Question

Light of appropriate wavelength can break chemical bonds. Light having \(\lambda<240 \mathrm{nm}\) can dissociate gaseous \(\mathrm{O}_{2}\). It requires light with \(\lambda<819 \mathrm{nm}\) to dissociate gaseous \(\mathrm{H}_{2} \mathrm{O}_{2}\) to \(2 \mathrm{OH}\). Assume that all of the photon energy is used solely for these dissociations. (a) Calculate the energy required to dissociate (i) \(\mathrm{O}_{2}\) and (ii) \(\mathrm{H}_{2} \mathrm{O}_{2}\). (b) Consider the results of part (a). How well do they correlate with the Lewis structures of \(\mathrm{O}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}_{2}\) ? Explain your answer.

Short answer

(i) 8.30 x 10^(-19) J, (ii) 2.43 x 10^(-19) J. Lewis structures support the energy differences.

Step-by-step solution

Calculate the Energy for O2

Using the equation for energy from wavelength, \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\) and \(c\) is the speed of light \(3.00 \times 10^{8} \text{ m/s}\). Convert the wavelength from nm to m: \(240 \text{ nm} = 240 \times 10^{-9} \text{ m}\). Substituting the values to calculate \(E\) for \(\mathrm{O}_{2}\):\[E = \frac{(6.626 \times 10^{-34} \text{ J s})(3.00 \times 10^{8} \text{ m/s})}{240 \times 10^{-9} \text{ m}} = 8.30 \times 10^{-19} \text{ J}\]

Calculate the Energy for H2O2

Convert the wavelength from nm to m for \(\mathrm{H}_{2}\mathrm{O}_{2}\): \(819 \text{ nm} = 819 \times 10^{-9} \text{ m}\). Again using the energy formula, substitute the values to find \(E\) for \(\mathrm{H}_{2}\mathrm{O}_{2}\):\[E = \frac{(6.626 \times 10^{-34} \text{ J s})(3.00 \times 10^{8} \text{ m/s})}{819 \times 10^{-9} \text{ m}} = 2.43 \times 10^{-19} \text{ J}\]

Analyze Lewis Structures Correlation

The Lewis structure for \(\mathrm{O}_{2}\) shows a double bond, which requires more energy to dissociate than a single bond. This aligns with the higher energy calculated for \(\mathrm{O}_{2}\). \(\mathrm{H}_{2} \mathrm{O}_{2}\) has weaker O-O and O-H single bonds, which correlates with the lower energy calculated for \(\mathrm{H}_{2} \mathrm{O}_{2}\). Therefore, the results correlate well with the bonding information from the Lewis structures.

Key concepts

Photon Energy

When it comes to breaking chemical bonds, photons play a critical role. Photons are particles of light that carry energy. Their energy is determined by the frequency of the light they represent. The relationship between photon energy and light frequency is given by the formula: \[ E = h \cdot f \]where \( E \) is the photon energy, \( h \) is Planck’s constant \((6.626 \times 10^{-34} \text{ J s})\), and \( f \) is the frequency. If we only know the wavelength, we can convert it to frequency using the speed of light \(c\) with the formula \(c = \lambda \cdot f\). This way, we can calculate the energy of photons, enabling us to understand how they can initiate chemical reactions, such as breaking bonds.
Light of lesser wavelengths, such as those less than 240 nm for \(\mathrm{O}_{2}\) or 819 nm for \(\mathrm{H}_{2} \mathrm{O}_{2}\), carries higher energy. The energy is used to overcome the binding energy of these molecules' chemical bonds, leading them to dissociate. Grasping photon energy helps in understanding not only light-based reactions but also broader areas in chemistry and physics where energy transformation is key.

Wavelength Calculation

Calculating the energy needed to dissociate molecules involves converting the given wavelength into energy. The wavelength \((\lambda)\) is inversely proportional to the energy. In simpler terms, shorter the wavelength, higher the photon energy.
The energy \(E\) required is given by the formula:\[ E = \frac{hc}{\lambda} \]where \(h\) is Planck's constant \((6.626 \times 10^{-34} \text{ J s})\) and \(c\) is the speed of light \((3.00 \times 10^{8} \text{ m/s})\).
To find the energy for wavelengths of 240 nm for \(\mathrm{O}_{2}\) and 819 nm for \(\mathrm{H}_{2} \mathrm{O}_{2}\), convert the wavelength from nanometers to meters (1 nm = \(10^{-9}\) m):

• For \(\mathrm{O}_{2}\), \(\lambda = 240 \times 10^{-9} \text{ m}\)
• For \(\mathrm{H}_{2}\mathrm{O}_{2}\), \(\lambda = 819 \times 10^{-9} \text{ m}\)

Plug these values into the formula to determine the photon energies required to break these bonds. Accurate wavelength calculation is crucial in spectroscopic techniques, playing a pivotal role in identifying substances chemically and physically.

Lewis Structures Correlation

Understanding Lewis structures is essential to explaining the energy differences observed in bond dissociation. Lewis structures give a visual representation of the bonds between atoms within a molecule, also indicating lone pairs of electrons.
For example, the Lewis structure of \(\mathrm{O}_{2}\) is represented with a double bond between the oxygen atoms. Double bonds are stronger than single bonds because they share more electrons, requiring more energy to break them.
In contrast, the Lewis structure for \(\mathrm{H}_{2} \mathrm{O}_{2}\) shows both O-O and O-H bonds, which are single bonds. These bonds are relatively weaker compared to the double bond in \(\mathrm{O}_{2}\), correlating well with the lower photon energy needed to dissociate \(\mathrm{H}_{2} \mathrm{O}_{2}\).
Therefore, the energy requirements calculated for dissociating \(\mathrm{O}_{2}\) and \(\mathrm{H}_{2}\mathrm{O}_{2}\) align with their Lewis structures, which is a perfect example of how chemical bonding theories help predict the behavior of molecules in the presence of energy sources like light. Using Lewis structures in conjunction with experimental data provides a clearer picture of chemical processes, facilitating better understanding and communication among scientists.