Eli Grabs Another Envelope

Bunnies are not really into complex calculations where the meaning is hidden in the math.  For one thing there is great opportunity to fool yourself, for another, it is pretty simple to mislead others.  To Eli math is something you grab after you have figured out what is happening, and then you grab it in shells, with the easy parts first.  That way if you mess up, you always can look at the inner shell to figure out where you went wrong.  EO Wilson had some recent thoughts about the role of mathematics

Fortunately, exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory. Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition.

and simple models help with forming those concepts.  In a recent previous life the Bunny Collective discussed the relative probabilities for emission by a single vibrationally excited  CO₂ molecule that has been vibrationally excited by absorbing a photon.  There are two fates for the little guy, emission of an IR photon.

[1]  CO₂^* -->  CO₂ + hν          Rate constant = k_R

and collisional de-excitation by another, say nitrogren or oxygen molecule

[2]  CO₂^* + M -->  CO₂ + M   Rate constant = k_M

Plugging in the numbers, at atmospheric pressure the Rabett showed that only one in a hundred thousand excited molecules would emit an IR photon.  As several pointed out, high up in the stratosphere this decreases in proportion to the pressure, but, for example, at ~17 km it is one in ten thousand, and at ~ 25 km one in a thousand.

On the other hand, thermal excitation of CO₂, the reverse of reaction [2] is just

[3]  CO₂ + M -->  CO₂^* + M   Rate constant = k_-M

so now, in addition to absorbing an IR photon CO₂^* can be excited by collisions, converting thermal translational energy into vibrational energy. 

How fast a reaction happens is called the reaction rate, and is proportional to the concentration of each reactant and a number called the rate constant which hides all the dynamics, e.g. the quantum chemistry, the probability of a collision having enough energy to actually cause molecular changes, etc.  If the rate of the forward reaction, equals the rate of the reverse reaction,  an equilibrium requires that as many molecules react in the forward direction as in the reverse




[4]  (k_M [M] + k_R )  = k_-M  [CO₂^*][M]

If  k_R < < k_M[M] it can be neglected and then [2] and [3] form a simple equilibrium

[5]  CO₂^* + M = CO₂ + M

which can be rearranged to yield

[6]    [CO₂^*]/[CO₂] = k_-M /k_M = K_eq

The next step is to show how thermodynamics allows a simple estimation of K_eq and [CO₂^*].  The emission rate in photons per second per unit volume will then simply be

[7]   k_R [CO₂^*]