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## Q. 17.1

Photon Energies

By how many kilojoules per mole is the energy of $O_{2}$ increased when it absorbs ultraviolet radiation with a wavelength of 147 nm? How much is the energy of $CO_{2}$ increased when it absorbs infrared radiation with a wavenumber of 2 300 $cm^{−1}$ ?

## Verified Solution

For the ultraviolet radiation, the energy increase is

$\Delta E = hv = h\frac{c}{λ}$

$= (6.626 × 10^{−34} J · s)[\frac{(2.998 × 10^{8} m/s)}{(147 nm)(10^{−9} m/nm)}] = 1.35 × 10^{−18}$ J/molecule

$(1.35 × 10^{−18} J/molecule)(6.022 × 10^{23} molecules/mol) = 814$ kJ/mol

This is enough energy to break the $O \xlongequal[]{} O$  bond in oxygen. For $CO_{2}$, the energy increase is

$\Delta E = hv = h\frac{c}{λ} = hc\widetilde{v}$           (recall that $\widetilde{v} = \frac{1}{λ}$)

$= (6.626 × 10^{−34} J · s)(2.998 × 10^{8} m/s)(2 300 cm^{−1})(100 cm/m)$

= 4.6 × $10^{−20}$ J/molecule = 28 kJ/mol

Infrared absorption increases the amplitude of the vibrations of the $CO_{2}$ bonds.

Test Yourself     What is the wavelength, wavenumber, and name of radiation with an energy of 100 kJ/mol? (Answer: 1.20 μm, 8.36 × $10^{3} cm^{−1}$ , infrared)