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.

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.

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 \(K\).

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.

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.