Question

In a future hydrogen-fuel economy, the cheapest sourceof ${\mathbf{H}}_2$willcertainlybe water. It takes467kJ to produce one mol of H atoms from water. What is the frequency, wavelength and minimum energy of a photon that can free an H atom from water?

Short answer

$\mathbf{1}\mathbf{.}\mathbf{17037}\mathbf{\times }{\mathbf{10}}^{15}\mathbf{Hz}$frequency,$\mathbf{256}\mathbf{.}\mathbf{3}\mathbf{nm}$wavelength and $\mathbf{7}\mathbf{.}\mathbf{755}\mathbf{\times }{\mathbf{10}}^{-19}\mathbf{J}$ minimum energy of a photon that can free an H atom from water.

Step-by-step solution

calculation ofminimum energy

One mole of any substance is equal to the Avogadro number.

To calculate the minimum energy of a photon that can free an H atom from water, the heat produced in the formation of H atoms from water is divided by the Avogadro number.

$\begin{array}{l}\mathbf{E}\mathbf{=}\frac{\mathrm{\Delta H}}{{\mathbf{N}}_A}\mathbf{=}\frac{467\mathrm{kJ}}{\mathbf{6}\mathbf{.}\mathbf{022}\mathbf{\times }{\mathbf{10}}^{23}\mathbf{mol}}\\ \mathbf{=}\frac{46700J}{\mathbf{6}\mathbf{.}\mathbf{022}\mathbf{\times }{\mathbf{10}}^{23}\mathbf{mol}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{755}\mathbf{\times }{\mathbf{10}}^{-19}\mathbf{J}\end{array}$

So, the minimum energy of a photon that is required to free an H atom from water is .$\mathbf{7}\mathbf{.}\mathbf{755}\mathbf{\times }{\mathbf{10}}^{-19}\mathbf{J}$

calculation of frequency

To calculate the frequency of a photon we use the Einstein-plank relation-

$\mathbf{E}\mathbf{=}\mathbf{h\upsilon }$

Where E is calledthe energy of the photon, h is calledplank’s constant, andiscalledfrequency.

$\mathbf{\upsilon }\mathbf{=}\frac{E}{h}$

The value of Plank’s constant is $\mathbf{6}\mathbf{.}\mathbf{626}\mathbf{\times }{\mathbf{10}}^{-34}\mathbf{J}\mathbf{.}\mathbf{s}$