METEO 300
Fundamentals of Atmospheric Science

6.3 How is energy related to the wavelength of radiation?

The energy associated with radiation is given by E = hν ,where E is the energy in Joules, h is Planck's constant (h = 6.626x10-34 J s), and ν is the frequency of the light in units of s-1 (or Hertz, Hz) (see figure below). Frequency is related to wavelength by λ=c/ν , where c, the speed of light, is 3.0x108 m s-1. Another unit that you will often see is wavenumber, σ=1/λ which has units of cm-1.

We can think of radiation either as waves or as individual particles called photons. The energy of a single photon that has the wavelength λ is given by:

E = hc/λ= ( 1.98x 10 16 λ(nm) ) J photo n 1 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

[6.2]
 

where λ is in units of nm. Note that 1 Joule = 6.242x1018 eV, 1 cal = 4.184 J (side note: Did you know that when a US food label says "100 calories," it really means "100 kilocalories"? ). Also note that as the wavelength of light gets shorter, the energy of the photon gets greater. If we consider the amount of energy as distributed in a mole of gas, we get the relationships:

1 eV = 23.06 kcal mol-1 = 96.48 kJ mol-1 = 1.60x10-19 J molecule-1 (or J photon-1)

Energy scales. See caption for more.
Energy scales: penetration through Earth’s atmosphere; radiation name by wavelength; physical object the size of that wavelength; frequency compared to wavelength; and temperature of an object that has its peak radiation at each wavelength.
Credit: Wikimedia Commons

In the lesson on atmospheric composition, you saw how solar UV radiation was able to break apart molecules to initiate atmospheric chemistry. These molecules are absorbing the energy of a photon of radiation, and if that photon energy is greater than the strength of the chemical bond, the molecule may break apart.

Check Your Understanding

Consider the reaction O3 + UV → O2 + O*. If the bond strength between O2 and O* (i.e., excited state oxygen atom) is 386 kJ mol-1, what is the longest wavelength that a photon can have and still break this bond?

Click for answer.

ANSWER: Change energy units to J photon-1 and then use equation [6.2]

E= 1.60x 10 19 96.48 386=6.4x 10 19  J photon -1 λ= 1.98x 10 16 6.4x 10 19 =309 nm This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.