03 November 2009

How CO2 matters

It turns out that the argument that there isn't a lot of CO2 (true, compared to total mass of the atmosphere) and therefore it can't matter much for climate (false) has been around longer than I had thought.  I was just reading Craig Bohren's book Clouds in a Glass of Beer: Simple Experiments in Atmospheric Physics and he's got reference to it (chapter 10, on the Greenhouse Effect),   The collection of experiments was published originally in 1987, and had evolved over some period before that.  So at least 22 years that the argument has been around.

From page 82 in my Dover edition:There seems to be little dispute that carbon dioxide concentrations in the atmosphere have been increasing because of increased burning of carbonaceous fuels such as coal and oil.  At present, for every one million molecules in the atmosphere, about 340 of them ar carbon dioxide (this is written 340 ppm, parts per million).  To those who snort that 340 ppm of anything must surely be of no consequence, I recommend 340 ppm of arsenic in their coffee.  I don't second the recommendation as the lethal dose is somewhere around 1 ppm.  Craig was being sarcastic, and blunt, two common words for describing him.   The 340 ppm was about the Mauna Loa station's reading for 1981, and the last year that would round to that (nearest 10 ppm rounding) is 1984, so it's probably 3-6 years before book publication that Craig was writing.  It's now past 385 ppm.

For climate purposes, we'll consider two different things.  First is, how can a rare thing (CO2) be important to the system?  Second is, is CO2 really all that rare?

As is obvious from the arsenic example, rare things can be important in some systems.  What we need to explore is the how.  For CO2, its importance comes from the fact that it is a greenhouse gas.  Most of the atmosphere, in fact the overwhelming majority of the atmosphere, has no great absorption for the energy emitted by the earth.  The three major gases are nitrogen (N2), oxygen (as O2), and Argon (Ar), which comprise well over 99% of the atmosphere, and none of which absorb energy emitted by the earth.  All greenhouse gases are trace gases, water vapor (H2O) included.

We'll get to a less-simplified notion of the greenhouse effect, but let's start with the oversimplified version.  In that version,
0) The sun throws energy at the earth
1) the earth emits energy towards space.
2) a greenhouse gas molecule captures a bit of that energy
3) it then spits it out in a random direction
3a) if it's towards space, no change from what was going to happen anyhow
3b) if it's towards the surface, the surface catches more energy than it would have otherwise
4) Because of 3b, the surface gets hotter.

This is correct as far as it goes, but it doesn't go very far.  An important thing missing is that step 3 almost never happens alone or immediately.  Related is that this picture only tells you about the temperature of the ground and the greenhouse gases -- not of the 99+% of the atmosphere that is not greenhouse gas.  The important missing part is between 2 and 3 -- A) most of the time, the greenhouse gas molecule that just absorbed some energy emitted by the earth will collide with a non-greenhouse gas molecule and pass the energy on to the other molecule.  The converse thing can also happen -- a greenhouse molecule get clobbered by a non-greenhouse molecule and then emits some energy (to space or the ground).  Greenhouse gases play an important role in setting the temperature of the non-greenhouse gases in the atmosphere, not just the surface.  And they do this in spite of being only a small fraction of the atmosphere.

Need to emphasize that, I think.  The image is out there that H2O is 4% (40,000 ppm) of the atmosphere.  That's only true in exceedingly warm air very near a water supply (ocean or lake).  Averaged through the entire atmosphere, it's more like 2000-4000 ppm.  I'll invite you to construct your own estimates, show the rationale and calculations as to the correct figure.  In any case, while water vapor is the most common greenhouse gas (in number) it is only 5-10 times CO2, on average, not 100 times.  Conversely, this leaves CO2 as 10-20% of greenhouse gas molecules.

Anyhow, we've got our answer to the first question -- these rare molecules (greenhouse gas molecules) are important because they set the temperature of the ground, and help set the temperature of the atmosphere itself (whether greenhouse gas molecules or otherwise).  Even though they're 'rare', they matter.

But, to the second part -- are they really 'rare'?  As a fraction of all molecules in the atmosphere, yes.  There are, however, other ways of deciding rarity.  I'll start with some farther afield.  385 ppm means that in a city of 1 million, you could find 385 people who were that unusual.  Refining it a little, in a group of 2600, you'd expect to find someone that unusual.  In a sports stadium with 52,000, there would be 20 people that unusual.  A key being, given our understanding from the first question, that the 20 are not hard to find -- they're the ones being exceptionally obnoxious, starting the fights in the stands, etc. -- you know that they're there because they bump in to you, or you see the fight start, or they're the 20 who start 'the wave' in the stands, and so on.

That suggests a different way of looking at 'rare'.  They're rare if they have no observable effect.  The one person in the stands who is reading a book, you don't know they're there unless you're extremely close by.  We already know that this isn't the case for greenhouse gases -- they do have effects and we do observe them.  But let's pretend we are a photon (energy packet) emitted by the earth towards space.  We could consider greenhouse gases rare if we could expect to get out to space without ever encountering one of them.

I'll put up my math for folks to check.  If you don't like the math, you can skip ahead a little.  But I think it's important to show that there's nothing up my sleeve here.  Over each square meter of the surface (at sea level) of the earth, there are about 10,000 kg of air.  CO2 is approximately evenly distributed throughout the atmosphere, so the current (2009) 385 ppm CO2 means that there are about 4 kg of CO2 over each square meter.  That's a fairly noticeable number to us as large bodies, but not necessarily to a photon.  So I'll continue.  Update: per carrot's comment, I had oopsed here.  Even though I pay attention to the fact that CO2 molecules don't weigh the same as average air does (44 vs. 29) in the next section, I failed to do so here.  That makes it 4*44/29 kg of CO2, for 6 kg CO2 over each square meter.  Corrected figure used for rest.

In chemistry, we learned about moles of things.  1 mole is a standard count for molecules (1 Avagadro's number of molecules; the chemists and others who know the size of this number already know how this story turns out).  1 mole of CO2 has a mass of 44 grams.  So the 6 kg of CO2 represent about 136 moles of CO2.  Again that seems large, but now think about yourself as a photon emitted by the earth.  Your 'size' is about 10 millionths of a meter (10 microns); that's your wavelength.  As you go speeding through the atmosphere, a CO2 molecule has to be sitting in that window only 10 microns wide before you're likely to notice each other.  I'll take a disk with 10 micron diameter to represent the zone a CO2 molecule has to be sitting in for you to be concerned.  The relevant area is not 1 square meter, but about 80 trillionths of a square meter.  So the relevant number of moles is this times the 136 moles over a full square meter, for about 11e-9 (11 billionths) of a mole.  [Throughout, I'm using more precise numbers than I'm quoting here.  Some of the math won't seem to line up because of the rounding.]

If you don't know Avagadro's number, 11 billionths of a mole looks like it's awfully small.  That 11 billionths of a mole gives us the number of CO2 molecules (it's worse for H2O, meaning larger -- more molecules to escape) that we have to hope don't notice us as we try to race off to space.  Photons can't dodge -- they have to move in straight lines at the speed of light through whatever medium they find themselves in.  Our only hope for the race to space is that the CO2 doesn't grab us.

The thing is, Avagadro's number is gargantuan.  It is about 6e22.  Try that again -- it is 6 billion (approximately number of humans on the planet) times 10 trillion (approximate gross domestic product of the US in dollars).  In other words, if every person in the world had as much money, themselves, as the entire US economy exchanged last year, they would have 1 Avagadro's number of dollars.
Update: Copied the number wrong.  Avagadro's number is 10 times bigger than that, 6e23.  Rest of note corrected for this.

For you as a photon trying to reach space, it means that there are about 6400 trillion CO2 molecules that have a chance to grab you (6.4 quadrillion).  This is not a small number!  The only reasons that any photons do reach space from the surface is that a) molecules are extremely selective about what colors they will absorb (your wavelength has to be exactly right) b) even if you have the right wavelength, molecules are typically extremely lazy and still probably won't grab you.

For our two questions, we see the answers now as
1) These very rare gases matter because the cause the entire greenhouse effect, and contribute to setting the temperature of the atmosphere (not just the ground).
2) Given 6400 trillion CO2 molecules that sit in the path of a photon trying to escape from the surface, it also isn't very reasonable to call them 'rare'.

I've tagged this note 'project folder'.  That's my flag for posts that include things that lend themselves better to projects, things for people to check me on, or things that you can take further.  In this note, the particular challenge is to come up with a way to compute the average atmospheric content of H20.  But you're also invited to recheck my math on how many CO2 molecules sit in the path of a photon from the earth's surface trying to escape to space.  An extension would be to look at the numbers for wavelengths of 4 microns and 15 microns (wavelengths CO2 is particularly likely to absorb -- it doesn't absorb at 10; I took 10 because it's the peak for what the earth emits, and it's between 4 and 15).


quasarpulse said...

correction on Avogadro's number: it's 6e23 (not 22). Which only serves to make your argument more convincing.

carrot eater said...

Hmm. This isn't how I'd go about it, but it's a way.

Some comments: Simply going from 10000 kg of air to 4 kg of CO2 seems to require all the components of air to have the same molecular weight, but they don't. Taking the average molecular weight to be 28.9 g/mol (dry basis), I get about 5.9 kg of CO2. Taking account of the average 0.4% or so water barely changes that. Doesn't affect the bottom line at all, but somebody might be annoyed by it.

Also, your image of the disc is really more the projected area of a cylinder of air. All those molecules aren't actually in the disc; they're in the column above the disc.

To keep it simple, you keep the quantum mechanics to vague verbal descriptions. But perhaps it'd be useful to just show the spectral molar absorptivities of CO2, water, nitrogen and oxygen, at whatever pressure. If nitrogen strongly absorbed at these wavelengths, nobody would care about a bit more CO2; then again, we also wouldn't be alive to worry about it.

carrot eater said...

In case you find it interesting, I thought of a different way of presenting it.

Say I hold up a 1 square meter piece of aluminum foil just over my head. I'll be most definitely be in the shade. Taking the thickness of a foil as 0.02 mm, I come up with 2 moles of aluminum in that foil. (hope my quick math was right).

Nobody doubts that those 2 moles are enough to effectively block visible light; we have everyday experience with the optical properties of metals. So one should not be surprised if 90 or so moles of CO2, which absorbs IR, would also have some effect. If a material is absorptive or reflective enough, very small amounts of it make a difference.

BCC said...

Good post (if a bit rambling toward the end). It's surprising how often your garden-variety internet skeptic reverts to the "390 ppm is small" argument.

I've been meaning to look up a toxic dosage that is << 390 ppm; arsenic in your coffee it is.

I also like the # of CO2 molecules vs. photons framing. A little harder to explain than poisoned coffee, but a good fallback.

R. Test said...

Not being a scientist I don't know if any part of this experiment makes any sense. Real Climate had something to say about this in another context a year or so ago.

I think it addresses the issue: CO2 constitutes a "small" portion of the atmosphere.

Experiment 1: a long tube containing x number of molecules of CO2 and no other gases. Shoot y number of photons into the bottom of the tube at a wavelength at which they will be captured by the CO2. Count how many come out the other end.

Experiment 2: The same tube containing x molecules of CO2 and 99x molecules of a mixture of oxygen, nitrogen and argon. Shoot y photons and count how many come out the other end.

I take it this would demonstrate something about the step between 2 and 3 in the less simplified model of the greenhouse effect.

Michael Hauber said...

Well now that you've proven that Co2 can matter even if its only 340 ppm of the sky, can you take on an even harder challenge?

Prove that the sun matters to us. Its diamter is roughly 0.5 degrees. The skies diameter is 180 degrees, and 0.5/180 ^2 is roughly 8 ppm....

Robert Grumbine said...

Carrot: Thanks for the correction. Sorry about being slow to incorporate it.

I tread lightly around the quantum mechanics stuff. It's hard to deal with both accurately and without the math. My hope is that by leaving it at vague verbal descriptions, readers realize that there's more going on -- rather than feeling a false sense of omniscience.

I do like your aluminum foil analogy.

Bcc: Thanks. The rambles, ..., well, I just hope that they cover interesting countryside. They're likely to happen. More than one friend has observed I could use an editor to keep the rambles in control.

R. Test:
You have 2/3rds of what I think would be a good experiment. The third part would be a tube of just Nitrogen, Oxygen, and Argon. Shine the same infrared light through that. Then we see that it's the CO2 that's vital to the heating of the gases. I also believe that just such experiments were done -- in the early 1900s. I have to check the citations for the details. Or, of carrot eater is a.k.a. Eli Rabett, he can post the details himself.

S2 said...

" I also believe that just such experiments were done -- in the early 1900s."

Actually I think it was John Tyndall in the 1860's.

What I find interesting is that the number of atoms in a molecule is a pretty good indicator of how good a greenhouse gas it is likely to be. Anything with less than three atoms is pretty poor, at three atoms we start to see an effect (CO2, H20, O3, etc.) CH4 has 5 atoms and is (per mole) much more effective than CO2. I can see that this is because the quantum levels available increase as the number of atoms in the molecule increase.

What I don't understand though is why nearly all the most potent GHGs contain Fluorine atoms.

Robert Grumbine said...

Tyndall (I read his 1861 paper) did study the absorption by a tube full of given gases. But as far as I saw, he didn't examine the temperature of the gases inside the tubes. If you have a citation to something of his (or anything else before 1900 that did) carrying out the experiment while checking temperature of the gas(es) inside the tube(s), please do post it.

carrot eater said...

There's a trade-off between transparency of corrections, and readability. You really don't need to cite me in the text at all.