Recent Advances in Climate Change Research: Part IX - How Carbon Dioxide Emits IR Photons
Skydive with us into the quantum world, where we provide to those unafraid of molecular energy transfer an answer to the question: what happens to Earth’s radiated infrared (IR) photons after they are absorbed by IR active CO2 molecules in the lower atmosphere? Part VIII (GEO ExPro Vol. 17, No. 3) showed how CO2 molecules absorb Earth’s IR radiation. Here, we show that the bulk background gases N2 and O2 are critical for the greenhouse effect because collisions of CO2 (and other greenhouse gases) with N2 or O2 both take away and add energy to the CO2 molecules. Every collision that adds energy gives the CO2 molecule a chance to undergo radiative decay and emit a photon.
“Thanks for the lonely night, for the hills, the rush of the darkness and the sea through my heart! This silence murmuring in my ears is the blood of all Nature seething; … the northern lights flare over the heavens to the north. By my immortal soul, I am full of thanks that it is I who am sitting here!”
From ‘Pan’ by Knut Hamsun (1859–1952), Norwegian winner of the Nobel Prize for Literature in 1920.
Crash Course in Molecular Energy Transfer
First, a brief review of the various mechanisms, definitions, and terms useful in understanding molecular energy transfer. Consider the bimolecular collision in which reactants A in quantum state i and B in quantum state j react to form products A in quantum state l and B in quantum state m,
Here ΔE is the exchanged energy during the collision process; quantum state 0 corresponds to the ground state. The forward process is described by the right arrow. An increase in the concentration [A(i)] or [B(j)] results in an increase in the rate of reaction. The concentration unit is measured in molecules per cubic centimeters per second. It stands to reason that the reaction rate is proportional to the increase in concentration, so rate = kf[A(i)][B(j)]. The rate constant kf is the proportionality constant relating the rate of the reaction to the concentrations of reactants.
A large kf means that the reaction is relatively fast, while a small value means that it is relatively slow. Raising the temperature of the reaction usually results in a higher rate of reaction; the particles move faster and faster, resulting in a greater frequency of collisions, so kf is temperature dependent. The reaction in equation 1 be shown to have exponential temporal behavior with a ‘time constant’ τ = 1/k[M]), where M equals A or B and τ is referred to as the relaxation time for the process.
Corresponding to the forward process is the reverse process (left arrow) with rate constant kr. When the rates of the forward and reverse reactions have become equal, the reaction has achieved a state of balance or equilibrium, kf[A(i)][B(j)] = kr[A(1)][B(m)], yielding the ratio (Denisov et al. 2003):
where K is the equilibrium constant, g denotes the degeneracy of the molecular quantum state, and kBT is the product of the Boltzmann constant and the temperature. The term ‘equilibrium’ indicates that the different forms of energy (rotational and vibrational) are characterized by one temperature T (energy equilibrium).
There are two kinds of vibrational energy exchange processes during bimolecular collisions: vibration-translation (V-T) and vibration-vibration (V-V). Let us consider a vibrationally excited molecule, A*, where the asterisk denotes one quantum of vibrational excitation. A* collides with a species M, and the vibrational energy is transferred into translational motion. This process may be expressed by the transfer equation:
where A represents the molecule in its ground state. Equation 3 is the special case of equation 1 when l=1, i=j=m=0, and B=M. In contrast to equation 1, here the left to right process is associated with the reverse rate constant, where the excited reactant A* undergoes vibrational deactivation. The V-T process results in energy transferring from molecular vibrational modes to molecular translation. One molecule loses one quantum while the vibrational state of the other molecule is unaltered. The process associated with the forward rate constant is that where the reactant A acquires enough energy to react by colliding with another molecule M. This process is called vibrational up-pumping.
How CO2 Relaxes in the Lower Atmosphere
Part VIII told us that the terrestrial radiation follows a blackbody distribution of characteristic temperature of 288K, with 98% of radiative power emitted in the 5–80 μm range. There is only one CO2 absorption band of importance in this range, at around 15 μm (667 cm−1). This band almost coincides with the spectral maximum of terrestrial radiation and therefore to a large extent determines the interaction of CO2 molecules with the radiation. CO2 is responsible for a large gap in the transmissivity of Earth’s IR radiation towards space, centered around 15 μm.
The process whereby a CO2 molecule absorbs an infrared photon of energy 667 cm–1 and goes to the vibrationally excited state CO2*(010) reads:
The photon transfers its energy to the IR active CO2 molecule and is removed from the radiation field, while the photon energy raises the CO2 molecule to a higher vibrational state. But since excited states are energetically unfavorable the molecule wants to return to the ground state by giving up energy. How? We provide the answer by following the respected physics tradition of ‘back-of-the-envelope’ calculations.
Finding the Winner
Vibrational energy can be transferred either radiatively by spontaneous or stimulated emission or non-radiatively by collision. Which process is the winner? To find out, one must compare the radiative lifetime of the excited level with the relaxation time of collisions. If the relaxation time is short compared with the average radiative lifetime of the excited level, then the collision process wins.
Radiative lifetime: The photon’s energy causes the CO2 molecule to elevate. The molecule releases this extra energy by emitting the photon, which is identical to the absorbed one, but emits in an arbitrary direction since it, like the drunken sailor, has no memory of its previous steps. Once the emitted photon has left, the molecule returns into its ground state. The radiative lifetime of the (010) molecular vibration is about 1.1s (Cheo, 1971). It is an eternity on a gas kinetic time scale.
Relaxation time by collisions: The collisional relaxation process occurs when the relaxation time can compete with the radiative lifetime of the excited energy levels. Even though the activated CO2 molecule, at a CO2 concentration of 400 ppm, is one among around 2,500 other molecules, it is moving very fast and it does not have to move far before it bumps into other molecules – usually N2 or O2 – and drops back into its ground state. The freed energy then adds speed to another molecule’s motion. When many collisions take place simultaneously, the faster speed of the molecules being bumped into raises the temperature of the gases in the atmosphere, since temperature is proportional to the average kinetic energy of the gas. Since the photon is permanently lost from the radiation field, this is absorption of photons.
How fast does this happen? The collision process for CO2 deactivation in the temperature range 300–140K against a number of gases has been studied by Siddles et al. (1994). Let M denote either the N2 or O2 molecule. The process of vibrational de-excitation from the 667 cm−1 level through collision with molecule M can be described by (see equation 3)
where kr(υB) is the V-T rate constant for relaxation of CO2(010) by M, where the vibrational energy ΔE resident in the CO2 bending-mode is transferred to M as translational kinetic energy, which is reflected on the macroscopic scale as a temperature increase.
The speed of the process depends on the temperature where the process runs. We select the altitude 3,550m where temperature is 265K (-8ºC). The number of molecules per cm3 in dry air at this height is [M]=1.79 × 1019, with 78% N2 and 21% O2. For N2 and O2 Siddles et al. (1994) give constants kr (N2) = 2.4 • 10–15 and kr (O2) = 3.6 • 10–15 cm3(molecule s)–1. The lifetime of collisional de-excitation for CO2 (010) in the atmospheric gas bath can be deduced as
The typical collision time through which a CO2 (010) molecule can transfer its energy to another gas molecule is about 20 μs in the lower atmosphere at altitude 3.5 km. Collisions take place more often than re-radiation. Therefore, when a CO2 molecule in air absorbs a photon, it is much more likely – on the order of 1s/20μs=0.5 × 105 times – to heat the surrounding air molecules with the energy it acquired from the absorbed photon than to re-radiate the photon. Statistically, the same CO2 molecule re-emits the photon energy two out of 100,000 times; but 99,998 times out of 100,000 the excited CO2 molecule is de-excited by collision.
In physics, thermalization is the process of physical bodies (e.g., molecules) reaching thermal equilibrium through mutual interaction (e.g., collisions). In general, the natural tendency of a system, like the atmosphere, is towards a state of equipartition of energy and uniform temperature. Since the collisional step is fast (about 20 μs at 3.5 km), the photon energy involved at 667 cm–1 is rapidly spread out among the surrounding air molecules – or thermalized into the ‘heat bath’ of the atmospheric gas. CO2 is then rapidly in thermal equilibrium with the rest of the gas molecules.
The Life Events of CO2 Molecules
The atmospheric bath receives an inflow of energy from Earth’s IR radiation where CO2 absorbs photons in the 667 cm–1-centered band but seemingly only negligibly emits photons. The gas bath increases its temperature, but a gas cannot easily increase its emissivity (Robitaille, 2014). As Earth continuously sends IR energy upwards, CO2 photon absorption would make the air get really hot … unless there is a process that is able to pass the received energy on. The gas needs to cool, and the question is how? This process must involve the creation of additional photons that can become the energy carriers for radiation. Of course, in steady state, under the assumption of local thermodynamic equilibrium, the reverse process to that we have considered is also ongoing, at equal rate, all the time (see equation 5). Therefore, collisions of ground-state CO2 molecules with air molecules may excite the former and cause them to radiate. The rate constant kf(vB) for vibrational up-pumping at gas temperature 265K, where kBT = 184.2 cm–1, can be found from kf(vB) by using equation 2:
The number 2 in this equation arises because of CO2 (010) being doubly degenerate; the two bending mode vibrations in CO2 have equal energy. Repeating the calculations as above with a new rate constant, the relaxation time for the reaction is found to be around 20 μs/0.0535 or about 400 μs.#
Recall that the radiative lifetime of CO2 de-excitation is around 1.1s. Since the collisional processes are much faster, a CO2*(010) molecule can de-excite in 20 μs and excite back to the (010) state in 400 μs. It is quite a pace! One trip back and forth takes 420 μs so that during 1.1s the number of possible trips is 1.1s/420 μs = 2,620. When 100,000 CO2*(010) molecules are available for de-excitation, it is likely that two of these will be reserved for radiation of a photon; the remaining 99,998 are back after 420 μs to offer 2/100,000 of these to radiate a photon while the rest de-excites through collisions. During a time interval of 1.1s, a back-of-the-envelope calculation indicates that a little less than 2,620 x 2 = 5,240 CO2*(010) are likely to participate in the photon radiation process. This is around 5% of the CO2*(010) molecules. We may thus conclude that around 5% of the CO2 molecules which absorb IR radiation from Earth’s surface tend to radiate IR photons (at altitude 3.5 km). The rest are busily colliding with N2 and O2 molecules.
The process of photon absorption and emission can be defined as follows: The photons in the band around 667 cm–1 from Earth’s surface are absorbed by CO2 molecules. Only a very small percentage re-radiate photons in a random direction and the rest lose that energy to the surrounding bath of atmospheric molecules. In turn, the atmospheric molecules collide with CO2 molecules so that they get excited to the (010) state. A very small percentage radiates new photons, again in a random direction, and the rest lose the energy by collision. It is a stressful life! The process repeats forth and back rapidly, so that in the timeframe of about a second, around 5% of the CO2 molecules radiate. Close to the surface the percentage is slightly higher since the temperature is elevated, while higher in the troposphere it is slightly less since the temperature is lower.
Further Reading in the 'Recent Advances in Climate Change Research' Series
Recent Advances in Climate Change Research: Part I - Blackbody Radiation and Milankovic Cycles
Martin Landrø and Lasse Amundsen, NTNU / Bivrost Geo
Geoscience will probably play an important role in mitigating carbon dioxide emissions. In part one of this series, we discuss some history and physics behind the topic of climate change including the concepts behind blackbody radiation and Millankovic Cycles.
This article appeared in Vol. 16, No. 2 - 2019
Recent Advances in Climate Change Research: Part II - Arrhenius and Blackbody Radiation
Martin Landrø and Lasse Amundsen, NTNU / Bivrost Geo
In Part II we look at Arrhenius’ seminal 1896 paper and see how it relates to blackbody radiation and absorption of infrared radiation by the atmosphere, taking a closer look at his model of the greenhouse effect.
This article appeared in Vol. 16, No. 3 - 2019
Recent Advances in Climate Change Research: Part III - A Simple Greenhouse Model
Martin Landrø and Lasse Amundsen, NTNU/Bivrost Geo
What would the temperature of Earth be without the atmosphere? By using simple physical models for solar irradiation and the Stefan-Boltzmans law for blackbody radiation, we can estimate average temperatures with and without atmosphere.
This article appeared in Vol. 16, No. 4 - 2019
Recent Advances in Climate Change Research: Part IV - Challenges and Practical Issues of Carbon Capture & Storage
Martin Landrø, Lasse Amundsen and Philip Ringrose
The basic idea behind CCS (Carbon Capture and Storage) is simple, but what are the main challenges and practical issues preventing a more global adoption of this method?
This article appeared in Vol. 16, No. 5 - 2019
Recent Advances in Climate Change Research: Part V - Underground Storage of Carbon Dioxide
Eva K. Halland, Norwegian Petroleum Directorate. Series Editors: Martin Landrø and Lasse Amundsen, NTU/Bivrost Geo
By building on knowledge from the petroleum industry and experience of over 23 years of storing CO₂ in deep geological formations, we can make a new value chain and a business model for carbon capture and storage (CCS) in the North Sea Basin.
This article appeared in Vol. 16, No. 6 - 2019
Recent Advances in Climate Change Research: Part VI - More on the Simple Greenhouse Model
Lasse Amundsen and Martin Landrø, NTNU/Bivrost Geo
We continue the discussion of the simple greenhouse model introduced in Part III.
This article appeared in Vol. 17, No. 1 - 2020
Recent Advances in Climate Change Research: Part VII - Arrhenius’ Greenhouse Rule for Carbon Dioxide
Lasse Amundsen and Martin Landrø, NTNU/Bivrost Geo
Here, we investigate the relationship between radiative forcing (heat warming) of carbon dioxide and its concentration in the atmosphere to better understand climate feedback and sensitivity.
This article appeared in Vol. 17, No. 2 - 2020
Recent Advances in Climate Change Research: Part VIII - How Carbon Dioxide Absorbs Earth’s IR Radiation
Lasse Amundsen and Martin Landrø, NTNU/Bivrost Geo
Skydive with us into the quantum world and find out how carbon dioxide molecules absorb thermal IR radiation.
This article appeared in Vol. 17, No. 3 - 2020