Q9.91CP - Chemistry: Molecular Nature Of Matter And Change

Question

In a future hydrogen-fuel economy, the cheapest source of ${\mathbf{H}}_2$will certainly be 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? 

Step-by-Step Solution

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 1: 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 .$\mathbf{7}\mathbf{.}\mathbf{755}\mathbf{\times }{\mathbf{10}}^{-19}\mathbf{J}$

 

Step 2: 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, and  is called frequency.

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

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