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).