Question

Ozone in the troposphere is formed by the following steps: $$ \begin{aligned} \mathrm{NO}_{2} & \longrightarrow \mathrm{NO}+\mathrm{O} \\ \mathrm{O}+\mathrm{O}_{2} \longrightarrow & \mathrm{O}_{3} \end{aligned} $$ The first step is initiated by the absorption of visible light \(\left(\mathrm{NO}_{2}\right.\) is a brown gas). Calculate the longest wavelength required for step 1 at \(25^{\circ} \mathrm{C}\). (Hint: You need to first calculate \(\Delta H\) and hence \(\Delta E\) for step \(1 .\) Next, determine the wavelength for decomposing \(\mathrm{NO}_{2}\) from \(\Delta E\).)

Short answer

The longest wavelength required for step 1 at 25°C is approximately 394 nm.

Step-by-step solution

Understand the Reaction

The reaction provided is the decomposition of NO2 into NO and O. This decomposition is initiated by absorbing a photon of light, which provides the energy needed for the reaction to proceed.

Calculate Enthalpy Change (ΔH)

To calculate ΔH, we use bond enthalpies. The reaction involves breaking the N-O bond in NO2 and forming an N-O bond in NO along with the formation of a new O atom. The bond enthalpy for N=O (in NO2) is approximately 305 kJ/mol. The bond enthalpy for N=O in NO is approximately 607 kJ/mol. Since an O atom is formed from O2 without bond formation, we only consider the bond broken to form single O atoms. The enthalpy change, ΔH, is roughly: \[ΔH = [ ext{Bonds broken}] - [ ext{Bonds formed}] = (305 ext{ kJ/mol}) - (607 ext{ kJ/mol})\] which simplifies to -302 kJ/mol.However, since only the bond breaking matters here for NO2, we focus on the breaking of the bond in NO2 alone for photon absorption consideration.

Calculate ΔE from ΔH

Since the reaction is at constant pressure, ΔH is approximately equal to ΔE (ΔE ≈ ΔH) because there is no non-PV work done. Thus, ΔE ≈ -305 kJ/mol. We take the magnitude for photon calculation, so use 305 kJ/mol for calculating energy per mole of photons.

Convert Energy to Per Photon Basis

To find energy per photon, use Avogadro's number (6.022 x 10^23/mol). Convert kJ/mol to Joules per photon: \[ ext{Energy per photon (J/photon)} = \frac{305,000 ext{ J/mol}}{6.022 imes 10^{23} ext{ photons/mol}} \] This gives approximately \(5.06 \times 10^{-19}\) J/photon.

Calculate the Longest Wavelength (λ)

Use the energy-wavelength relationship, \(E = \frac{hc}{λ}\), to find the longest wavelength. Here, \(h\) (Planck's constant) is \(6.626 \times 10^{-34} \, \text{Js}\) and \(c\) (speed of light) is \(3.00 \times 10^8 \, \text{m/s}\). Rearranging gives:\[\lambda = \frac{hc}{E} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{5.06 \times 10^{-19}}\]Calculating gives \(λ ≈ 3.94 \times 10^{-7} \, \text{m}\) or approximately 394 nm.

Key concepts

Troposphere Chemistry

The troposphere is the lowest layer of Earth's atmosphere, where we live and breathe. It hosts various chemical reactions crucial for air quality and climate. One essential process in this layer is the formation of ozone (\(\text{O}_3\)).
The formation of ozone involves chemical reactions that play a significant role in atmospheric chemistry. In the troposphere, nitrogen dioxide (\(\text{NO}_2\)) decomposes into nitrogen oxide (\(\text{NO}\)) and an oxygen atom. This oxygen atom can then react with diatomic oxygen (\(\text{O}_2\)) to form ozone.
Understanding this process is key to studying the effects of pollution and its regulation, as tropospheric ozone is a significant air pollutant impacting human health and ecosystems.

Reaction Mechanism

The mechanism of ozone formation from \(\text{NO}_2\) involves a series of steps initiated by the absorption of light. This light absorption provides the energy needed to activate \(\text{NO}_2\), resulting in its decomposition.

• First Step: \(\text{NO}_2\) absorbs a photon, breaking down into \(\text{NO}\) and an oxygen atom. This step is essential because it releases the reactive oxygen atom necessary for ozone formation.
• Second Step: The free oxygen atom reacts with \(\text{O}_2\) to produce ozone (\(\text{O}_3\)). This second step completes the transformation into ozone.

This reaction mechanism is vital, as it highlights how external energy sources, like sunlight, initiate significant chemical processes in the troposphere.

Photon Absorption

Photon absorption is the process in which a molecule absorbs a light particle, or photon, providing the energy needed for a chemical reaction. In the case of \(\text{NO}_2\), light absorption is crucial for its decomposition.
Light energy is measured in wavelengths, ranging across the electromagnetic spectrum. Different compounds absorb varying wavelengths based on their structure.
For \(\text{NO}_2\), absorbing a photon of the correct wavelength splits it into \(\text{NO}\) and an oxygen atom. This process demonstrates how molecules can undergo significant changes simply by absorbing energy from light, crucially affecting atmospheric chemistry.

Wavelength Calculation

Calculating the wavelength of light necessary to decompose \(\text{NO}_2\) involves understanding the energy dynamics of photon interactions with molecules. This energy is derived from Planck's equation:\[ E = \frac{hc}{\lambda} \]where \(E\) is the energy of the photon, \(h\) is Planck’s constant \(6.626 \times 10^{-34}\,\text{J}\cdot \text{s}\), \(c\) is the speed of light \(3.00 \times 10^8\,\text{m/s}\), and \(\lambda\) is the wavelength.

By calculating the energy required to break the \(\text{NO}_2\) bond, we can use the energy equation to find the longest wavelength of light that can provide this energy. For this process, the energy per photon is calculated by dividing the energy change per mole (from bond energies) by Avogadro's number.
Finally, plugging these values into the equation yields the wavelength, indicating which portion of the light spectrum can drive this specific reaction. This understanding is essential for predicting and monitoring reactions driven by sunlight in the troposphere.