Consider the following reaction: $$ \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{Cl}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{HOCl}(g) \quad K_{298}=0.090 $$ For \(\mathrm{Cl}_{2} \mathrm{O}(g)\), $$ \begin{aligned} \Delta G_{\mathrm{f}}^{\circ} &=97.9 \mathrm{~kJ} / \mathrm{mol} \\ \Delta H_{\mathrm{f}}^{\circ} &=80.3 \mathrm{~kJ} / \mathrm{mol} \\ S^{\circ} &=266.1 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol} \end{aligned} $$ a. Calculate \(\Delta G^{\circ}\) for the reaction using the equation \(\Delta G^{\circ}=-R T \ln (K)\) b. Use bond energy values (Table 8.4) to estimate \(\Delta H^{\circ}\) for the reaction. c. Use the results from parts a and \(\mathrm{b}\) to estimate \(\Delta S^{\circ}\) for the reaction. d. Estimate \(\Delta H_{\mathrm{f}}^{\circ}\) and \(S^{\circ}\) for \(\mathrm{HOCl}(g)\). e. Estimate the value of \(K\) at \(500 . \mathrm{K}\). f. Calculate \(\Delta G\) at \(25^{\circ} \mathrm{C}\) when \(P_{\mathrm{H}_{2} \mathrm{O}}=18\) torr, \(P_{\mathrm{Cl}_{2} \mathrm{O}}=\) \(2.0\) torr, and \(P_{\mathrm{HOCl}}=0.10\) torr.

Step by step solution

01

Calculate ΔG°

First, we need to calculate the change in Gibbs free energy (ΔG°) for the reaction using the equation ΔG°=-RT ln(K), where R is the gas constant, T is the temperature in Kelvin, and K is the equilibrium constant. Given K=0.090 at T=298 K, we will use the following formula to calculate ΔG°: $$ \Delta G^{\circ}=-R T \ln (K) $$ Use the gas constant value, R=8.314 J/mol K, to get numerical values for the variables. $$ \Delta G^{\circ}=-8.314\times 298 \times \ln(0.090) $$ After performing the calculations, find the value of ΔG°. #b. Use bond energy values (Table 8.4) to estimate ΔH° for the reaction.#

02

Estimate ΔH° Using Bond Energies

To estimate the change in enthalpy (ΔH°) for the reaction, we will use bond energy values from Table 8.4 (not provided here). We will subtract the bond energies of the reactants from the bond energies of the products. Compare the bond energies to find ΔH° for the reaction. #c. Use the results from parts a and b to estimate ΔS° for the reaction.#

03

Calculate ΔS° for the Reaction

Now, we need to calculate the change in entropy (ΔS°) for the reaction using the values we found in parts a and b. We can find ΔS° using the following equation: $$ \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} $$ Rearrange the formula to isolate ΔS°: $$ \Delta S^{\circ} = \frac{\Delta H^{\circ} - \Delta G^{\circ}}{T} $$ Plug in the values we found in parts a and b, and the given temperature (298 K) to calculate the value of ΔS°. #d. Estimate ΔHf° and S° for HOCl(g).#

04

Estimate ΔHf° and S° for HOCl(g)

To estimate the standard enthalpy of formation (ΔHf°) and the standard entropy (S°) for gaseous hypochlorous acid, we will use the values we obtained in parts a, b, and c, as well as the provided thermodynamic properties for Cl2O(g). The balanced reaction equation will aid in determining the stoichiometric coefficients for each species. Once we have the stoichiometric coefficients, we can use the following equations: $$ \Delta H_{\text{rxn}}^{\circ} = \sum n_i \Delta H_{\text{f},i}^{\circ} $$ $$ \Delta S_{\text{rxn}}^{\circ} = \sum n_i S_{i}^{\circ} $$ where \(n_i\) are the stoichiometric coefficients, and \(\Delta H_{\text{f},i}^{\circ}\) and \(S_{i}^{\circ}\) are the standard enthalpy of formation and standard entropy for each species i, respectively. Based on the stoichiometric coefficients and the given data for Cl2O(g), find the values of ΔHf° and S° for HOCl(g). #e. Estimate the value of K at 500 K.#

05

Estimate the Value of K at 500 K

To estimate the value of the equilibrium constant (K) at a new temperature (500 K), we can use the van't Hoff equation: $$ \ln\frac{K_2}{K_1} = \frac{-\Delta H^\circ}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) $$ Here, \(K_1\) and \(K_2\) are the equilibrium constants at temperatures \(T_1\) and \(T_2\), respectively. We are given \(K_1 = 0.090\) at \(T_1 = 298\) K, and we are trying to find \(K_2\) at \(T_2 = 500\) K. Use the calculated value of \(\Delta H^\circ\) from part b and the gas constant R to solve the van't Hoff equation for \(K_2\). #f. Calculate ΔG at 25°C when PH2O=18 torr, PCl2O=2.0 torr, and PHOCl=0.10 torr.#

06

Calculate ΔG at 25°C

Finally, we will calculate the change in Gibbs free energy (ΔG) at 25°C (298 K) when the partial pressures of H2O, Cl2O, and HOCl are given as 18 torr, 2.0 torr, and 0.10 torr, respectively. To do this, we can use the following equation: $$ \Delta G = \Delta G^\circ + RT\ln Q $$ Here, Q is the reaction quotient, which can be found using the given partial pressures: $$ Q = \frac{(P_{\text{HOCl}})^2}{P_{\text{H2O}} P_{\text{Cl2O}}} $$ Calculate Q using the given partial pressures, and then use the value of ΔG° calculated in part a, the gas constant R, and the temperature T=298 K to find ΔG.

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