−1

23. wayne says:

Stephen and tallbloke, that’s a needed post and I’ve been out of place for a while so it it’s going to take a while to digest what everyone is saying, or trying to say. I have already noticed some statements that might be misconstrued without knowing some of the previous discussion. TB, thanks for the kind words, you do understand what has been said, sometimes it hard to tell if the words are going anywhere at all.

Gonna take a while to catch up, I’d like to understand the points Bill Gilbert is raising, most of his statements sound right on the surface and he might be able to help undercover why the heat capacity ratios seem a bit odd when matched to the atmospheric profile.

24. suricat says:

Bill Gilbert says: January 13, 2013 at 9:51 pm

I whole heartedly concur Bill. Nice explanation. :)

The ‘mole’ “n” should be a true representation of the ‘kinetic energy equivalence’ of the gas mixture. My god, it’s years since I did this stuff, but this brings it all back. However, this is also representative of the ‘chemical equivalence’ of a substance (the reason why it was first brought into being).

There’s a problem with this approach. I don’t see a problem with ‘chemical equivalence’, but there’s something ‘missing’ from the ‘thermodynamic’ approach. How to represent ‘internal molecular energy’? Perhaps I’ll address this later.

This is where ‘classical physics’ fails and we need to look to ‘micro physics’ for an explanation. All well and good, but there’s a ‘bridge’ that needs to be built between these two disciplines and it isn’t completed yet (if it ever can be). Classical physics explains the ‘general’ nature of things, whereas micro physics explains the ‘reality’ of an observation, ‘in small detail’, and offers a divergent ‘classical’ (macroscopic) outcome. :(

The Clausius Clapyron relationship ‘rules’ for ‘humidity’ at the surface, but atmospheric physics is ‘something else’. It’s a nice innovation to show, in your linked ‘.pdf’, that the 1-2k altitude is warmer than the surface atmosphere (adiabat excluded). AFAIK, this is consistent with Ferenc’s current investigations (to my understanding) and critical for Earth’s radiation sig.

One thing I may add. Upper atmosphere ‘rains’ into a lower atmosphere. Thus, a weak precipitation ‘parcel’ becomes ‘stronger’ with altitude, whereas a strong precipitation ‘parcel’ becomes ‘weaker’ with altitude (perhaps this is the ‘difference’?).

Best regards. Ray.

25. Stephen Wilde says:

Bill Gilbert said:

“R only changes when the gas deviates from “ideal” and demonstrates intermolecular forces. Remember that R is also based on “moles” not mass. The “specific gas constant”, by definition, will vary with molecular weight, but that is not what is used in PV = nRT. R has nothing to do with work lifting mass off the surface of the planet.”

The point is that composition variations cause the gas to deviate from ‘ideal’ and so R does change.

The definition of R is set out above and summarised as:

“The physical significance of R is work per degree per mole”

and you will see that ‘length’ is mentioned so just substitute height for length.

So composition variations can move the gas away from ‘ideal’ which causes height (length) to change which alters R because more work is required to achieve the changed height and at the same time V changes in order to keep the equation balanced.

“As V increases, the density does decrease, but “n”, the number of molecules in the system, remains constant. “n” is not density. “n” does not decrease when V increases.”

I note that and I think I have dealt with it.

Density is the number of moles in a given volume.

Moles represent the amount of substance of all types and characteristics and are therefore an accurate indication of total mass.

“the quantity which describes the mass of a given amount of substance is the molar mass”

from here:

http://en.wikipedia.org/wiki/Amount_of_substance

The number of molecules in the system does remain constant which is why P has to be held steady despite the reduction in density.

However that reduction in density must still have a physical effect so R and V change instead of n and P.

The change in R deals with both the increase in V and the reduction in n to keep the equation balanced.

I explained that in my article.

26. Stephen Wilde says:

Of course, if n does not change when density reduces that could simplify my case because the change in R would only have to match the change in V rather than covering the reduction in n as well.

My interpretation of n, though, was the amount of substance in each unit of volume rather than in the entire system in which case it would be relevent on a parcel by parcel of air basis if not in relation to the entire system.That must be right because the Ideal Gas Law is usually applied for free floating parcels of air so that reduced n causes a fall in density resulting in a rise in height and a reduction in P.

The Ideal Gas Law is a rather odd construct because it deals with individual parcels of air very well but when considering an entire atmosphere it requires reconsideration because in that latter case P is constrained from varying by the persistence of total n despite reduction in density within the atmosphere.

It is that inflexibility of P for an entire atmosphere that creates the need for R to vary instead.

For example if P drops when V increases there is no need for R to change because the reduction in pressure reduces the work required to lift a molecule to the increased height for a zero change in R.

It is the holding of P steady when V changes for a planetary atmosphere that forces R to change in order to maintain equilibrium.

So what we have here is the fact that composition variations can upset the Ideal Gas Law by making the Gas non ideal.

That change cannot alter P because the mass in an atmosphere is fixed. It can only alter V so a matching change is needed on the other side of the equation.

If V changes but P stays the same then more work R is required to lift the molecules to the new height so it is R that must change and not T because more work creates more PE at the expense of KE. Shifting KE to PE obviates the need to add any more energy to T.

That is consistent with the fact that T represents total energy coming in from outside the atmosphere so additional energy arising for any reason within the atmosphere cannot be added to T whereas AGW theory requires just that.

Holding P steady whilst altering V can only alter R and not T.

Allowing P to vary whilst altering V needs no change in either T or R because P and V vary in inverse proportions to each other for a zero net effect on their side of the equation.

27. Stephen Wilde says:

R is a universal constant with a specific, well defined value INDEPENDENT of gravity, temperature, etc.
R = 8.3144621(75) J K−1 mol−1

My source says:

“The physical significance of R is work per degree per mole”

That does not preclude there being a universal constant out in the open universe.

However, in relation to a specific planet with specific characteristics the calculated value will vary along with those characteristics.

The Ideal Gas Law describes the fixed relationship between the various terms and all of them can vary as necessary to ensure that PV = nRT at equilibrium.

The thing that seems to have caused confusion and error is the effect of holding P steady when other terms change. One must hold P steady on a planet with an atmosphere because surface pressure remains the same regardless of V.

Within the atmospheric column the problem of fixed P is resolved by moving parcels of air up and down which does allow P to vary within the parcel of air.

Having effectively removed the variability of Pfor the planet as a whole it becomes necessary for something else to ‘give’ on the other side of the equation when V alters if the Law is to be complied with.

That somethingelse is R, not because the universal constant is changed but because the physical characteristics of the planet change so as to interact differently with the universal constant.

R in the Ideal Gas Law is not the universal constant in itself. It is in fact the outcome of the interaction between the planet and the universal constant.

28. JCrew says:

Many seem to see the Gas Law without the presence of gravity. Gas in the atmosphere is constantly doing work. The presence of gravity is a base factor to this work. So which parameter in the Ideal Gas Law is most affected? I think Steve is helping us think this through.

Picture a tennis match where the tennis ball flies through the air. Is gravity playing a role in how the air swirls?

29. tallbloke says:

Does this comes back to Loschmidt and the question of whether the isotropic distribution of energy in the equilibrium atmosphere has a thermal gradient due to gravity or not?

30. Stephen Wilde says:

I don’t know much about Loschmidt but what it does come back to is the pressure gradient determining how much of the total energy at any given height needs to be in the form of KE which does affect temperature and PE which does not.

That results in a thermal gradient as a side effect because the amount of energy in KE form tapers down with height and the temperature as measured by sensors that measure only KE must follow.

The energy is still all present and correct but the non sensible portion (PE) increases with height.

Is that accommodated by Loschmidt’s work ?

31. michael hart says:

My understanding is in correspondence with Bill Gilbert. n=number of moles, not mass, not density. The molar gas constant is for a “perfect” gas, an approximation which sometimes produces acceptable results, depending on circumstances. The perfect-gas equation is also described as being derived from the “kinetic theory of gases”. I think this a better starting point to understand the explicit assumptions because it does so at the molecular level (perfectly elastic colliding spheres of negligible volume, no higher-order interactions, a hypothetical container without walls….etc). It helps understand why a mole of hydrogen gas (molecular weight=2) occupies exactly the same volume as a heavier, larger mole of carbon dioxide (molecular weight=44) if treated as a perfect gas.

Regarding “work”, I think one needs to careful with words/definitions. A satellite not subject to frictional losses requires no input of work (units Joules, as with heat and energy) to maintain it’s orbit. Work is required to raise the orbit further. If it collides with another, perfectly elastic satellite it may go up, and the other may go down, net result no work. But the Second Law says that a perfectly elastic collision is not possible at any temperature above absolute zero. Even Helium does not make a perfectly elastic sphere. And the Third Law says you cannot reach absolute zero. So energy will be lost as a result of “satellite” collisions, possibly by radiation to space etc.

Sometimes, more complicated events occur: more than two “satellites” colliding at the same location, phase changes, chemical reactions, changes due to EM radiation, or interacting with the “walls” of the big round container called “Ground”…

32. Stephen Wilde says:

Thanks to michael hart for the refinement of definitions but does any of that alter the outcome ?

I think we should beware of overcomplexity in what is at base a pretty simple proposition.

As regards the comments of Bill Gilbert I addressed them at 10.04am

33. Bill Gilbert says:

Let’s get back to basics. R is not a variable in the equation PV = nRT. The variables are P, V, T and n. R is just a constant correction factor that adjusts for the various units of measure used in the four variables.

R = k Na

Where k is the Boltzmann constant = 1.38066 x 10-23 J/K. This is just a correction factor to convert the microscopic kinetic energy of atoms/molecules (measured in Joules) to the arbitrary macroscopic measurement of T established by Kelvin. Na is the Avogadro’s constant = 6.0221 x 1023 /mol which just defines the number of atoms/molecules in a “mole”. This is just another unit of measure correction factor. It was based on the number of atoms in 12g of carbon-12. (Chemists like to use moles so they can balance chemical equations which involve discrete atoms/molecules).

Thus R is just a unit of measure correction factor. It does not vary unless the units of measure in the equation are changed. The SI units of measure for R is J/mole K. The only reason that the numerator (joules) is expressed as work energy in the Wikipedia example is that the left side of the Ideal Gas Law equation is PV, which is a measure of work (force x distance) – but the unit is simply joules, a measure of energy.

R is not a variable, it is a constant.

If one wants to deal with gravity and work in the atmosphere, use the first law of thermodynamics, not the ideal gas law (see my paper link above).

Bill

34. Stephen Wilde says:

Bill.

If R is a measure of work done in the Ideal Gas Law then it varies if more work per degree per mole needs to be done as a result of changes in the other terms.

“The physical significance of R is work per degree per mole”

Clearly, if the atmosphere expands resulting in greater height then more work must be done to maintain that height and since the temperature change is the same but across a greater distance then the work per degree per mole must be greater against the gravitational field.

Likewise a lower atmospheric height would result in less work per degree per mole against the gravitational field because the height to be maintained would be less

I don’t understand your assertion that R does not change unless the units of measure in the equation are changed.

” It may be expressed in any set of units representing work or energy.”

from here:

http://en.wikipedia.org/wiki/Ideal_gas_constant

35. Stephen Wilde says:

Bill.

I’ve had a look at your paper and amongst other things you say this:

“When a higher concentration of surface water vapor exists, which subsequently
condenses thus releasing more latent heat, more work energy (PdV) will be
needed to convectively cool the parcel and move the temperature profile closer to

Presumably that means that the parcel must rise to a greater height and as I implied in my article one needs the greater height to move the temperature profile closer to the DALR.

If R represents work per degree per mole doesn’t your comment impliedly acknowledge that R must be variable ?

I actually agree with much of what you said in that article. Is it really inconsistent with a variable R ?

36. Bill Gilbert says:

Stephen,

You have the right ideas as to what is physically occurring in the troposphere, but you can’t get there using the Ideal Gas Law. You can’t make R what it is not. It is simply a correction factor for all the historically independent and diverse decisions made on units of measure for the various state functions (T, P, V). The biggest culprit was temperature.

I quote from your Wikipedia reference:

“Physically, the gas constant is the constant of proportionality that happens to relate the energy scale in physics to the temperature scale, when a mole of particles at the stated temperature is being considered. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of the energy and temperature scales, plus similar historical setting of the value of the molar scale used for the counting of particles.”

“It is equivalent to the Boltzmann constant, but expressed in units of energy (i.e. the pressure-volume product) per temperature increment per mole (rather than energy per temperature increment per particle).”

Perhaps this example from the kinetic theory of gases will make this clearer:

KE = 3/2 k T (joules/molecule)

KE = 3/2 R T (joules/mole)

As you can see, the Boltzmann constant, k, just converts temperature to energy per molecule. The gas constant, R, just converts temperature to energy per mole. They both are just conversion factors and are constant, not variable. R has nothing to do with “work”.

What you are trying to do is easily done using the first law. In fact I have derived a first law equation that predicts the height of the tropopause by using the concept of equivalent potential temperature. I have tested it out using radiosonde data and it works. Maybe I’ll try to get it published some day if I can find somebody to peer review that understands atmospheric thermodynamics.

Bill

37. tallbloke says:

Bill: We managed to assemble some good people here (including you!) for the Loschmidt discussion. Would you consider open peer review here? You would be in the Vanguard along with Hans Jelbring and Nikolov and Zeller. Good company!

38. wayne says:

Stephen, I had given you enough information and hints in the past few threads. The problems you are having is your are trying to lay blame on ‘R’, a constant, in the ideal gas law and instead you need to look into the Cp/Cv heat capacity ratio and how that play’s out in the potential temperature equation when that exponent makes the lapse rate identical to the atmospheric profile in the standard atmosphere… it all leads to Cp and Cp/Cv, not R. You seem to be barking up the wrong tree.

If this has to do with your loop, I’ll call it the inter-atmosphere energy loop instead of an adiabatic loop, a no energy transfer loop, you also need to look at what happens after condensation occurs in clouds. What happens to that moisture? Most of the time there are clouds forming and dissipating but no precipitation so most of the evaporation is not from the surface, it is recycled moisture from the clouds that have dissipated as a huge heat pipe. A dissipated cloud just becomes part of the area between clouds that is sinking, but since the area between clouds is usually much greater that the area of the thermals carrying energy upward it sinks slowly heating by the potential temperature curve and re-evaporating as it sinks. That moisture is eventually back at the surface absorbing solar and surface radiation and conduction and up it goes again.

The air that is right at the surface of the oceans is close to saturation unless actual precipitation has occurred nearby making it drier and actual surface evaporation will then occur. When that is true, that the surface air is saturated, RH near 100%, there is no net evaporation from the surface for there is just as much condensation from air to the water, it’s a balance in the bottom millimeter where the air touched the water. But your loop is still grinding away recycling the moisture that is already in the air and the energy is the only thing with a net loss, upwards and out.

39. wayne says:

Stephen, you know, that thought in my last comment probably has a tie to the meteorological visibility. When more humidity is present the visibility lowers, from 20 mile to 10 mile to 5 miles (not speaking of smoke of smog), and what is limiting you visibility is the micro-droplets as the air between thermals sinks and the temperature rises and causing these micro-droplets or multi-molecule clumps of water vapor to evaporate, split apart, soaking up energy, that will eventually get caught up again in a thermal and carried upward.

Some days a sailplane can get a mediocre day even if no clouds are visible but the thermals are still there though weak. But if I remember correct (that sailplane short jaunt occurred decades ago), as you go upward the visibility decrease and gets noticeably hazy on such days depending on your altitude even though there are no clouds. That haze is the condensed micro-droplets.

Just trying to tie it to actual observations I have experience in the past.

40. Stephen Wilde says:

So this is wrong ?

“Dimensions of R

From the general equation PV = nRT we get

R = PV/nT or (pressure × volume) / (amount × temperature).

As P is defined as force per unit area, so we can also write the gas equation as

R = [(force/area) × volume] / (amount × temperature).

Again, area is simply (length)2 and volume is equal to (length)3. Therefore,

R = [force / (length)2] (length)3 / (amount × temperature).

Since force × length = work,

R = (work) / (amount × temperature).

The physical significance of R is work per degree per mole. It may be expressed in any set of units representing work or energy.”

From here:

from here:

http://en.wikipedia.org/wiki/Ideal_gas_constant

41. Stephen Wilde says:

Reading that link in more detail I wonder whether the term R in the real world of a mixed non ideal gaseous atmosphere refers to the specific gas constant and not the universal gas constant.

If the former then obviously it would vary with composition unlike the universal gas constant.

And the further the specific gas constant diverges from the universal gas constant the more the other terms would need to adjust to maintain equilibrium.

The amount of work must vary according to the scale of any divegence.

So, as before, any divergences from the ideal gas lapse rate are compensated for in the way I set out.

42. Stephen Wilde says:

This extract appears to resolve wayne’s objection by linking the specific gas constant to Cv and Cp:

“Another important relationship comes from thermodynamics. This relates the specific gas constant to the specific heats for a calorically perfect gas and a thermally perfect gas.

Rspecific = Cp – Cv

where cp is the specific heat for a constant pressure and cv is the specific heat for a constant volume.

It is common, especially in engineering applications, to represent the specific gas constant by the symbol R. In such cases, the universal gas constant is usually given a different symbol such as R to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.”

I also see that the term R in the Ideal Gas Law does seem to be the symbol for the specific gas constant and not the universal gas constant.

43. Stephen Wilde says:

The correct sequence of events would therefore seem to be that composition variations first cause the specific gas constant R to diverge from the universal gas constant which potentially destabilises the lapse rate so in order to maintain equilibrium volume increases to compensate for the non ideal lapse rate.

44. suricat says:

Stephen Wilde says: January 14, 2013 at 8:00 pm

“I actually agree with much of what you said in that article. Is it really inconsistent with a variable R ?”

Absolutely! Though you may be confounding ‘a constant’ with ‘a variable’ from/with ‘phase transition':

http://en.wikipedia.org/wiki/Phase_transition

You have my sympathy Stephen, TB really did publish your post too soon. :(
[Reply] I have a lot on my plate, and open peer review is the quickest and most effective way to develop ideas and resolve misunderstandngs over definitions. There is no shame in making mistakes, or collaborating to correct them and make progress. Rog

Your approach is OK for a ‘DALR’ (Dry Adiabatic Lapse Rate), but a DALR is a rare beast in ‘nature’ (which is what we ‘observe’). The ‘lapse rate more naturally encountered’ (ELR, or environmental lapse rate), almost always encounters the production of ‘WV’ (water vapour) ‘condensate’ (liquid), or ice, at some altitude.

Hm! Yes, clouds are a real pain, and I think you have tried to include these with PV=nRT, but there isn’t a place for ‘phase transition’ in this equation. Thus, the thermodynamics are incomplete.

Best regards, Ray.

45. Ray C says:

wayne says:
January 14, 2013 at 10:51 pm
“……. That haze is the condensed micro-droplets.”

Wayne, I think atmospheric haze is water vapour condensed on aerosols. The WV needs a surface to change phases on or rather finds it so much easier to so. Micro-droplets are water molecules collecting on the relatively flat surface of an aerosol, that is how they become bigger multi-molecular clumps,(one way at least). Thing is there are a lot of existing aerosol clumps. Aerosols “play” with water vapour and energy!!

46. Stephen Wilde says:

I think I must disagree with the nay sayers here for the following reasons:

i) It is clear that R is only the universal constant for an ideal gas which never exists in reality. For all non ideal gases R is the specific gas constant and has a unique value for every mixture of non ideal gases. Thus it is variable depending on composition. That deals with Bill Gilbert’s objections.

ii) The specific gas constant bears a relationship to Cv and Cp so that deals with wayne’s objections.

What happens is that in so far as any non ideal gas mixture forces the lapse rate away from the ideal lapse rate then V must change because P is held steady.

The increase in height offsets the effect of the changed lapse rate and all additional energy goes to PE and not KE for no change in temperature.

As regards CO2 the position depends on whether it does or does not affect the lapse rate. If it does then V changes to cancel the effect. If it does not then it has no thermal effect in the first place.

Whether CO2 does have net effect or not depends on the balance between absorption capability and radiative capability but that really doesn’t matter because whatever the net effect the lapse rate and V simply change to negate it.

47. Stephen Wilde says:

The significance of water vapour and its phase transitions is simply to provide an alternative way of regulating lapse rate and V.

It is herefore included in my proposition already as is every other mechanism that affects lapse rate and V.

Once properly interpreted the Ideal Gas Law provides a complete solution.

48. tallbloke says:

Stephen, that’s a good clear response which Bill and wayne should be able to address.

“The increase in height offsets the effect of the changed lapse rate and all additional energy goes to PE and not KE for no change in temperature.”

But it doesn’t offset it completely if the actual average environmental lapse rate is 30% lower than the DALR does it? So is that difference the expression of the water vapour (plus co2?) component of the greenhouse effect? (along with all the other factors we’ve identified which contribute to the Earth’s surface being warmer than the Moon’s)

49. Stephen Wilde says:

Rog,

I think you have taken it that the process restores the actual lapse rate back to the ideal lapse rate but it does not. I admit that until recently I thought that must be the case too.

Having analysed the Ideal Gas Laws as above I now realise that the actual lapse rate can diverge from the ideal lapse rate and remain diverged from it so long as V changes appropriately.

The actual lapse rate remains diverging from the ideal lapse rate but the change in height corrects the change in the rate of flow by allowing more time (distance) for the energy to reach the top of the atmosphere. Such additional time (distance) is required because of the less steep slope which represents a slower flow of energy from surface to space.

Furthermore the change in V is a result of the net change in lapse rate caused by all other factors combined.

So, if the composition changes the lapse rate but other processes offset that change in part or in whole then the amount of change in V that is required will reduce.

The process that I have described is a ‘mopping up’ process that effects a final adjustment when all other factors have been netted out.

If R and V change then temperature does not need to because each changes its own side of the equation by the same amount leaving P (at the surface), n (for the entire atmosphere) and T (energy from outside sources) all unchanged.

The extra energy within the atmosphere is all used up in maintaining the change in V and to do that it is converted from KE to PE.

The biggest error has been in people not realising until now that R (being specific and not universal) varies with composition and V follows to maintain thermal balance. Everyone mistakenly thought that V and T changed but T doesn’t change if there is no change in energy supply from outside the atmosphere.

When R changes due to composition varying the only possible correction is via V because P at the surface is fixed however much the atmosphere expands.

50. wayne says:

Ray C, that’s very true, on the seed aerosols. Thanks for the clarification.

But I just didn’t want it to distract from the entire point of what happens to the moisture when clouds dissipate on cloudy cumulus rain-less days which is more the normal on the hemisphere having summer. If your patient in the summer, you can watch this happening over and over and over again in any given hour but it’s slow as molasses, clouds form, clouds disappear… what is really going on with the thermal energy we keep speaking of and the moisture? The moisture is not falling out as rain so the moisture forming the clouds must not be coming from evaporation from the surface to any large degree, it’s recycled, but the condensation/evaporation cycle keeps a chugging along all day long.

51. Stephen Wilde says:

Note that a higher atmosphere results in a more vigorous water cycle (part of the adiabatic loop) but because a complete circuit takes correspndingly longer there is no more KE returning to the surface than before, the excess having been converted to PE and retained in that form as long as the height is maintained.

However, the phase changes of the water cycle on Earth do then shift energy higher for an overall enhancement of the water cycle.

It also raises the height of GHGs which can then radiate to space more easily.

Thus, on Earth, the cooling effect of the water cycle and GHGs increases radiative flow to space and that partly offsets the need for an expansion of the atmosphere. It doesn’t seem to offset it completely though because the actual lapse rate is observed to still be less steep than the ideal lapse rate and so the atmosphere remains higher than it otherwise would be.

As I said previously, more GHGs change radiative flow to space by changing atmospheric heights (via R affecting V) so that the adiabatic loop works more hard or less hard depending on the net thermal effect of GHGs.

That net thermal effect from CO2 in particular is not yet clear.The net thermal effect seems to differ from one GHG to another because of their other varied physical characteristics.

But in the end whatever their physical characteristics it is Rspecific that changes and has a knock on effect on V in the way I set out above.

52. Edmonton Al says:

Stephen Wilde’s explanation is perfectly clear to me.
A greenhouse has a fixed V; the earth doesn’t.
Increase the T and V goes up to balance it.
The atmosphere holds more heat[PE] but the T stays about the same.
What did Occam say?

53. Stephen Wilde says:

Look here:

http://www.aos.wisc.edu/~aalopez/aos101/wk4.html

“R is a constant, its value changes with each gas.
For dry air , R = 287 J/kg K.”

Its VALUE CHANGES with each gas.

Thus a non ideal gas has a different value of R to an ideal gas and every other non ideal gas.. Work is required to change V. The energy expended on work is converted to PE. PE does not register as heat. No change in T.

54. Stephen Wilde says:

Correction of typos – too many distractions here at work :)

“Increase the T and V goes up to balance it”

No. Increase R and V goes up to balance it. No change in T.

“The atmosphere holds more heat [PE] but the T stays about the same.”

The atmosphere holds more energy (KE + PE) but the additional energy is in PE form which does not affect T.

55. Tim Folkerts says:

Stephen!

You are getting things SO muddled!

“Its VALUE CHANGES with each gas.
In the equation you were referring to, they had written the ideal gas law in terms of MASS not MOLES. IE

PV = mRT, not PV = nRT

The first R = PV/mT is the specific gas constant, which depends on the molar mass of the gas (which is the whole meaning of the word “specific”). The second R = PV/nT is the universal gas constant, which is — well — “universal” for all ideal gases. For the gases in the atmosphere, the ideal gas approximation is good enough for almost any purpose (other than the condensation of H2O).

You can’t jump back and forth from one R to the other R and make any sort of logical sense!

You have now taken one of the most basic equations in thermodynamics — PV = nRT — and tried at various times to redefine n, R and T!

56. Stephen Wilde says:

Tim,

How would your point about mass and moles make any difference to the proposition (not mine) that the value of R changes with each gas ?

I certainly can treat the specific gas constant for non ideal gases as separate from the universal constant for ideal gases because it is well understood that non ideal gases cause the lapse rate to diverge from the DALR yet you seem to suggest that that should be ignored.

Combining the change in slope with the change in volume so as to alter the proportions of KE and PE is what prevents T from changing so it isn’t a surprise that you want to ignore it.

57. Stephen Wilde says:

Where Tim may have a point is that perhaps my article should have used PV= mRT for the specific gas constant rather than PV = nRT for the universal constant.

That is the whole point of this open peer review but I don’t see that it makes any difference to the basic proposition if I do shift to PV=mRT

58. oldbrew says:

My 1959 physics textbook by Norman Feather says it should be:

[quote] pV = m/M * RT
where ‘m’ is the mass of an ideal gas of molecular weight, or, strictly, of molar mass, ‘M’ [end quote]

59. wayne says:

True oldbrew and n = m/M and also Rs = Ru/M.

Many times just R is used and that can be more than a bit confusing if it’s not specified.

It’s a lot easier to use the fact that ρ = V/m and let V=1 cubic meter as a unit volume and just let
P/ρ = Rs•T, no volume involved thereby not trying to describe an entire column at once but instead, two unit volumes at different altitudes. Then you have just four equations to deal with.

Stephen’s post seems to be caught in a circularity in the tornado of proper ideal law equations:

P = (m / V) • (Ru / M) • T
m = P • V • M / ( Ru • T)
M = m • Ru • T / (P • V)
P • V = (m / M) • Ru • T
Rs = P / ((m/V) • T)
m = P • V / ( Rs • T)
Ru = P • V / (n • T)
Ru = P • V / (n • T)
P • V = n • Ru • T
P = (ρ) • (Rs) • T
T = P / (ρ • Rs)
Rs = P / (ρ • T)
ρ = P / (Rs • T)
Rs = (Ru / M)
n = (m / M)
ρ = m / V
m != M
!!!!!!!
!!!!!
!!!!
@

Or you can just use kB and the molecular mass with Na and get an equal number of equations or mix them all up and the number of correct equations explodes! ;)

Maybe slowing down a bit and specify Stephen, use some number examples even if the numbers are not perfect or have too much precision with a calculator and everyone would be able to follow your examples closer.

You say you want some suggestions, well there’s mine. Keep it simple and crystal clear with examples and not too many equations.

60. Stephen Wilde says:

I’m glad it isn’t just me who is confused by the plethora of alternative terms of expression.

I think we should stay well away from unnecessary detail because it really is so simple.

Just regard R as the specific gas constant unique to each compositional variant for an atmosphere as a whole. As soon as one changes composition then the value of R changes.

As soon as one does that on one side of the Ideal Gas Law equation one needs an adjustment on the other side to restore balance.

T is not involved at that point.

P on the other side is fixed because there is no change in total atmospheric mass or the strength of the gravitational field so only V can vary to match the change in R.

Therefore any change in composition immediately affects volume (V), still without affecting T.

The change in volume is what prevents a change in T because a larger volume always results in a lower T as per the Ideal Gas Law.

It is true that the new composition holds more total energy so why is T not affected ?

Well, if T were affected having changed R on one side and V on the other then the system would be out of balance again so how is that extra energy in the atmosphere to be dealt with without changing T ?

A change in V results in a reapportionment of the energy within the system between KE and PE because a greater or lesser height without changing the inflow of energy from outside the atmosphere must result in an increase or decrease of PE relative to KE.

So as soon as composition changes (in any way) then the value of R changes to reflect the new value of the specific gas constant, then V changes virtually instantly and all the additional energy retained within the atmosphere goes straight to more PE which does not register as heat so no change in T.The total of PE + KE has increased but KE remains the same so T is unchanged.

There are however changes in circulation within an atmosphere of changed height.

Now one might say that this supports AGW theory in so far as it links atmospheric composition changes with circulation changes and circulation creates climate zones so a changed circulation will change the sizes positions and intensities of those zones.

However, the entire atmospheric mass is involved in determining the value of R as the specific gas constant so changes from human sources are infinitesimally small.

AGW theory proposes that more CO2 (a compositional change) affects T whereas in fact all compositional changes affect R and not T.

The value of R changes constantly to reflect changing atmospheric composition so the balance between PE and KE also changes constantly to maintain stability..

61. tallbloke says:

Stephen,
If R is defined by the speed of sound, then R will be affected by altitude too. Speed of sound changes at different pressures for non-ideal gases. Also note that Wayne’s ‘degrees of freedom’ comes into this, in the section I have bolded below.

http://en.wikipedia.org/wiki/Speed_of_sound

The speed of sound in an ideal gas is independent of frequency, but it weakly depends on frequency for all real physical situations. It is a function of the square root of the absolute temperature, but is independent of pressure or density for a given ideal gas. Sound speed is slightly dependent on pressure only because air is not quite an ideal gas. In addition, for different gases, the speed of sound is inversely dependent on square root of the mean molecular weight of the gas, and affected to a lesser extent by the number of ways in which the molecules of the gas can store heat from compression, since sound in gases is a type of compression. Although (in the case of gases only) the speed of sound is expressed in terms of a ratio of both density and pressure, these quantities cancel in ideal gases at any given temperature, composition, and heat capacity. This leads to a velocity expression in ideal gases using only the latter independent variables.

62. tallbloke says:

Great article on the stratosphere temperatures by Doug Hoffman at Resilient Earth:

http://theresilientearth.com/?q=content/science-gets-stratosphere-wrong

63. Stephen Wilde says:

Rog.

The conversion of changes in composition (via a change in the value of R) to an increase in V would presumably happen at the speed of sound because it is a mechanical proces rather than a radiative one and so is affected by density.

Unless someone can suggest a different speed and justify it.

Not sure whether that means that R itself is defined by the speed of sound.

R is work done per degree per mole so an increase in altitude means more work required per degree per mole to lift the molecules higher against the force of gravity.

However, the force of gravity declines with height so as one goes up the additional work required for each further degree per mole becomes less.

That is what makes expansion so easy from any change in R.

Which would fit with your comment:

“for different gases, the speed of sound is inversely dependent on square root of the mean molecular weight of the gas, and affected to a lesser extent by the number of ways in which the molecules of the gas can store heat from compression”.

But I’m not sure how that advances the main issue though it is interesting.

64. tallbloke says:

Stephen,

http://en.wikipedia.org/wiki/Ideal_gas_constant

“As of 2006, the most precise measurement of R is obtained by measuring the speed of sound”

So for non-ideal gases such as our atmosphere, empirical experiments to determine the speed of sound at various altitudes would seem to be in order.

65. Stephen Wilde says:

An interesting thing about the Ideal Gas Law is that it only holds true at equilibrium.

In other words at the very same time that PV=nRT or PV=mRT the balance of radiation in from space and radiation out to space at top of atmosphere is zero. Both will be the same.

So, radiation within the atmosphere can be ignored because when the Ideal Gas Law is being observed at equilibrium the net effect of radiative energy transfers within the atmosphere must be zero too otherwise the top of atmosphere balance could not be zero.

That means that if a change in the balance of radiation transfers within the atmosphere becomes non zero for any reason then the Ideal Gas Law simply results in an immediate change in one or more of its other terms to remove that imbalance and restore balance at top of atmosphere.

In other words an immediate and complete negative system response.

If the Ideal Gas Law were ever to fail to effect a return to equilibrium then over time the residual imbalance must result in the loss of the atmosphere because cooling or warming would just go on indefinitely until the atmosphere were boiled off to space or frozen to the surface or the cause of the initial imbalance were reversed.

But as we all know the Ideal Gas Law works out in every case for a planet with an atmosphere.That is why it has the staus of a Law.

All the windbaggery about positive feedbacks within an atmosphere therefore comes to nought.

The feedback to any attempt to disrupt radiative balance at the top of the atmosphere is always immediate, negative and complete by virtue of the Ideal Gas Law.

That is why the AGW theorists try to ignore the Ideal Gas Law, the gradients of lapse rates, the volumes of atmospheres and especially the flexibility of the value of Rspecific in atmospheres that are not composed only of ideal gases (all of them).

As soon as one recognises that the Ideal Gas Law can accommodate non ideal gases by substituting a specific gas constant for the universal gas constant all consideration of radiative physics within an atmosphere becomes irrelevant for the reasons set out in my article.

66. Stephen Wilde says:

“As of 2006, the most precise measurement of R is obtained by measuring the speed of sound”

Well spotted. I hadn’t noticed that but oddly it does ring a bell in my memory.

67. Stephen Wilde says:

That must be it.

The usual approach of AGW proponents when referred to the Ideal Gas Law is to say that it doesn’t apply in a non ideal atmosphere so that radiative physics can be applied instead.

Of course no atmosphere is comprised of ideal gases so that gives them a free pass.

Until now everyone has been focused on the universal gas constant which is of course fixed for an ideal gas so that on the face of it the Ideal Gas Law seems not to work for a non ideal atmosphere.

The breakthrough has been to spot that the Ideal Gas Law can/must be applied to non ideal gas atmospheres by simply substituting the Specific Gas Constant for that atmosphere in place of the universal gas constant.

If the Ideal Gas Law is then applied with the Specific Gas Constant as proposed by me above then the flexibility of R gives the additional variable to explain why and how T fails to increase when composition is changed.

68. suricat says:

Stephen Wilde says: January 15, 2013 at 8:59 am

“I think I must disagree with the nay sayers here for the following reasons:

i) It is clear that R is only the universal constant for an ideal gas which never exists in reality. For all non ideal gases R is the specific gas constant and has a unique value for every mixture of non ideal gases. Thus it is variable depending on composition. That deals with Bill Gilbert’s objections.”

No Stephen. ‘”R” ‘IS’ THE UNIVERSAL constant for an IDEAL gas’! From THIS, the ‘SPECIFIC’ GAS CONSTANT can be calculated for any SINGULAR gas. From there, EACH ‘SPECIFIC GAS CONSTANT’ (the ‘singular’ gas derivation) can be summed to give the ‘macroscopic’ “R” factor for any ‘GAS MIXTURE’ and can be calculated for any ‘mixture’, but I think I’m beginning to realise where you are ‘coming from’. It’s the wrong perspective, but I think my thoughts are running ahead of this post!

Question! What’s the ‘”R” factor/index for ‘ice’ (or its equivalent for water)’ WRT ‘the gas laws’??? The question is rhetorical because ‘phase transition’ sets these phenomena ‘worlds apart’. To ‘re-connect’ these phenomena we need to go up a ‘science directory’ ‘LEVEL’ and return to the basics of ‘PVT’ (‘Boyle’s & Charles’ Laws and perhaps something even more ‘retro’), but that’s just in IMHO. :)

Do you “really” understand the ‘volumetric’ implication of the ‘phase transition changes’ in Earth’s atmosphere for H2O??? Thoughts towards ‘degrees of freedom’ and ‘attractors’ (that ‘suck up’ and ‘dump’ energy) are most important in these scenarios.

I’ll try to help if I can. :)

TB has my e-mail address if you need a personal dialogue. TB now has my permission to disclose this address to you (please don’t abuse it).

The rate of ‘thermal transport’ through Earth’s ‘lower atmosphere’ ‘ISN’T’ ‘represented’ by your criteria, so if you’re trying to ‘dis’ ‘radiative energy transport’, you’re doing a good job! :)

Best regards, Ray.

69. Stephen Wilde says:

suricat:

I see that you agree that the Specific Gas Constant is different from the universal gas constant and that it varies with composition so the first part of your post is a misunderstanding about my use of symbols.

See here:

“It is common, especially in engineering applications, to represent the specific gas constant by the symbol R. In such cases, the universal gas constant is usually given a different symbol to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.”

from here:

http://en.wikipedia.org/wiki/Ideal_gas_constant

So I have switched to the Specific Gas Constant and where I refer to R I now mean Rspecific. Using it in the Ideal Gas Law allows the necessary variability to avoid raising T by raising Rspecific instead of T when V is increased.

Indeed it is the raising of the Specific Gas Constant by changing composition that causes the increase in V in the first place and the subsequent loss of energy to PE.

I am applying it to the ENTIRE atmosphere, not just the tiddly portion where the physical properties of H2O become relevant.

We do not need to consider the phase changes of anything since those phase changes themselves affect Rspecific because they represent changes in atmospheric composition between solid, liquid and gas and Rspecific simply adjusts accordingly with the appropriate response from V.

V will change however necessary to mop any residuals following a change of composition and after all other non radiative processes including the water cycle have done their part.

It is that mopping up exercise by V that reapportions PE and KE globally so as to ensure top of atmosphere radiative balance without affecting T if the change is from composition alone.

70. Tim Folkerts says:

Stephen says: “Therefore any change in composition immediately affects volume (V), still without affecting T.

This is where I strongly disagree with you. You are simply ASSUMING that only volume will change and the temperature will stay the same. There is nothing in the equations so far that would suggest how much each of these two variables will change. Other constraints would be needed to determine how much each would change.

Also, I think you are putting way more emphasis on R (the gas constant) than you should (and on the ideal gas law). As you say, the specific R will indeed change with composition (because different molecules have different mass). But how much does composition change? Humidity varies from ~ 0-4 % of the gas, but the rest is pretty constant (4/5 N2, 1/5 O2). The “effective molar mass” will only vary by ~ 2% as humidity changes, so the specific R will also only change by ~ 2%. This is not going to make huge changes to the Ideal Gas Law.

A much bigger effect is the fact that the air is not an Ideal gas. Condensation of water is NOT part of the ideal gas law, but is a BIG part of the climate. This condensation cannot be modeled well by a variable R.

71. Stephen Wilde says:

Tim.

i) I considered the possibility that a change in R might change both R and T leading to a change in V

The trouble with that is that both R and T are on the same side of the equation and so would need to be multplied by one another.

The problem then is that the only change being in R and T being previously dependent on energy coming in from outside it just doesn’t make sense for T to rise as well unless more energy comes in from outside but it doesn’t.

The question then is as to how the extra energy arising from an increase in R could possibly provide enough energy to both raise T AND account for the extra PE needed when V rises after multiplying the increase in R by the increase in T.

It just isn’t possible. The energy that would have caused T to rise is needed to fuel the expansion provoked by the change in R unless you can prove otherwise.

ii) I agree with your point about compositional changes having other effects including those relating to the phase changes of water but I dealt with that in my post at 3.51 in response to suricat.where I said this:

“We do not need to consider the phase changes of anything since those phase changes themselves affect Rspecific because they represent changes in atmospheric composition between solid, liquid and gas and Rspecific simply adjusts accordingly with the appropriate response from V.

V will change however necessary to mop any residuals following a change of composition and after all other non radiative processes including the water cycle have done their part.

It is that mopping up exercise by V that reapportions PE and KE globally so as to ensure top of atmosphere radiative balance without affecting T if the change is from composition alone.”

So I disagree that I am placing too much reliance on the change in R. That change is the key to it all as long as one uses the specific gas constant rather than the universal gas constant.in the Ideal Gas Law

Radiative issues are irrelevant as per my post at 1.26pm where I said:

“at the very same time that PV=nRT or PV=mRT the balance of radiation in from space and radiation out to space at top of atmosphere is zero. Both will be the same.

So, radiation within the atmosphere can be ignored because when the Ideal Gas Law is being observed at equilibrium the net effect of radiative energy transfers within the atmosphere must be zero too otherwise the top of atmosphere balance could not be zero”.

72. Bill Gilbert says:

Stephen,

I’ll try one more time and then I’m done. I don’t think you understand the difference between R, the universal gas constant, and the specific gas constant, Rspecific. More bluntly, I don’t think you understand gas constants. Both R and Rspecific are simply proportional conversion factors, based on the units of measure being used, and nothing more.

R is a universal gas constant because the mass unit is expressed in moles. It is universal because a given volume V of any given ideal gas will contain the same number of molecules (with moles being the unit of measure that describes the given number of molecules). This is true no matter the composition of the gas. It is universal.

Rspecific is the same gas constant but with the units of mass being expressed in traditional mass units such as kg, g, pounds, ounces, etc. Since each molecule of any given gas component has a unique molecular weight, the gas constant has to reflect the molecular weight of the given component(s). The weight of any given volume will vary according to its composition (but each composition will contain the same number of molecules). The gas constant must be “specific” to the molecular weight of the component(s) filling that given volume. That is why it is different (specific) for different component compositions.

R does not vary depending on whether the gas is ideal or non-ideal. What varies is the relationship between T. P and V. R just makes sure the units of measure are compatible between the three variables and the mass unit being used. Both sides of the ideal gas law represent energy but the two sides have variables with different units of measure. R makes sure that the units of measure for energy are the same on both sides of the equation. That’s it! It has no effect on the independent, empirical values of T, P and V.

Your hypothesis does not work. But it was a very creative attempt.

Bill

73. tallbloke says:

Bill,
If you’ve read the foregoing comment from me, you’ll have seen that i found that Wiki says the most accurate way to determine R is via the speed of sound. But wiki also says that the speed of sound for non ideal gases varies with pressure. How then, can Rspecific be a constant in the real world?

74. Stephen Wilde says:

Bill:

“R is a constant, its value changes with each gas.
For dry air , R = 287 J/kg K.”

Its VALUE CHANGES with each gas.

That is not simply a matter of using different terms such as moles or kilograms for the same value.

ii) R = (work) / (amount × temperature).

The physical significance of R is work per degree per mole. It may be expressed in any set of units representing work or energy.

from here:

http://en.wikipedia.org/wiki/Gas_constant

So obviously if more work is required such as in a higher atmosphere then the value of R will be greater.

I am not making this up. I have given you the sources.

75. Stephen Wilde says:

And in both sources R involves work:

i.e. R = work (287J) / kg (amount of mass which could be expressed in moles)/ K (per degree Kelvin)

So I think you should reconsider.

76. Stephen Wilde says:

The reason that R involves work is because it represents the physical relationship between a gravitational field and matter held within or moving within that field.

I am becoming astonshed at the level of misunderstanding in these matters. It was all well understood 50 years ago that radiative characteristics have nothing to do with the equilibrium temperatures of atmospheres and gas clouds but here I am being forced to re invent the wheel because of years of misinformation from people who should have known better.

The term R converts energy to work to height and that is what stops KE from increasing. The extra height converts all extra energy from anything other than more mass, more gravity or more incoming energy to PE instead of KE

77. wayne says:

“Just regard R as the specific gas constant unique to each compositional variant for an atmosphere as a whole. As soon as one changes composition then the value of R changes.”

Agreed Stephen.

You also now stated clearly that you are speaking of the “entire atmosphere” as a whole and I take this to mean we can now narrow it down and speak of an ‘averaged whole-Earth atmospheric column’, one square meter in area, so we can use some well defined figures to prove your point. Is that ok to get the specifications tightly defined and more simple?

R.air (specific gas constant of dry air) is defined in the US76SA at (R*=8.31432)/0.0289644 or 287.053 J/kg/K.

At 1% WV it would be (R*)/(99%*0.0289644 + 1%*0.018) or 288.144
At 2% it would be (R*)/(98%*0.0289644 + 2%*0.018) or 289.243
At 3% it would be 290.350
At 4% it would be 291.466

I assume that is part of your possible changing composition and how it affects R.air and columnar volume. The other is a doubling of halving of co2. Right?

Also there would be an equal change in the density that by P/ρ = R.air × T whatever increase in R.air you have and equal linear decrease in density (ρ) so P and T do not change at all. Is that one point you have been trying to make in just words?

With a doubling of co2, R.air would be about 8.31432/((1-0.00039)*0.0289644 + (0.00039)*0.044) or 286.995 instead of 287.053 J/kg/K. Not very much difference at all between the two but is that not also what you are speaking of in the “changing air composition” as to R.air in relation to co2 concentration?

While we are concentrated on these numbers, WV’s 18 g/mol is just as far below air’s 29 g/mol as co2’s 44 g/mol is above air’s 29 g/mol. One ratio is 18/29, the other 29/44, very close together. So if WV specific concentration decreases in the same proportion as co2’s concentration increases you end up with zero T change because P never changes, right? (well, the added C12 atoms do raise the mass a very tiny bit but I think we all agree it is so tiny compared to the 10300 kg/column it can be ignored at a fraction of 0.000394)

Are we not getting close to the point where we can actually calculate an example of your increased or decreased in volume of the column at some pre-defined pressure, let’s say something like at the 10 mb level altitude and prove your point? Or maybe I have already basically proved your entire point. I’ll have to think on what I just wrote. it might need one more step through ρ’s definition of kg/m3 a gravitation field, less dense does expand and flexes with R.air.

78. tallbloke says:

Wayne: Excellent. It is very encouraging to see firm understanding and complimentary talents being brought to bear on this central hypothesis. I sense real progress, and great potential for a new methodological paradigm enabling a better understanding of atmospheric processes.

More power to you both, let’s carry this forward.

79. Stephen Wilde says:

Thanks wayne.

I’m happy for someone else to help with figure work due to the demands of my day job and lack of experience with numbers.

If it turns out that I’m wrong then so be it but I would be surprised.

“so P and T do not change at all. Is that one point you have been trying to make in just words?”

Yes.

“So if WV specific concentration decreases in the same proportion as co2’s concentration increases you end up with zero T change because P never changes, right?”

(Sounds like Miskolczi’s finding).

Yes. The fact that P holds steady despite the volume change is what throws the whole system response onto V. Normally density reducing with expansion results in less pressure for an individual parcel of air but that is not possible for an atmosphere around a sphere.Pressure at the surface stays the same despite expansion.

I don’t follow all your figures but it looks ok so far.

Why would WV decline with increase in CO2?

Perhaps because CO2 radiates out and has a net cooling effect which slows down the water cycle ?

But why such a precise relationship ?

I don’t claim to have it all sorted out, just the general principle. I hope.

80. wayne says:

“Why would WV decline with increase in CO2?”

You must take the time to read Miskolczi’s two papers.

81. wayne says:

tallbloke January 16, 2013 at 10:45 pm:

Thanks to both you and Stephen. Tallbloke, I’m afraid you were confusing that the degrees of freedom has something to do with R* or R.air but I’ve never seen the tie, you are remembering back to the cp/cv or Cp/Cv discussion. R* is always Cp – Cv — yet how Cp is manifests does definitely have to do with the degrees and the molecular aspects of the composition gases. If you ever were able to read my earlier comment on an earlier post tying the different equations that must all simultaneously hold, the γ = Cp/Cv was in there as a component in the exponent as (γ-1)/ γ. I hope both of you will spend some time reading on both of these:

Ideal Gas Law:
P/ρ = R. air ×T
&
Potential Temperature (Poisson’s Equation):
Φe = T.base × (P / P.base)^(R.air / cp) or equally
Φe = T.base × (P / P.base)^(R* / Cp) or equally
Φe = T.base × (P / P.base)^( (γ-1)/ γ)
&
Lapse Rate:
T = T.base + h × Γ, Γ being 0.0065 K/m.

All must hold at every point the ‘P’ and ‘T’ within the linear lapse of a troposphere.

However by Φe = T.base × (P / P.base)^(R.air / cp), the cp is 50% higher than the cp you will find in any table about air, like under Wikipedia’s ‘Heat Capacity’ page near the bottom, listing it as 1.003 J/g/K or 1003 J/kg/K where to match our troposphere is has to be about 1508 J/kg/K for R.air and P.base and T.base are all well defined. That is the missing 2/3rd from my post. The only pertinent variables are P and cp, and P, it is the independent variable leaving cp as the culprit. ;)

That is for the ‘Whole-Earth Averaged Column’. But to match the profile over the Sahara desert in summer on a very dry day the cp has to be – ta da – you guessed it – right at 1003 J/kg.K. Now tell me that is not only water vapor’s influence and not a bit having to do with co2 or other GHG’s. Not! It’s totally the state change ability of water vapor.

Btw: cp is per kilogram, Cp is per mole, also best to not confuse these two even though I have caught myself not capitalizing (or not) when I should. This is a very picky nomenclature area! R* or R.air, Cp or cp. Brother!

82. suricat says:

Stephen Wilde says: January 16, 2013 at 3:51 pm

Oh Stephen. :( ‘R’ is only ‘altered’ by the ‘reduction/increase’ of ‘n’ Moles of the WV that has ‘condensed/evaporated’! PV=nRT is a ‘dry gas’ “formula” that can’t show the freedom of ‘phase transition’ and ‘latent heat’. However, ‘WV’, ‘water aerosol’ and ‘ice’ are prevalent throughout the troposphere and above. We need a modification to PV=nRT before the atmospheric hydrological cycle can be properly audited WRT ‘energy transfer’.

In truth? For a ‘cooling scenario’, ‘n’ changes (reducing P and n, but adding to T [this removes a ‘volume’ from the atmosphere, but adds KE from ‘latency’ which acts to negate the deficiency of P]). Thus, the original ‘Molar’ sample has altered and requires another ‘dimension’ (DON’T tell me ‘you’ve already done this’ because it ‘just ain’t true’).

Let’s take the scenario of an asscending colunm of air with a lot of WV in it and ‘CCNs’ (Cloud Condensation Nuclei) are abundant. The column will begin to condense the WV when the ‘vapour pressure’ for WV becomes intollerable for the support of WV as a gas and condenses it into an ‘aerosol’ phase.

Where have we heard this before? The ‘partial vapour pressure’ argument in the Clausius Clapyron ‘relationship’.

Best regards, Ray.

83. wayne says:

“But why such a precise relationship ?”

Sorry Stephen, I forgot to an attempt to answer that other question. Don’t think anyone knows yet and even though the specify humidity as gone up in the lower boundary layer it has more than enough gone down elsewhere. Is it a one-to-one releation? Don’t know. I’ve never taken the time to explore that question to get a close-enough-to-correct answer to satisfy myself but I can’t imagine it being a perfect fit in the chaos case of weather but physics many times does do such things — equal and opposite.

84. Bill Gilbert says:

Stephen,

I’m sorry you do not understand. Yes, I said Rspecific (e.g., joules/kg T) changes with composition and your link was in kg. But R (e.g., joules/ mole T) does not change with composition. Doesn’t it bother you that your hypothesis depends on whether you measure in kg or moles? That takes Heisenberg’s uncertainty principle to new heights (or lows).

Oh, and R does no work.

Rog,

The calculation for R using the speed of sound was done at very specific conditions and at different pressures and then extrapolated to the zero pressure limit. But there are many ways of measuring R (and k and Na; remember that R = k Na)) and they all come out with the same answer plus or minus the 10th decimal place or so.

Wayne,

I think your problem with your Cp calculation is due to your use of the moist adiabatic lapse rate (6.5 K/km) versus the dry adiabatic lapse rate (9.8 K/km). 6.5/9.8 = 0.663 and 1003/1508 = 0.665. Potential temperature does not include water vapor and follows the dry adiabatic lapse rate when brought to the surface. Equivalent potential temperature includes water vapor. Try that.

As to your question as to why humidity goes up in the lower boundary layer but goes down elsewhere, see my paper linked above. That’s what the paper is all about. (Miskolczi liked it).

Bill

85. wayne says:

Bill, I’m very sorry, that was YOUR paper that put that information in my head. I read your paper just three days ago, my age is showing. ;)

That is one good paper and it’s on my queue to read much deeper as I get time. No wonder Ferenc liked it, it ties an explanation to his papers. Should have pointed my words to it.

A very interesting paper to write would be one on what happens in between thermals, the convection updrafts that everyone talks about but no one mentions the opposite downdrafts. I was lucky to fly a sailplane for a few years long ago and most don’t realize how that part of the science operates, but leave it to a sailplane pilot for he is in directly interaction every hour he is up with that environment.

One thing that amazed me was that before you every left the ground you could hold up your fingers and measure the cloud size that day and could then measure the average distance between the clouds. I’ll call that cloud spacing. Square that. Now once you get up and get a handle on how strong the thermals are, let’s say +1000 ft/min you already know how fast the sink is in-between the clouds. The plane I rented would drop on a gray lift-less day at -200 ft/min so the real lift was really +1200 ft/min upward and with a cloud spacing of a bit over three you knew the downdrafts would be about 1/10th the uplift, so your variometer should read about -320 ft/min if you find no lift. Amazing… that is what you would experience, ~320 ft/min downward and could keep the airport in proximity by that guess measured of clouds with the widths of your fingers!

Most people immediately think the downdraft equals the uplift and when viewing the mass balance, that is correct, but the rate between the two is much different, due to the difference in the area covered by each. Now that must have something to do with the cooling process.

On wet-soil hot days, clouds were much closer together and the downdraft was much greater, even dangerous if you didn’t watch it. On such a day, cloud spacing of one, I got caught in one of those, the meter pegged at -1200 ft/min but my estimate by the spinning altimeter was -2000 ft/min and almost was forced to land off base. That I take is what Stephen is speaking of… how fast the hydrologic process operates given the energy input.

Bill, on the potential temperature, you seem to have missed prior posts here on the subject but I was using Φe to mean ‘effective’ to differentiate between the actual ‘Φ’ potential temperature that matches the DALR. I know that’s not very kosher to many in this area of science but by merely altering the ‘cp’ the potential temperature equation then exactly matches the standard atmosphere’s profile. That doesn’t means anything not already known but it sure is a simple way to know any temperature at any pressure just as the ELR makes it easy to know any temperature at any altitude, but then, water is involved in the equation (though hidden).

86. Stephen Wilde says:

Bill Gilbert said:

“Oh, and R does no work.” and ” But R (e.g., joules/ mole T) does not change with composition”

I have given you two sources saying different i.e. that “The physical significance of R is work per degree per mole. It may be expressed in any set of units representing work or energy.”

and that the value of R changes with each gas (or mixture of gases)

so I cannot accept your simple assertions to the contrary.

Suricat said:

‘”R’ is only ‘altered’ by the ‘reduction/increase’ of ‘n’ Moles of the WV that has ‘condensed/evaporated’! PV=nRT is a ‘dry gas’ “formula””

The value of R changes from any change in any mixture of gases. That change is clearly not a phenomenon limited to the presence or otherwise of water vapour.

Every discrete mix of gases has its own value of R.

I have already addressed your overly close focus on the phase changes of water. They are adequately accommodated in a wider scenario involving the change of atmospheric heights or volumes and the creation or destruction of PE relative to KE.

87. Stephen Wilde says:

wayne.

Could you explain it in words ?

Is my hypothesis still supported by your figures ?

88. tallbloke says:

Stephen: “The fact that P holds steady despite the volume change is what throws the whole system response onto V.”

It’s noticeable that when the atmosphere contracts due to solar change, a lot of rain gets dumped. That’s a lot of mass loss from the atmosphere, which must surely affect pressure?

89. tallbloke says:

Bill, thanks for your patience and continued involvement here. I know you’ve spent a lot of time and brain cycles on atmospheric thermodynamics and we appreciate your input. I think I shall post your paper in full so we can better appreciate the points of linkage and divergence with Stephen and Wayne’s approaches.

90. wayne says:

Sure Stephen,

I had to re-read your post one more time and the answer would be yes and no. You say some very correct statements but a few are clearly not correct. I for one forgave you immediately for mixing the R’s ;) and I just interpret them in their correct form as needed as I read, universal and specific, I am not speaking of that matter.

You talk as if all GHGs are the same and in using the Ideal Gas Law for the thrust of your post they definitely are not the same — because of their molar mass. H2O is 18 g/mol, CO2 is 44 g/mol. H2O is lighter that the mean molar mass of air, co2 is heavier, so R.specific goes up with water vapor concentration but goes down with an increase in co2’s concentration. That is where yours logic fails in speaking of the volume expansion. You are trying to combat the AGW script but in using the IGL it actually supports it a bit (but really a very, very tiny amount). Additional water vapor expands the atmosphere at a constant temperature, co2 contracts the column. See the slip?

But let’s look at how much that is going top matter using the IGL. Look back at my comment at 10:29 when I say “286.995 instead of 287.053” for co2, make that a ratio, 287.053/286.995, and multiply by 288 K surface temperature. That gives a temperature of 288.0582 K. Gee… that’s only a 0.06 °C rise for an entire doubling of co2, and personally, I can live with that as a fact and think that is all we will ever see caused specifically by co2 itself, radiative effects having no impact in our basically totally opaque atmosphere to all GHG’s IR lines, bands and continuums anywhere near the surface.

It was the sun’s influence we saw in the 80’s and early ‘90s. I was an early SOHO enthusiast and saw it happening in all of the filters, really frightening, the raw furor during the previous cycle. One day NASA might just find that the pyrometers and bolometers are not recording the absolute value of the TSI as they should. That’s my guess. I’ve waited daily for that press release for quite a while now.

But carrying that further on co2, that 0.06 °C will be partially or completely compensated by the near IR 2.7 µm and 4.3 co2 lines that also absorb from solar incoming radiation and never reach the surface. Therefore, I don’t think even that 0.06 °C will ever be totally manifested as temperature at the surface.

But you do have it pretty correct when your speaking of the hydrological loop and it speeding up and slowing down with the available energy… that I have personally experienced and it happens at all times of any day somewhere on this globe where the sun is high above, which is every second and always. That also has to do with the atmosphere’s local columnar volume expanding and contracting due to both water vapor concentrations and temperature.

91. Stephen Wilde says:

Hi wayne.

i) I did recognise that different GHGs have different effects by linking their effects to the extent that they cause the actual lapse rate to dverge from the ideal lapse rate. Water vapour in the troposphere and ozone in the stratosphere have very different effects for example. I also mentioned that it is unclear what effect CO2 has since it could be net warming, net cooling or neutral but that whatever it is the volume adjustments deal with it. I currently favour net cooling.

ii) My logic supports an effect from CO2 but limited to a miniscule circulation change rather than a temperature change though there would be some change in T from the extra mass but so tiny as to be ignored.

iii) I also impliedly acknowledged that GHGs and other gases or materials can have thermal effects other than from their addition of mass. The phase changes of water are one such example and I think suricat has missed that part of my proposition too. My logic allows for such other mechanisms with the volume change just mopping up the net residuals to maintain top of atmosphere radiative balance.

So what I have been looking for is something other than T that can change as necessary to keep the system stable. The variability of Rspecific gives us that additional flexibility but works alongside other non radiative mechanisms to achieve the final net outturn.

Note too that R appears to involve the amount of work required to lift a specific amount of substance (however defined) to the point higher up in the atmosphere where it is 1Kelvin cooler.

That value can therefore be affected by other factors arising from other characteristics of GHGs or any other gases apart from simple mass but in the end everything has to net out to zero for the Ideal Gas Law to be complied with.

So somehow the value of R actually depends upon the netted out value of every physical feature of every substance in the atmosphere whether solid (aerosols) liquid (cloud droplets) or gases.

In fact from this definition:

“The physical significance of R is work per degree per mole. It may be expressed in any set of units representing work or energy.”

one could argue that R represents work done to lift 1 mole (or whatever) to a height where it is 1K cooler so that 1K’s worth of KE has been converted to PE.

Thus R is at base the energy cost of converting KE to PE.

and that energy cost depends on every possible variable throughout an atmosphere combined and netted out in so far as they interact with the gravitational field.

Pretty much every aspect of the physical properties of all the disparate solids liquids and gases within an atmosphere would have a bearing on the value of R.

The truth is that for any given atmosphere it must be impossible to actually calculate what the value of R really is but nonetheless it is there hidden in the Ideal Gas Las as the ultimate balancing factor for top of atmosphere radiative equilibrium after mass, gravity, density, pressure and volume have all been quantified.

92. oldbrew says:

SW says: ‘for any given atmosphere it must be impossible to actually calculate what the value of R really is’

If you know what the composition of the given atmosphere is, as we do for Earth, why would it not just be a calculation? The molecular weights of the relevant gases are here.

http://www.engineeringtoolbox.com/individual-universal-gas-constant-d_588.html

‘The Individual Gas Constant depends on the particular gas and is related to the molecular weight of the gas. The value is independent of temperature.’

93. Stephen Wilde says:

Yes oldbrew but we only know the composition to a rough approximation but. that is good enough for most day to day purposes.

I was just playing devils advocate with my own hypothesis and thinking aloud.

.

94. Stephen Wilde says:

In fact I disagree with my own comment to this effect:

“That value (R) can therefore be affected by other factors arising from other characteristics of GHGs or any other gases apart from simple mass ”

The lack of an edit facility is a problem.

Other characteristics just net out to zero within the atmosphere as a result of changes in the speed of non radiative processes.

And that includes the radiative characteristics of GHGs.

95. Stephen Wilde says:

I’ve found an error in my narrative and am crossing fingers that the correction will be helpful.

Apologies but it is difficult starting from first principles when there is so much contradictory information around..

I have been looking at the system response to a compositional change that does NOT involve a change in R and linking it to a compositional change that DOES involve a change in R.

So let’s look at the two scenarios separately in light of the Ideal Gas Law containing Rspecific which I will refer to as R since that seems to be quite common.

1) If the compositional change involves a change in the value of R due to an increase in the molecular weight of the gas then the system response will be a higher temperature because the greater value of R will offset the lower density and so both V and T can increase.

2) If the compositional change does not involve a change in the value of R, say from greater absorption and radiative characteristics then there will be no change in anything because any potential change in V on one side of the equation will be cancelled by the potential reduction in density offsetting the potential rise in T on the other side of the equation.

That is why changes in radiative characteristics have no effect on temperature.

Instead, the air circulation simply runs faster which accelerates energy flow through the adiabatic loop to offset the slowdown in the diabatic loop.

Where does that leave the PE/KE exchange?

There are inevitable time lags especially concerning the rates of energy release from the oceans. During the equalisation process the atmosphere will expand and contract so that energy is converted between KE and PE as necessary to retain top of atmosphere energy balance.

That fits with the long established science that only mass, gravity and energy input can affect T.

Does that make more sense?

96. Bill Gilbert says:

Wayne,

Yes, it would be an interesting exercise to evaluate the entire convective and subsidizing field of play rather than just the convection funnels. I have played around with that to some degree and it is primarily a mass transfer rather than a heat transfer problem. But it does point out the great importance of mass transfer in fully understanding the dynamics of the atmosphere. Those “radiation only” guys never understand that fact and that is why their reasoning is so flawed. You can’t use only two dimensions (W/m2) when playing in a three dimensional world.

No, I did not see your past posts concerning your potential temperature calculations. But I see that you adjusted Cp in Poisson’s equation to go from the DALR to the US Standard Atmosphere lapse rate. That’s why the Cp and lapse rate ratios matched perfectly. I haven’t seen that done before. I would be interested in reading your posts if you can give me a link. I also noticed you are investigating Cp/Cv ratios and I am interested to see what your issues are.

“Additional water vapor expands the atmosphere at a constant temperature, co2 contracts the column.”

This gets back to the problem I am having with the use of Rspecific in this article. Yes, if you add water vapor to the system Rspecific increases and vice versa for the addition of CO2. This is due to the changing molar mass. But you have to remember that the term “m” in PV = mR(specific)T is also changing. “m” increases in both cases since you are adding mass (kg) – and the average molar mass changes in the process. But in both cases V will increase since you are adding molecules to the system. For an ideal gas at constant T and P, the number of atoms/molecules in a given volume is the same regardless of the composition of the particles. If you add particles, the volume increases. Thus your statement above is incorrect. (P will also increase, but let’s ignore that for now).

The change in Rspecific has no effect on the system. The change on the system is due to the change in “m”. Rspecific is only converting “m” in mass units (kg) to moles so that the equation can work. If you change the molar mass of “m” you have to change Rspecific to compensate. But it all ends up as moles in the end.

Bill

97. The simplification PV/T = k (constant) seems to work for any particular atmosphere of suitable density, don’t you think? Thanks to the “point source, perfectly elastically colliding” nature of gas molecules. With water phase change actively moving surface-adsorbed energy to the tropopause? Brett Keane, New Zealand

98. Bill Gilbert says:

Brett,

Yes, PV = NkT is the micro equivalent to the macro expression PV = nRT. The difference being N = number of atoms/molecules and n = number of moles. The first is derived from statistical mechanics and the second is derived from classical thermodynamics. Both are universal expressions that are independent of the mass composition.

Neither deal with water phase change and mass transfer. That is best left to the first law of thermodynamics.

Bill

99. suricat says:

Stephen Wilde says: January 17, 2013 at 9:02 pm

“Does that make more sense?”

It’s beginning to. Let’s go back to the original equation of “PV=nRT”. This works fine where the ‘Molar’ constituents remain unchanged, but this isn’t the case for Earth’s atmosphere. WV is added to the atmosphere at the Earth’s surface and precipitated out at some altitude or other. This ‘equation’ needs to be modified/edited for a better representation that can model Earth’s atmosphere.

We need to add a P2, V2, n2 and T2 to accommodate the ‘phase transition’ elements.

Best regards, Ray.

100. Stephen Wilde says:

Thanks suricat,

I think I need to think more about the reasons for the lapse rate changes away from the ideal lapse rates in the tropopause from water vapour and in the stratosphere from ozone.

If it is the case that a change in radiative characteristics cannot raise V or T because of the constraint imposed by the potential reduction in density when no extra energy is coming in from outside the atmosphere then we need to look elsewhere for the reasons for the changes in V and the gradient of the lapse rate in those two layers.

The observation of changed volumes and lapse rate slopes maybe has nothing to do with the radiative characteristics of water vapour and ozone after all and so might have been an unnecessary complication in my initial narrative.

I think suricat and maybe someone else has been saying that the lapse rate change in the troposphere is due to the phase changes of water rather than the radiative characteristics of water vapour.

In the stratosphere ozone is heated by the sun from above rather than by upward longwave from below so are the absorption capabilities of ozone the real reason for the changed slope in the stratosphere rather than the radiative characteristics of ozone ?

My point about increasing PE reducing the net fund of KE could still hold as a means of preventing an increase in T as a result of any such non radiative features of water vapour and ozone that do affect the lapse rate slope and V.

My article seems to have been a useful starting point but to get a more complete solution we need to split up the different characteristics of GHGs more than I have done and thereby isolate the consequence of a change in radiative characteristics alone.

If we can remove the issues of volume changes and lapse rate changes to other causes than radiative capability then that does simplify things.

Is it sufficient therefore to simply say that changes in just the radiative characteristics of GHGs are constrained from having any effect on T by the fact that the consequent density reduction prevents it and that therefore the only effect is to accelerate all other non radiative processes proportionately ?

101. Stephen Wilde says:

i) The enhanced radiative capabilities of GHGs cannot directly raise T because to do so would involve an expansion of the atmosphere and a reduction of density (by virtue of the Ideal Gas Law) that would immediately negate their effects.

iii) All those processes then fail to raise T because they do cause an expansion of the atmosphere which converts a larger proportion of the available energy (PE + KE) to PE rather than KE so as to keep T stable.

iv) The reason for that indirect route for the negation of radiative effects is that atmospheric volume is the result of mechanical rather than radiative processes so that radiative energy must first be converted to mechanical energy before its thermal influence can be neutralised.

v) Thus does the Ideal Gas Law regulate the radiative energy flow through any atmosphere so as to maintain top of atmosphere radiative equlibrium.

102. tallbloke says:

I think Stephen’s five point summary is the right sort of way to communicate a description of atmospheric reality. Now we need the simplest possible exposition of the equations which underpin it. There is value in working from both ends towrds the middle like this. Ultimately, numerical solutions have to be describable in terms of commonly worded summaries such as this. Einstein’s popular science book on relativity was accessible to many more people than his technical papers.

103. Stephen Wilde says:

As regards that 5 point summary the main problem I have with AGW proponents about it is their concept that T at the surface must rise before any atmospheric expansion can occur.

That perception is an inevitable consequence of focusing primarily on radiation in which case that would follow.

However, out in the real world atmospheric height is maintained by mechanical processes involving work being constantly done to maintain atmospheric height whilst solar input flows through and out again pretty much undisturbed at equilibrium apart from the wavelength shift.

Those mechanical processes are primarily comprised of the constant decompression of rising air and compression of descending air in the adiabatic loop. That process locks away variable amounts of energy in the form of PE which is then not available to be measured as heat.

That process of compression and decompression provides the basic amount of energy locked into the surface / atmosphere energy exchange that must be retained to keep the atmosphere suspended at any given height above the surface but all other mechanical (non radiative) processes are then superimposed onto that baseline amount of energy and will cause variation in the amount of energy locked away as PE as the different non radiative processes ebb and flow as a result of other factors within the system.

We have seen in the discussion about Rspecific that certain changes in composition involving mass and molecular weights do have an effect on the Ideal Gas Law equation such that an increase in the value of Rspecific is needed to offset the reduction in density when the atmosphere expands otherwise there is not enough extra energy from the change in composition to raise both V and T.

The point then is what happens when a change in composition occurs that does not involve a rise in the value of Rspecific. Such a change would be an increase in the amount of radiatively active gases.

Since the value of Rspecific cannot rise there is no extra energy to offset the reduction in density if volume were to increase so there is not enough energy to raise T as well.

Thus such changes in composition can have no effect on equilibrium temperature and can only go to increasing or decreasing the speed and vigour of non radiative processes.

Those radiatively active gases are not at the surface. They are floating about in the atmosphere so their additional energy causes expansion in the parcels of air containing them and not at the surface.

That is a mechanical process and not a radiative process so the local expansion can occur away from the surface without needing to project any change in the lapse rate back down to the surface to establish surface temperature which is what AGW proponents do.

We can see that happening most clearly in the stratosphere as a result of ozone which is a non condensing GHG like CO2 and as such is a more suitable comparator than the water vapour in the troposphere where phase changes complicate and obscure the basic truth about non condensing radiative gases.

In the stratosphere the ozone present warms and expands the air locally to the extent that the lapse rate actually reverses.

Nevertheless no one proposes that to establish surface temperature one should then extrapolate the stratosphere lapse rate back to the surface. That would be an obvious nonsense yet AGW proponents insist on going through that procedure from the so called effective radiating height in order to decide what the surface temperature ‘should’ be.

The truth must be that radiatively active gases warm the air locally around them away from any contact with the surface without affecting surface temperature at all. That air then expands and the gases in it rise higher due to reduced density. That parcel of air then cools as more of its energy is converted to PE due to the lower pressure at the increased height.

What has happened is that the additional absorption capability of the GHG molecule has enhanced the mechanical process of convection which is a cooling process and has converted the extra energy it absorbed to PE which does not register as heat.

Meanwhile the surface temperature remains the same throughout because it is set by density, pressure and insolation at the surface in accordance with the Ideal Gas Law and not by DWIR from above.

So one can have atmospheric expansion from additional radiative gases without having to raise surface T first.

The point being that mechanical non radiative processes do the necessary work locally within the atmosphere well away from the surface and are self negating in the absence of more mass, gravity or incoming energy.

Any attempt to increase T by the enhanced radiative capability of non condensing GHGs is cancelled by an increase in V which converts all the additional energy absorbed by those gases to PE.

104. suricat says:

High Stephen.

This’ll be a ‘slog’, because ‘mass’ (mechanical) systems are chaotic in the tropo. :(

i) a) Surely, as I understand it, the ‘enhancement’ you describe is one of “vibrating/rotating atomic bonds” which hold the molecule together? If so, I concur. T remains virtually ‘unchanged’, because the ‘internal molecular vibrations’ were “robbed” from the KE of the prevailing T in the first instance.
b) However, if you imply that ‘photon emission’ from the constituent atoms of the molecule, when they are ‘stimulated’, due to atomic stresses applied by the molecular vibration? I’m uncertain, but I’m persuaded that this probability is enhanced by low pressure. Thus, photon emission is enhanced in the upper atmosphere and any emission ‘distance to extinction’ is also enhanced in this way. This also implies that ‘photon emission’ can only belong to the KE ‘loss attractor’ description, for energy transport to ‘EM energy’.
I don’t believe the Ideal Gas Law covers photon emission, does it?

ii) See i). IMHO. Photon emission can only ‘dilute’ a T ‘hot spot’ of transmissible energy and the ‘flapping bonds’ scenario is insignificant to total KE.

iii) I find this mark confusing by definition. Previously you’ve referred to PE as both ‘gravitational potential energy’ and ‘inertial potential energy’. Relativistically, they are the same, but each has its place in differing scenarios. Which is which? In terms of atmospheric circulation they play different roles in different systems.

iv) This depends on the atmospheric system under observation and the altitude observed.

v) No it doesn’t. Other factors need to be addressed before this conclusion can be reached.

I hope you can understand my prose and that it helps you arrive at some conclusion. :)

BTW. Stephen Wilde says: January 18, 2013 at 2:50 am

Best regards, Ray.