This Is Where Eli Came In

One of the useful things the Rabett used to do was to explain what happens to the energy when a molecule, say CO2 (carbon dioxide) although you could also say H2O (water vapor) or CH4 (methane) absorbs light. For the purpose of this post, the photon would be in the infrared region of the spectrum.  This is an evergreen for two classes of bunnies

1. Bunnies who don't realize that the molecule can also emit light.  This is a popular one amongst organikers and analytical chemists whose experience with IR spectroscopy is in an absorption spectrum for analysis of samples
2. Bunnies who think that the only way that an excited molecule can get rid of the energy is to emit a photon.  

 For every CO2 molecule there are roughly 3000 2500 other molecules in the same volume of air.  When a CO2 molecule collides with one of the other molecules, almost certainly an oxygen or nitrogen molecule, energy transfer occurs.  Each CO2 molecule can be described as having translational, vibrational and rotational energy and the same is true of the collision partner.  Any collision can in principle change the amount of any of these forms of energy by any amount subject to conservation of energy and momentum.  The probability of this happening depends on the relative translational energy of the collision, the relative orientation of the molecules, their distance of closest approach and the distribution of energy in each of the collision partners prior to the collision.  The detailed study of such effects is called collision dynamics or molecular dynamics. 

Fortunately, we can take thermal averages over many of these variables, either theoretically or experimentally which makes life, theory and experiments much simpler and a hell of a lot less expensive and time consuming.  That sort of thing usually goes under the rubric of reaction (when there is one) kinetics or energy transfer studies when there isn't.

The fate of the energy in a vibrationally  excited CO2 molecule can be thought of as a race between emission of a photon taking essentially all of the vibrational energy away and a collision that does the same.  Here Eli is going to tell you who wins the race, something that is generally handwaved, correctly handwaved, but handwaved none the less.

For the lowest vibrational level, the 667 cm^-1 doubly degenerate bend, labelled as (0,1⁰,0), the only lower state is the unexcited ground state (0,0⁰,0)

 CO2(0,1⁰,0) + M --> CO2(0,0⁰,0) + M

where M is any other molecule.  The probability of this happening depends on M, but it can be measured by exciting a small amount of CO2 mixed into a large bath of M and then watching how long it takes for the excited CO2 to decay away.   This behavior is captured by the rate equation

d[CO2^*]/dt = - kR [CO2^*] - kM[CO2^*][M]    (1)

where kR is the radiative emission rate in the absence of collisions and kM is the collisional quenching rate, the rate at which collisions with M de-excite the CO2.  The square brackets indicate the concentration of whatever species is inside.  If [M] is very large compared to [CO2] we have what is called pseudo first order conditions, because the effect of the collision on the quantum state of M will be very small and Eq. 1 is a first order differential equation whose solution is

[CO2^*] =  [CO2^*]_t=0 exp (-{kR  + kM[M]}t)

where t is time and we are assuming that the CO2 is being excited during a very short time so we can neglect the competition between excitation and de-excitation which simplifies the kinetic analysis.

The best study of this process for CO2 was The vibrational deactivation of the (00o1) and (0110)Modes of CO2 measured down to 140 K by Siddles, Wilson, Simpson, Chemical Physics 189 (1994) 779-91.  They measured collisional quenching cross-sections for CO2 (0,1⁰,0) and the higher lying (0,0⁰,1) levels against a number of gases, including carbon dioxide, the rare gases, nitrogen, oxygen and deuterated methane and deuterium.  Oxygen and nitrogen have quenching rate constants of 5.5 and 3.1 x 10^-15 cm³/molecules-sec respectively at 295K, decreasing roughly linearly to about a factor of five less at 140K.  To make sense of this number, we can compare it to what is known as the gas kinetic quenching rate, e.g. what the rate would be if every collision were 100% effective, of the order of 10^-10 cm³/molecules-sec.  More to the point, at atmospheric pressure where the concentration is ~2.6 x 10¹⁹ molecules/cc (Loschmidt's number)
the rate of collisional de-excitation for CO2(0,1⁰,0) by N2 will be

kM[M] ~ 5 x 10^-15 cm³/molecules-sec x 2.5 x 10¹⁹ molecules/cm³ ~ 10⁵ s^-1

Note the ~ meaning oom or order of magnitude.  It's not that Eli can't afford a calculator, it's that who needs one in this sort of calculation when toes and ears are available.  The lifetime will be the reciprocal of this, 10^-5 s or 10 us.

What about the radiative lifetime?  Radiative lifetimes of vibrationally excited states are long.  Insight into why this is so can be found by looking at the relationship between the Einstein A and B coefficients.  Roughly put, the A coefficient is proportional to the rate of spontaneous emission and the B coefficient to the rate at which photons can be absorbed by the molecule.  assuming that the transitions are between two isolated levels

A/B = 8 π h ν³/c³

 The ν³ is what makes emission from low lying vibrational levels so slow.  The isolated radiative lifetime of CO2(0,1⁰,0) is ~1.1 s.  The (0,0⁰,1) lifetime is 2.4 x 10^-3, but remember it occurs at 2350 cm^-1 where the ν³ factor can do its work.

Comparing the radiative rate kR (the inverse of the lifetime) to the collisional deactivation rate kM[M], provides a quick estimate that only one out of 100,000 CO2 molecules excited into the (0,1⁰,0) by collision or absorbing a photon, will emit.