Question

The stratospheric ozone \(\left(\mathrm{O}_{3}\right)\) layer helps to protect us from harmful ultraviolet radiation. It does so by absorbing ultraviolet light and falling apart into an \(\mathrm{O}_{2}\) molecule and an oxygen atom, a process known as photodissociation. $$\mathrm{O}_{3}(g) \longrightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)$$ Use the data in Appendix \(C\) to calculate the enthalpy change for this reaction. What is the maximum wavelength a photon can have if it is to possess sufficient energy to cause this dissociation? In what portion of the spectrum does this wavelength occur?

Short answer

The enthalpy change for the photodissociation of one mole of \(\mathrm{O}_{3}\) molecules is \(106.9 \:\mathrm{kJ/mol}\). The maximum wavelength a photon can have to possess sufficient energy for this process is \(184.8 \:\mathrm{nm}\).

Step-by-step solution

Calculate the enthalpy change for the reaction.

We need to find the enthalpy change for the reaction using the given enthalpies of formation for the products and reactants. Appendix C data: \(\Delta_\mathrm{f}H^\circ (\mathrm{O}_3) = 142.3 \:\mathrm{kJ/mol}\) \(\Delta_\mathrm{f}H^\circ (\mathrm{O}_2) = 0 \:\mathrm{kJ/mol}\) (since it's the most stable form of this element) \(\Delta_\mathrm{f}H^\circ (\mathrm{O}) = 249.2 \:\mathrm{kJ/mol}\) The formula for calculating enthalpy change is: \[ \Delta H^\circ = \sum \Delta_\mathrm{f}H^\circ (\text{products}) - \sum \Delta_\mathrm{f}H^\circ (\text{reactants}) \] Substituting the given values: \[ \Delta H^\circ = [\Delta_\mathrm{f}H^\circ (\mathrm{O}_2) + \Delta_\mathrm{f}H^\circ (\mathrm{O})] - [\Delta_\mathrm{f}H^\circ (\mathrm{O}_3)] \]

Calculate the enthalpy change value.

Plug in the values and calculate the enthalpy change: \[ \Delta H^\circ = [0 + 249.2] - [142.3] = 106.9 \:\mathrm{kJ/mol} \] The enthalpy change for the reaction is \(106.9 \:\mathrm{kJ/mol}\).

Determine the maximum wavelength using Planck's equation.

Planck's equation relates the energy of a photon to its wavelength: \[ E = \dfrac{hc}{\lambda} \] Here, \(E\) is the energy of the photon, \(h\) represents Planck's constant (\(6.626 \times 10^{-34} \:\mathrm{J\:s}\)), \(c\) is the speed of light (\(2.998 \times 10^8 \:\mathrm{m/s}\)), and \(\lambda\) is the wavelength of light. We need to determine the wavelength (\(\lambda\)) that corresponds to the energy needed for the reaction: \[ \lambda = \dfrac{hc}{E} \] Remember, we want the energy needed for the process in J/mol and substitute the value: \[ \lambda = \dfrac{hc}{106.9 \times 10^3 \:\mathrm{J/mol}} \]

Calculate the maximum wavelength.

By plugging the values and calculating the result: \[ \lambda = \dfrac{(6.626 \times 10^{-34} \:\mathrm{J\:s})(2.998 \times 10^8 \:\mathrm{m/s})}{106.9 \times 10^3 \:\mathrm{J/mol}} = 1.848 \times 10^{-7} \:\mathrm{m} \] To express the wavelength in nanometers, multiply the result by \(10^{9}\): \[ \lambda = 1.848 \times 10^{-7} \:\mathrm{m} \times 10^9 = 184.8 \:\mathrm{nm} \] Thus, the maximum wavelength a photon can have to possess sufficient energy to dissociate one mole of \(\mathrm{O}_{3}\) molecules is \(184.8 \:\mathrm{nm}\).

Key concepts

Stratospheric Ozone Layer

The stratospheric ozone layer acts like a protective shield around Earth. It absorbs harmful ultraviolet (UV) radiation from the sun. Without this layer, life on Earth would be at risk from increased UV exposure.
Ozone in the stratosphere undergoes a process called photodissociation. In this process, ozone molecules (\(\mathrm{O}_3\)) break down into oxygen molecules (\(\mathrm{O}_2\)) and oxygen atoms (\(\mathrm{O}\)).
This absorption of UV radiation helps prevent skin cancer, cataracts, and other health issues in humans. It also protects crops and marine ecosystems, essential for life on Earth.
The stratospheric ozone layer is situated between 10 to 50 kilometers above Earth’s surface. Its critical role could be compromised by pollutants like chlorofluorocarbons (CFCs), hence the importance of environmental conservation efforts.

Enthalpy Change

Enthalpy change, represented by \(\Delta H\), measures the heat change during a chemical reaction at constant pressure. It can indicate whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).
To determine the enthalpy change for the ozone decomposition, assess the energy needed to break ozone into \(\mathrm{O}_2\) and \(\mathrm{O}\). Use the formula:
\[\Delta H^\circ = \sum \Delta_\mathrm{f}H^\circ (\text{products}) - \sum \Delta_\mathrm{f}H^\circ (\text{reactants})\]
In this case, the values for ozone (\(\mathrm{O}_3\)), oxygen molecule (\(\mathrm{O}_2\)), and oxygen atom (\(\mathrm{O}\)) are used.
The enthalpy change for the reaction \(\Delta H^\circ\) is \(106.9 \:\mathrm{kJ/mol}\), meaning the process is endothermic as it requires energy input to break the bonds.

Maximum Wavelength

The maximum wavelength of light sufficient for a chemical reaction refers to the longest wavelength that can provide enough energy to cause the reaction. For photodissociation, it's crucial to know this to understand what kind of light breaks down substances like ozone.
To find out, use the calculated enthalpy change of the reaction to help determine the corresponding photon energy. The formula derived from Planck’s equation is:
\[\lambda = \dfrac{hc}{E}\]\(\lambda\) is the wavelength of the photon, \(h\) is Planck’s constant, \(c\) is the speed of light, and \(E\) is the energy in Joules required to cause the reaction.
In the given exercise, the maximum wavelength calculated was \(184.8 \:\mathrm{nm}\), placing it in the ultraviolet part of the spectrum. This demonstrates which regions of light are capable of causing photodissociation.

Planck's Equation

Planck's equation helps in understanding the relationship between a photon's energy and its wavelength. It is essential for calculating the maximum wavelength that can drive a photochemical reaction like ozone photodissociation.
The formula for Planck's equation is:
\[E = \dfrac{hc}{\lambda}\]
Where \(E\) is energy in Joules, \(h\) (Planck's constant) is \(6.626 \times 10^{-34} \:\mathrm{J\:s}\), \(c\) (speed of light) is \(2.998 \times 10^8 \:\mathrm{m/s}\), and \(\lambda\) is the wavelength in meters.
This equation is crucial for determining not only the energy but also the potential effects of light on compounds, especially in the stratospheric ozone layer. Knowing how to calculate and apply these values helps in studying environmental and chemical processes.