Question

(a) Draw the Lewis structure for hydrogen peroxide, \(\mathrm{H}_{2} \mathrm{O}_{2}\). (b) What is the weakest bond in hydrogen peroxide? (c) Hydrogen peroxide is sold commercially as an aqueous solution in brown bottles to protect it from light. Calculate the longest wavelength of light that has sufficient energy to break the weakest bond in hydrogen peroxide.

Short answer

The Lewis structure for H2O2 is: ``` H O O H \ || || / ``` The weakest bond in hydrogen peroxide is the O-O bond with a bond dissociation enthalpy (BDE) of 207 kJ/mol. The longest wavelength of light capable of breaking the weakest bond in hydrogen peroxide is approximately \(5.73 * 10^{-6}~\text{m}\) or 5730 nm.

Step-by-step solution

Draw the Lewis structure for hydrogen peroxide

To draw the Lewis structure for H2O2, first count the total number of valence electrons. Oxygen has 6 valence electrons and hydrogen has 1 valence electron. Since there are 2 oxygen atoms and 2 hydrogen atoms, the total number of valence electrons is 2(6) + 2(1) = 14. Now, arrange the electrons around the oxygen and hydrogen atoms to satisfy the octet rule. The oxygen atom forms a single bond with one hydrogen atom and another oxygen atom. The remaining oxygen atom forms a single bond with the other hydrogen atom. The Lewis structure for H2O2 is: ``` H O O H \ || || / ```

Determine the weakest bond in hydrogen peroxide

To determine the weakest bond, we need to consider the bond dissociation enthalpy (BDE) for each bond. Bonds with a higher BDE are stronger and require more energy to break. The bond dissociation enthalpies for the O-H and O-O bonds in hydrogen peroxide are: - O-H: 467 kJ/mol - O-O: 207 kJ/mol From these values, we can conclude that the O-O bond is the weakest bond in hydrogen peroxide.

Calculate the longest wavelength of light that has sufficient energy to break the weakest bond

Having identified the weakest bond with a BDE of 207 kJ/mol, we can calculate the energy required to break the bond. To convert this value to J/mol, use the conversion factor: \(207~\text{kJ/mol} * \left(\frac{1000~\text{J}}{1~\text{kJ}}\right) = 207000~\text{J/mol}\) Next, use the Planck's equation to determine the energy of a photon: \(E = h\nu\), where \(E\) is the energy of a photon, \(h\) is Planck's constant (\(6.626 * 10^{-34}~\text{J} \cdot \text{s}\)), and \(\nu\) is the frequency of the light. We can also relate the frequency to the wavelength using the speed of light equation, \(c = \lambda\nu\), where \(c\) is the speed of light (\(3 * 10^8~\text{m/s}\)) and \(\lambda\) is the wavelength. Substituting the wavelength equation into the energy equation, we get: \(E = \cfrac{hc}{\lambda}\) We want to calculate the longest wavelength (\(\lambda\)) of light capable of breaking the weakest bond with an energy of 207000 J/mol. Rearrange the equation and solve for the wavelength: \(\lambda = \cfrac{hc}{E}\) Plugging in the values, we have: \(\lambda = \cfrac{(6.626 * 10^{-34}~\text{J} \cdot \text{s}) (3 * 10^8~\text{m/s})}{207000~\text{J/mol}}\) Now, we should convert the energy from J/mol to J/photon to have the same units. To do this, we utilize Avogadro's number which is \(6.022 * 10^{23}~\text{mol}^{-1}\): \(\cfrac{207000~\text{J/mol}}{6.022 * 10^{23}~\text{mol}^{-1}} = 3.45 * 10^{-19}~\text{J/photon}\) Finally, we can solve for the wavelength: \(\lambda = \cfrac{(6.626 * 10^{-34}~\text{J} \cdot \text{s}) (3 * 10^8~\text{m/s})}{3.45 * 10^{-19}~\text{J/photon}} = 5.73 * 10^{-6}~\text{m}\) The longest wavelength of light that has sufficient energy to break the weakest bond in hydrogen peroxide is approximately \(5.73 * 10^{-6}~\text{m}\) or 5730 nm.

Key concepts

Valence Electrons

Valence electrons refer to the outermost electrons of an atom that are involved in forming chemical bonds. They play a crucial role in determining an element's chemical properties and reactivity. In the context of Lewis structures, knowing the number of valence electrons can help you predict how atoms will connect within a molecule.

To find the number of valence electrons, simply refer to the group number of the element in the periodic table for main-group elements. For example:

• Hydrogen, which is in Group 1, has 1 valence electron.
• Oxygen, in Group 16, has 6 valence electrons.

In hydrogen peroxide ( H₂O₂ ), each hydrogen atom contributes 1 valence electron, and each oxygen atom contributes 6 valence electrons. In total, the molecule has 14 valence electrons to work into its Lewis structure. Understanding this helps in arranging the atoms correctly to satisfy each atom's valency, ensuring the molecule's stability.

Bond Dissociation Enthalpy

Bond dissociation enthalpy (BDE) is the energy required to break a bond between two atoms in a molecule, measured in kilojoules per mole (kJ/mol). It is a critical concept when analyzing chemical reactions since it gives insight into the stability and strength of different bonds.

In hydrogen peroxide, there are O-O and O-H bonds. Each type of bond has associated energy values which reflect how much energy is needed to break them:

• The O-H bond has a BDE of 467 kJ/mol.
• The O-O bond has a BDE of 207 kJ/mol.

These values indicate that the O-O bond is weaker in comparison to the O-H bond since it requires less energy to break. This information is crucial when predicting chemical behaviors or transformations such as decomposition or reactions when energy supply is limited. Knowing the weakest bond helps in calculating the energy requirements to induce a chemical change.

Planck's Equation

Planck's equation is a formula used to relate the energy of photons to their frequency. It is expressed as:\[E = h u\]where:

• \(E\) is the energy of the photon,
• \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{J} \cdot \text{s}\)),
• \(u\) is the frequency of the light.

This equation is foundational in quantum mechanics and essential for calculations involving spectral lines or energy transitions of photons in physics and chemistry.

Planck's equation helps derive the energy required from light to break specific chemical bonds. When evaluating hydrogen peroxide, one leverages this equation to determine the frequencies of light that possess enough energy to sever its weakest bonds. This enables the calculation of the longest wavelengths capable of performing such chemical transformations.

Wavelength Calculation

Wavelength calculation is a step that connects the energy of a photon to its frequency and subsequently to its wavelength using the equation connecting Planck's equation to the speed of light:\[E = \frac{hc}{\lambda}\]where:

• \(E\) is the photon's energy,
• \(h\) is Planck’s constant,
• \(c\) is the speed of light (\(3 \times 10^8 \text{m/s}\)),
• \(\lambda\) is the wavelength of the light.

To solve for wavelength, rearrange the formula:\[\lambda = \frac{hc}{E}\] This calculation allows us to identify the longest wavelength of light that can break a specific bond in hydrogen peroxide by using the energy calculated from bond dissociation enthalpy.

Assuming we convert the energy requirement from J/mol to J/photon, the equation simplifies the determination of wavelengths in practical applications. The outcome is significant for understanding light's impact on chemical substances, such as decomposing hydrogen peroxide under photochemical reactions.