Lets start again. When a greenhouse gas molecule is excited by infra-red, the next event is most likely to be a collision with a surrounding air molecule (any of them). Thus the energy is shared and the greenhouse gas is said to be thermalised at the local temperature. Now the greenhouse gas molecules will continue to do what they always do i.e radiate infra-red at just the same frequencies at which they absorb best. This is basically an example of Kirchoff’s law. A similar effect at short wavelengths was observed by the older Angstrom when looking at the darker absorption lines of e.g hydrogen and helium in the Sun.

Returning to the Earth, The difference between the emission from the ground and the atmosphere is the following: first the re-radiation is weaker because it is from a colder source and secondly it appears to have doubled in amount because it is upwards as well as downwards. In practice these two effects balance. There is nothing in the physics to prohibit multiple bouncing but you do not need to consider it, if you do the accounting in the simplest way. I do not think that there is a significant leaking from the main absorption bands to extreme long wavelengths as suggested. That would be an example of fluorescence. In fact the process is governed by Schwartzschilds equation (warning: that is seriously mathematical) which deals with one wavelength at a time.

References.
I’m sorry that I could not find an elementary one quickly.
http://geosci.uchicago.edu/~rtp1/ClimateBook/ClimateVol1.pdf
(look for Kirchoff’s law and Section 4.2.1 and p.164, Eqs.4.6, 4.7 where the frequency (Greek letter nu) is specified)

or alternatively
http://atmos-chem-phys-discuss.net/2/289/2002/acpd-2-289-2002.pdf

Or Google Scholar greenhouse+gas+Schwarzschilds

The next question you should ask a chem major, how is adding more CO2 to the troposphere different from adding a mirror behind a mirror.

That’s actually a good question. It is very different, but don’t ask a chem major to explain you that. It has to do with the existence of a vertical (negative) temperature gradient (lapse rate) in the atmosphere. Adding more CO2 makes the atmosphere more opaque, so the longwave rays escaping to space come from a higher — thus colder — level in the atmosphere, and the system must warm up to compensate. The gradient spans a greater vertical distance.

One could say, it’s not like adding a mirror behind a mirror; it’s like adding a garment over a garment.

Even the logarithmic behaviour follows naturally from this conceptual model.

Quoth John Tyndall (1862):

”As a dam built across a river causes a local deepening of the stream, so our atmosphere, thrown as a barrier across the terrestrial rays, produces a local heightening of the temperature at the earth’s surface.”

The change in CO2 concentration is commonly accepted as linear, not exponential, as given in
IPCC-4 Ch. 2 page 138

I’ve ran a little model of IR absorption based on 0x, 1x, 2x, 10x CO2, you can see result here
The important part to note is how close the lines are at 1x, 2x, and 10x CO2, and the difference in absorption (area under the curves) between 1x and 2x CO2 is only like 0.03% for this case

http://global-warming.accuweather.com/2009/06/a_follow_up_on_that_co2_and_te.html

Basically, if you compare the concentration of CO2 to the resulting warming, there is a logarithmic relationship, as you already know. And when you compare our emissions of CO2 to the actual concentration, the relationship is exponential, due to feedbacks (especially the saturation of carbon sinks). So when you compare our emissions to the resulting warming, the graph is actually linear, as the exponential graph is inverse to the log graph, and thus offsets it.

Eventually the spectrum would become saturated, and adding more CO2 would have no effect. But I’d venture a guess that our species has plenty of time to screw itself over before that happens.

Now, the tricky thing is, the near-IR spectrum is already saturated in the atmosphere, actually, it is already saturated in the lower troposphere. The next question you should ask a chem major, how is adding more CO2 to the troposphere different from adding a mirror behind a mirror.

I wasn’t sure if it was a one-time affair or not, or if other greenhouse gases could pick up those bands, so I put “multiple” instead of “twice” to be on the safe side. Thanks for your input, I’m still quite early on in physics so it’s nice to have my misconceptions corrected.

Radiative heat transfer in the atmosphere is a very complicated matter. In my efforts to understand the nuances of it (I’m ok with the basics from 3rd-year thermo), I’ve so far come to disappointing and underwhelming explanations on many fora (including RC), which lead me to suspect that climatologists are not the best experts to talk about this subject. Thermo guys and optics physicists are probably your best bet.

The number of atoms (and their mass) per molecule.

O2 and N2 have just 2 atoms per molecule. So, they are limited in how they can vibrate. C02, H20, CH4, N0x and CFC have more than 2 atoms and can vibrate in many differant ways. The additional vibrational states allow these molecules to resonate with more frequencies, which tend to be in the infrared range.