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Chapter 19: Problem 78

Consider the reaction 2 \(\mathrm{NO}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{O}_{4}(g) .(\mathbf{a})\) Using data from Appendix \(\mathrm{C},\) calculate \(\Delta G^{\circ}\) at 298 \(\mathrm{K}\) . (b) Calculate \(\Delta G\) at 298 \(\mathrm{K}\) if the partial pressures of \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) are 0.40 atm and 1.60 atm, respectively.

Short Answer

Expert verified
The standard Gibbs free energy change (ΔG°) for the reaction 2 NO₂(g) → N₂O₄(g) at 298 K is -4.71 kJ/mol. When the partial pressures of NO₂ and N₂O₄ are 0.40 atm and 1.60 atm, respectively, the Gibbs free energy change (ΔG) at 298 K is 2.61 kJ/mol.

Step by step solution

01

Calculation of Standard Gibbs Free Energy Change (ΔG°)

We'll use the provided standard Gibbs free energies of formation (ΔGf°) from Appendix C to calculate ΔG° for the reaction. For NO₂(g): \( \Delta G_{f}^{\circ} = 51.30 \thinspace \dfrac{kJ}{mol} \) For N₂O₄(g): \( \Delta G_{f}^{\circ} = 97.89 \thinspace \dfrac{kJ}{mol} \) Using the formula for ΔG°: \( \Delta G^{\circ} = \Sigma n_i \Delta G_f^{\circ}(products) - \Sigma n_j \Delta G_f^{\circ}(reactants) \) \( \Delta G^{\circ} = [\Delta G_f^{\circ}(N_{2}O_{4})] - 2[\Delta G_f^{\circ}(NO_{2})] \) Calculating ΔG° using the given values: \( \Delta G^{\circ} = 97.89 - 2 \times 51.30 = -4.71 \thinspace kJ/mol \)
02

Calculation of Reaction Quotient (Q)

In order to calculate ΔG, we first need to find the reaction quotient Q, which is given by: \( Q = \dfrac{P_{(N_{2}O_{4})}}{P_{(NO_{2})}^2} \) Substituting the given partial pressures of NO₂ (0.40 atm) and N₂O₄ (1.60 atm) into the equation: \( Q = \dfrac{1.60}{(0.40)^2} = 10 \)
03

Calculation of Gibbs Free Energy Change (ΔG)

Now that we have Q and ΔG°, we can use the following equation to calculate ΔG: \( \Delta G = \Delta G^{\circ} + RT \ln Q \) Where: R = 8.314 J/(mol·K) T = 298 K Substitute the calculated values: \( \Delta G = (-4.71 \times 10^3 J/mol) + (8.314 J/(mol·K)) \times 298 K \times \ln (10) \) Calculating ΔG: \( \Delta G = -4.71 \times 10^{3} + 8.314 \times 298 \times 2.303 = 2.61 \thinspace kJ/mol \) So, the Gibbs free energy change (ΔG) for the given reaction at 298 K with partial pressures of NO₂ and N₂O₄ as 0.40 atm and 1.60 atm, respectively, is 2.61 kJ/mol.

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Most popular questions from this chapter

Calculate \(\Delta S^{\circ}\) values for the following reactions by using tabulated \(S^{\circ}\) values from Appendix \(\mathrm{C} .\) In each case, explain the sign of \(\Delta S^{\circ} .\) $$ \begin{array}{l}{\text { (a) } \mathrm{HNO}_{3}(g)+\mathrm{NH}_{3}(g) \longrightarrow \mathrm{NH}_{4} \mathrm{NO}_{3}(s)} \\ {\text { (b) } 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \longrightarrow 4 \mathrm{Fe}(s)+3 \mathrm{O}_{2}(g)} \\ {\text { (c) } \mathrm{CaCO}_{3}(s, \text { calcite })+2 \mathrm{HCl}(g) \rightarrow} \\\ {\mathrm{CaCl}_{2}(s)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(l)}\\\ {\text { (d) } 3 \mathrm{C}_{2} \mathrm{H}_{6}(g) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{6}(l)+6 \mathrm{H}_{2}(g)}\end{array} $$

In chemical kinetics, the entropy of activation is the entropy change for the process in which the reactants reach the activated complex. Predict whether the entropy of activation for a bimolecular process is usually positive or negative.

(a) Which of the thermodynamic quantities \(T, E, q, w,\) and \(S\) are state functions? (b) Which depend on the path taken from one state to another? (c) How many reversible paths are there between two states of a system? (d) For a reversible isothermal process, write an expression for \(\Delta E\) in terms of \(q\) and \(w\) and an expression for \(\Delta S\) in terms of \(q\) and \(T .\)

For each of the following pairs, predict which substance has the higher entropy per mole at a given temperature: (a) \(\operatorname{Ar}(l)\) or \(\operatorname{Ar}(g),(\mathbf{b}) \operatorname{He}(g)\) at 3 atm pressure or \(\operatorname{He}(g)\) at 1.5 atm pressure, (c) 1 mol of \(\mathrm{Ne}(g)\) in 15.0 \(\mathrm{L}\) or 1 \(\mathrm{mol}\) of \(\mathrm{Ne}(g)\) in \(1.50 \mathrm{L},(\mathbf{d}) \mathrm{CO}_{2}(g)\) or \(\mathrm{CO}_{2}(s) .\)

Predict which member of each of the following pairs has the greater standard entropy at \(25^{\circ} \mathrm{C} :(\mathbf{a}) \mathrm{C}_{6} \mathrm{H}_{6}(l)\) or \(\mathrm{C}_{6} \mathrm{H}_{6}(g)\) (b) \(\mathrm{CO}(g)\) or \(\mathrm{CO}_{2}(g),(\mathbf{c}) 1 \mathrm{mol} \mathrm{N}_{2} \mathrm{O}_{4}(g)\) or 2 \(\mathrm{mol} \mathrm{NO}_{2}(\mathrm{g})\) (d) \(\mathrm{HCl}(g)\) or \(\mathrm{HCl}(a q) .\) Use Appendix \(\mathrm{C}\) to find the standard entropy of each substance.
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