6.3 How is energy related to the wavelength of radiation?

We can think of radiation either as waves or as individual particles called photons. The energy associated with a single photon is given by E = hν , where E is the energy (SI units of J), h is Planck's constant (h = 6.626 x 10–34 J s), and ν is the frequency of the radiation (SI 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 2.998 x 108 m s–1. Another quantity that you will often see is wavenumber, σ=1/λ, which is commonly reported in units of cm–1.

The energy of a single photon that has the wavelength λ is given by:

E = hc λ = 1.986× 10 −16 J nm photon −1 λ MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaabbaaaaaaaaIXwyJTgapeGaamyraiaabckacqGH9aqpcaqGGcWaaSaaaeaacaWGObGaam4yaaqaaiabeU7aSbaacqGH9aqpcaqGGcWaaSaaa8aabaWdbiaaigdacaGGUaGaaGyoaiaaiIdacaaI2aGaey41aqRaaeiOaiaaigdacaaIWaWdamaaCaaaleqabaWdbiabgkHiTiaaigdacaaI2aaaaOGaaeiOaiaabQeacaqGGaGaaeOBaiaab2gacaqGGcGaaeiCaiaabIgacaqGVbGaaeiDaiaab+gacaqGUbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaaak8aabaWdbiabeU7aSbaaaaa@5AA2@

[6.2a]
 

Note that as the wavelength of light gets shorter, the energy of the photon gets greater. The energy of a mole of photons that have the wavelength λ is found by multiplying the above equation by Avogadro's number:

E m = hc N A λ = 1.196 × 10 8 J nm mol −1 λ MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaabbaaaaaaaaIXwyJTgapeGaamyra8aadaWgaaWcbaWdbiaad2gaa8aabeaak8qacaqGGcGaeyypa0JaaeiOamaalaaabaGaamiAaiaadogacaWGobWdamaaBaaaleaapeGaamyqaaWdaeqaaaGcpeqaaiabeU7aSbaacqGH9aqpcaqGGcWaaSaaa8aabaWdbiaaigdacaGGUaGaaGymaiaaiMdacaaI2aGaaeiOaiabgEna0kaabckacaaIXaGaaGima8aadaahaaWcbeqaa8qacaaI4aaaaOGaaeiiaiaabQeacaqGGaGaaeOBaiaab2gacaqGGcGaaeyBaiaab+gacaqGSbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaaak8aabaWdbiabeU7aSbaaaaa@5A32@

[6.2b]
 
wave size-radio: building, microwave: butterfly, infrared: needle, visible: protazoa, ultraviolet: molecule, x-ray: atoms, gamma: nucleus
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: Credit: EM Spectrum Properties reflected by Inductiveload from Wikimedia Commons is licensed under CC BY-SA 3.0.

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?

ANSWER: Solve for wavelength in equation [6.2b]

λ= 1.196 × 10 8 J nm mol −1 E m = 1.196 × 10 8 J nm mol −1 386 × 10 3 J mol −1 =309 nm MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@817C@