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By Charles Rhodes, P.Eng., Ph.D.


In order to understand the infrared emission by the Earth it is necessary to understand the absorption/emission characteristics of the greenhouse gases and how these gases affect the infrared spectrum of the Earth. The gases of most interest are water vapor, carbon dioxide, ozone, methane and nitrous oxide.

When the earth is viewed from outer space with an infrared thermal emission spectrometer, the spectrum consists of radiation from species that will readily interact with electromagnetic radiation within the thermal emission spectral range. Oxygen and nitrogen do not have charge separation in their molecules and hence do not interact with photons in the thermal emission spectral range and hence are not seen. Water molecules, which have charge separation, interact with photons across most of the thermal infrared emission spectrum. Carbon dioxide and ozone also have strong interaction bands.

The water is concentrated in the lower atmosphere. Hence, from the perspective of a space craft, carbon dioxide and ozone in the upper atmosphere filter the thermal infrared emission from water that originates in the lower atmosphere.

The thermal infrared emission spectrum of the Earth, as recorded from deep space by the Mars Global Surveyor Spacecraft in November 1996, is shown in Figure 1 below.

Figure 1

Reference: Initial Data from the Mars Global Surveyor Thermal Emission spectrometer Experiment: Observations of the Earth

Most of the solar energy incident on the Earth that is not reflected by clouds and ice is absorbed by sea water. This absorbed energy causes heating and hence evaporation. Most of the absorbed solar energy initially becomes latent heat of vaporization of water. The water vapor molecules rise in the atmosphere and then condense losing translational motion kinetic energy by collisions with N2 and O2 molecules. On condensation the H2O molecules clump topgether to form liquid water droplets. These liquid water droplets then cool by emission of infrared radiation and freeze. Basic physics can be used to calculate the range of emitted photon radiation frequencies.

When water vapor condenses the latent heat of vaporization lost to other molecules is: 2257 J / g. The corresponding photon radiation frequencies are much higher than the thermal emission band so the primary energy loss is via inter molecular collisions.

When liquid water cools the maximum energy loss before freezing occurs is:
(100 deg C) X (1 cal / gm deg C) X (4.18 J / cal) = 418 J / gm

The latent heat of fusion of water is:
334 J / gm

The enthalpy released in cooling ice from 0 C to - 10 C is 20.27 J / g
The enthalpy released in cooling ice from -10 C to - 20 C is 19.72 J / g
The enthalpy released in cooling ice from -20 C to - 30 C is 19.13 J / g
The enthalpy released in cooling ice from -30 C to - 40 C is 18.51 J / g
The enthalpy released in cooling ice from -40 C to - 50 C is 17.84 J / g
The enthalpy released in cooling ice from -50 C to - 58 C is 14.00 J / g

The total enthalphy release in cooling ice from 0 C to - 58 C is: 109.47 J / g

Each mole contains 6.023 X 10^23 molecules

Hence the enthalpy of fusion per molecule is:
(334 J / gm) X (18 gm H2O / mole) X (1 mole H2O / 6.023 X 10^23 molecules) = 0.998174 X 10^-20 J / molecule

Assume that liquid water freezes by emitting one infrared photon per molecule. Then when a water molecule transitions from 0 degrees C liquid to 0 degrees C solid by emitting infrared photons the photon energy is:
E = 0.998174 X 10^-20 J

The energy of each such photon is given by:
E = h F
where h = Planck's constant
= 6.626 X 10^-34 J-s.

F = E / h
= (0.998174 X 10^-20 J) / (6.626 X 10^-34 J-s)
= 0.150645 X 10^14 Hz
= 15.0645 THz

The speed of light is:
299,792,458 m / s

Hence the wave number of the far infrared radiation emitted by 0 degree C liquid water transitioning to 0 degree C ice is:
(Wave Number) = F / C
= (0.150645 X 10^14 Hz) / (299,792,458 m / s)
= .050249758 X 10^6 / m
= 502.5 X 10^2 / m
= 502.5 / cm

This wave number is at the peak of Earth's experimentally measured thermal infrared emission spectrum shown on Figure 1.

The wave number of the far infrared radiation emitted by 100 degree C water transitioning to freezing at -58 C is:
[(109.47 J / g + 418 J / g + 334 J / g) / (334 J / g)] (502.5 / cm) = 1296.07 / cm

This calculation explains the experimentally observed sharp drop in thermal emission at a wave number of about 1250 / cm shown in Fig.1. Note that water vapor does not emit infrared radiation at wave numbers immediately above 1300 / cm because the large heat of vaporization of water provides no accessible energy state transitions in that spectral range.

As the CO2 concentration in the Earth's upper atmosphere increases the infrared radiation emission by freezing of water droplets must increase for thermal radiation to remain in balance with absorbed solar radiation.. In order to provide more warm water to drive this infrared radiation emission process the temperature of the lower atmosphere must increase.

This web page last updated July 2, 2016.

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