Photodissociation of water $$ \mathrm{H}_{2} \mathrm{O}(l)+h \nu \longrightarrow \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) $$ has been suggested as a source of hydrogen. The \(\Delta H_{\mathrm{rxn}}^{\circ}\) for the reaction, calculated from thermochemical data, is \(285.8 \mathrm{~kJ}\) per mole of water decomposed. Calculate the maximum wavelength (in \(\mathrm{nm}\) ) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?

First convert the energy change per mole (\(\Delta H_{\mathrm{rxn}}^{\circ}\)) into energy per molecule. To do this, divide it by Avogadro's number (6.022 x 10^23 molecules/mole). Then convert the energy per molecule from joules to electron volts (eV) using the conversion factor 1 eV/1.602 x 10^-19 Joules: \[\frac{285.8 \times 10^3 \, \mathrm{J}}{6.022 \times 10^{23} \, \mathrm{molecules}}\] Converting from Joules to eV, gives:\[\frac{1.89044 \, \mathrm{eV}}{\mathrm{molecule}}\]

Planck’s equation (E = hν) links the energy of a photon to its frequency; where h is Planck’s constant and ν is the frequency of the radiation. As the wavelength λ is related to frequency by c = λν, where c is the speed of light in vacuum, we can rewrite the equation in terms of λ: E = hc/λ. To find the maximum wavelength we need to supply to initiate the reaction (λ), we rearrange the equation to λ = hc/E. Substituting the known values (h = 4.135 x 10^-15 eV.s, c = 3.0 x 10^8 m/s, and E is the energy per molecule we calculated in step 1) we get:\[λ = \frac{(4.135 \times 10^{-15} \, \mathrm{eV} \cdot \mathrm{s}) (3.0 \times 10^8 \, \mathrm{m}/\mathrm{s})}{1.89044 \, \mathrm{eV} / \mathrm{molecule}}\] to find that \[λ \approx 657.97 \, \mathrm{nm}\]

The maximum wavelength of sunlight is about 2500 nm. As the energy of a photon is inversely related to the wavelength, this means that photons of sunlight can definitely achieve the energy required for the given reaction to happen because the maximum wavelength required for the reaction (657.97 nm) is within the range of wavelengths present in sunlight.