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Chapter 19: Problem 23

From the data given in the following table, determine \(\Delta S^{\circ} \quad\) for the reaction \(\quad \mathrm{NH}_{3}(\mathrm{g})+\mathrm{HCl}(\mathrm{g}) \longrightarrow\) \(\mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) .\) All data are at \(298 \mathrm{K}\) $$\begin{array}{lcc} \hline & \Delta H_{f}^{\circ}, \mathrm{kJ} \mathrm{mol}^{-1} & \Delta G_{f,}^{\circ} \mathrm{kJ} \mathrm{mol}^{-1} \\ \hline \mathrm{NH}_{3}(\mathrm{g}) & -46.11 & -16.48 \\ \mathrm{HCl}(\mathrm{g}) & -92.31 & -95.30 \\ \mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) & -314.4 & -202.9 \\ \hline \end{array}$$

Short Answer

Expert verified
\(\Delta S^{\circ} = -0.284 \ K^{-1} mol^{-1}\)

Step by step solution

01

Identify the relevant thermodynamic relation

The relationship between Gibbs Free Energy (\( \Delta G \)), Enthalpy (\( \Delta H \)), Temperature (T) and Entropy (\(\Delta S\)) is given by the equation: \[ \Delta G = \Delta H -T \Delta S \]Considering we are dealing with standard conditions (298 K), this can be adapted to:\[ \Delta G^{\circ} = \Delta H^{\circ} -T \Delta S^{\circ} \]To find \(\Delta S^{\circ}\), we rearrange the equation like so:\[ \Delta S^{\circ} = (\Delta H^{\circ} - \Delta G^{\circ}) / T \]
02

Calculate the change in enthalpy and Gibbs free energy of the reaction

Next, calculate the enthalpy (\(\Delta H_f^{\circ}\)) and Gibbs free energy (\(\Delta G_f^{\circ}\)) for the reaction. This can be done using the provided tables and the formulas:\[\Delta H^{\circ} = \sum \Delta H_f^{\circ} (products) - \sum \Delta H_f^{\circ} (reactants)\]\[\Delta G^{\circ} = \sum \Delta G_f^{\circ} (products) - \sum \Delta G_f^{\circ} (reactants)\]For our reaction:\[\Delta H^{\circ}= \Delta H_f^{\circ}(NH_{4}Cl) -(\Delta H_f^{\circ}(NH_{3}) + \Delta H_f^{\circ}(HCl)) = -314.4-(-46.11-92.31) = -175.98 \,kJ/mol \]\[\Delta G^{\circ}= \Delta G_f^{\circ}(NH_{4}Cl)- (\Delta G_f^{\circ}(NH_{3}) + \Delta G_f^{\circ}(HCl)) = -202.9 -(-16.48-95.3) = -91.12 \, kJ/mol \]
03

Determine the standard entropy change

Finally, calculate the standard entropy change (\(\Delta S^{\circ}\)) using the formula from Step 1:\[\Delta S^{\circ} = (\Delta H^{\circ} - \Delta G^{\circ}) / T = (-175.98 - (-91.12)) / 298 = -0.284 \ K^{-1} mol^{-1} \]Notice that the units of \(\Delta S^{\circ}\) are K^{-1}mol^{-1}, which are the proper units for entropy

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Most popular questions from this chapter

Briefly describe each of the following ideas, methods, or phenomena: (a) absolute molar entropy; (b) coupled reactions; (c) Trouton's rule; (d) evaluation of an equilibrium constant from tabulated thermodynamic data.

For the reaction \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}_{2}(\mathrm{g})\) all but one of the following equations is correct. Which is incorrect, and why? (a) \(K=K_{\mathrm{p}} ;\) (b) \(\Delta S^{\circ}=\) \(\left(\Delta G^{\circ}-\Delta H^{\circ}\right) / T ;\left(\text { c) } K_{\mathrm{p}}=e^{-\Delta G^{\circ} / R T} ;(\mathrm{d}) \Delta G=\Delta G^{\circ}+\right.\) \(R T \ln Q\).

Arrange the entropy changes of the following processes, all at \(25^{\circ} \mathrm{C},\) in the expected order of increasing \(\Delta S,\) and explain your reasoning: (a) \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{atm}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 1 \mathrm{atm})\) (b) \(\mathrm{CO}_{2}(\mathrm{s}, 1 \mathrm{atm}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g}, 10 \mathrm{mm} \mathrm{Hg})\) (c) \(\mathrm{H}_{2} \mathrm{O}(1,1 \mathrm{atm}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}, 10 \mathrm{mmHg})\)

At \(298 \mathrm{K}, \Delta G_{\mathrm{f}}^{\mathrm{p}}[\mathrm{CO}(\mathrm{g})]=-137.2 \mathrm{kJ} / \mathrm{mol}\) and \(K_{\mathrm{p}}=\) \(6.5 \times 10^{11}\) for the reaction \(\mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons\) \(\mathrm{COCl}_{2}(\mathrm{g}) . \quad\) Use these data to determine \(\Delta G_{f}\left[\mathrm{COCl}_{2}(\mathrm{g})\right],\) and compare your result with the value in Appendix D.

Use thermodynamic data from Appendix D to calculate values of \(K_{\mathrm{sp}}\) for the following sparingly soluble solutes: (a) \(\operatorname{AgBr} ;\) (b) \(\operatorname{CaSO}_{4} ;\) (c) \(\operatorname{Fe}(\text { OH })_{3}\). [Hint: Begin by writing solubility equilibrium expressions.
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