Question

Using data from Appendix 4, calculate \(\Delta H^{\circ}, \Delta G^{\circ}\), and \(K\) (at 298 K) for the production of ozone from oxygen: $$3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}_{3}(g)$$ At \(30 \mathrm{~km}\) above the surface of the earth, the temperature is about 230\. \(\mathrm{K}\) and the partial pressure of oxygen is about \(1.0 \times 10^{-3}\) atm. Estimate the partial pressure of ozone in equilibrium with oxygen at \(30 \mathrm{~km}\) above the earth's surface. Is it reasonable to assume that the equilibrium between oxygen and ozone is maintained under these conditions? Explain.

Short answer

In summary, we calculated the values of \(\Delta H^{\circ}\) and \(\Delta G^{\circ}\) at 298 K for the given reaction using the data from Appendix 4. We then found the equilibrium constant, \(K\), at 298 K and subsequently at 230 K using the van 't Hoff equation. The partial pressure of ozone at 30 km altitude was estimated based on the equilibrium condition and the known partial pressure of oxygen. Then we discussed the validity of the equilibrium assumption under the given conditions by considering atmospheric factors and comparing our estimate with experimental data.

Step-by-step solution

Calculating the Enthalpy Change \(\Delta H^{\circ}\) and Gibbs Free Energy Change \(\Delta G^{\circ}\) at 298 K

From Appendix 4, we can find the standard molar enthalpies of formation and molar Gibbs free energies of formation for O2 and O3. Let's denote these values as \(\Delta H^{\circ}_f(\mathrm{O_2})\), \(\Delta H^{\circ}_f(\mathrm{O_3})\), \(\Delta G^{\circ}_f(\mathrm{O_2})\), and \(\Delta G^{\circ}_f(\mathrm{O_3})\). The overall enthalpy change and Gibbs free energy change for the reaction are thus given by: $$\Delta H^{\circ} = \sum \Delta H^{\circ}_f(\text{products}) - \sum \Delta H^{\circ}_f(\text{reactants}) = 2\Delta H^{\circ}_f(\mathrm{O_3}) - 3\Delta H^{\circ}_f(\mathrm{O_2})$$ $$\Delta G^{\circ} = \sum \Delta G^{\circ}_f(\text{products}) - \sum \Delta G^{\circ}_f(\text{reactants}) = 2\Delta G^{\circ}_f(\mathrm{O_3}) - 3\Delta G^{\circ}_f(\mathrm{O_2})$$ With the values from Appendix 4, plug in the numbers and calculate.

Calculating the Equilibrium Constant \(K\) at 298 K

With the \(\Delta G^{\circ}\) value at 298 K, we can now calculate the equilibrium constant using the following equation: $$\Delta G^{\circ}=-RT \ln K$$ Where \(R = 8.314\ \mathrm{J\ mol^{-1}\ K^{-1}}\) is the universal gas constant, and \(T = 298\ \mathrm{K}\) is the temperature. Rearranging the equation and solving for K: $$K = e^{\frac{-\Delta G^{\circ}}{RT}}$$ Plug in the values of \(\Delta G^{\circ}\), \(R\), and \(T\) and calculate \(K\).

Estimating the Partial Pressure of Ozone at 230 K using the Equilibrium Condition

To estimate the partial pressure of ozone at a height of 30 km above the surface of the Earth, we first need to calculate the equilibrium constant at 230 K. According to the Van 't Hoff equation, the relation between the equilibrium constant (\(K\)) and the temperature (\(T\)) is as follows: $$\ln \frac{K_2}{K_1} = \frac{-\Delta H^{\circ}}{R}\left(\frac{T_2-T_1}{T_1T_2}\right)$$ Where \(K_1\) and \(T_1\) are the known equilibrium constant and temperature, and \(K_2\) and \(T_2\) are the unknown equilibrium constant and temperature. In our case, \(K_1 = K\), \(T_1 = 298\ \mathrm{K}\), and \(T_2 = 230\ \mathrm{K}\). Using the above equation, solve for \(K_2\). Now, considering the stoichiometry of the reaction, we can write the expression for \(K_2\): $$K_2 = \frac{(P_{\mathrm{O_3}})^2}{(P_{\mathrm{O_2}})^3}$$ We know that \(P_{\mathrm{O_2}} = 1.0 \times 10^{-3}\ \mathrm{atm}\) at 30 km altitude. We can now solve for the partial pressure of ozone, \(P_{\mathrm{O_3}}\), at this altitude.

Discussing if the Equilibrium Assumption is Reasonable

After calculating the partial pressure of ozone at 30 km altitude, we can determine whether the equilibrium assumption is valid by comparing the partial pressures of oxygen and ozone. It's crucial to consider that the equilibrium condition can be affected by other factors, such as UV radiation and chemical reactions with other trace constituents in the atmosphere. Additionally, we can compare the calculated partial pressure of ozone with available experimental data. If the calculated partial pressure of ozone is reasonable, considering all these factors, the equilibrium assumption can be considered valid under the given conditions.

Key concepts

Enthalpy

Enthalpy is a measure of the total energy of a thermodynamic system, encompassing both internal energy and the energy required to make room for it in its environment. In atmospheric chemical reactions, like the formation of ozone from oxygen, knowing the change in enthalpy (delta H^{\circ}) is essential. It helps assess if the process is endothermic (absorbing heat) or exothermic (releasing heat). For instance, the enthalpy change for the production of ozone is calculated using the formula:\[ \Delta H^{\circ} = \sum \Delta H^{\circ}_f(\text{products}) - \sum \Delta H^{\circ}_f(\text{reactants}) \]This equation sums up the standard molar enthalpies of formation for both reactants and products. By comparing these values, students gain insight into whether the reaction absorbs or releases energy, which is crucial for applications in atmospheric chemistry.

Gibbs Free Energy

Gibbs Free Energy (delta G^{\circ}) is a valuable concept in thermodynamics that indicates the "useful" work obtainable from a thermodynamic process at constant temperature and pressure. It's particularly useful in chemistry to determine whether a reaction can occur spontaneously.To calculate Gibbs Free Energy change for a reaction, the formula used is:\[ \Delta G^{\circ} = \sum \Delta G^{\circ}_f(\text{products}) - \sum \Delta G^{\circ}_f(\text{reactants}) \]For the conversion of oxygen to ozone, understanding delta G^{\circ} helps predict if the process occurs naturally under certain conditions. A negative delta G^{\circ} indicates a spontaneous process, while a positive value suggests non-spontaneity. In atmospheric chemistry, considering Gibbs Free Energy is vital to understanding the behavior and transformation of atmospheric gases.

Equilibrium Constant

The Equilibrium Constant (K) exemplifies the state of balance in a chemical reaction. It quantifies the concentrations of products and reactants at equilibrium. For the transformation of oxygen into ozone, the equilibrium constant is calculated using Gibbs Free Energy through:\[ K = e^{\frac{-\Delta G^{\circ}}{RT}} \]Where R is the universal gas constant and T is temperature. This equation shows the relationship between the energy aspects of a reaction and its yields at equilibrium. A higher value of K implies a larger concentration of products compared to reactants. Students often use K to predict the direction and extent of chemical reactions, facilitating a deeper understanding of how atmospheric reactions balance and change.

Partial Pressure

Partial Pressure is a key concept when discussing gas reactions, particularly in applications such as atmospheric chemistry. It refers to the pressure that an individual gas component exerts in a mixture of gases.In the ozone formation reaction, the partial pressures of oxygen (P_{\mathrm{O_2}}) and ozone (P_{\mathrm{O_3}}) at different altitudes help determine the equilibrium state. The equilibrium constant expression for the reaction at a different temperature includes partial pressures:\[ K_2 = \frac{(P_{\mathrm{O_3}})^2}{(P_{\mathrm{O_2}})^3} \]Knowing the partial pressures, along with the equilibrium constant, allows scientists to estimate the distribution of different gases in the atmosphere. This knowledge is indispensable in understanding how reactions occur in the upper layers of the Earth's atmosphere.

Atmospheric Chemistry

Atmospheric Chemistry is the study of chemical processes in the Earth's atmosphere, focusing on the composition, transformations, and effects of chemical compounds. Understanding atmospheric chemistry, especially regarding the ozone layer, is crucial in gauging our planet's environmental health. The formation and destruction of ozone protect life on Earth by absorbing harmful UV radiation. Chemical reactions at high altitudes, like those producing ozone from oxygen, play significant roles. Factors affecting these reactions include:

• Temperature changes
• Pressure variations
• Presence of other chemicals or pollutants

Such knowledge assists scientists in predicting atmospheric reactions and developing strategies to address issues like ozone depletion and climate change. It underlines the importance of chemical equilibriums and energy considerations in environmental science.