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Chapter 18: Problem 67

Consider the reaction $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g) $$ Given that \(\Delta G^{\circ}\) for the reaction at \(25^{\circ} \mathrm{C}\) is 173.4 \(\mathrm{kJ} / \mathrm{mol}\), (a) calculate the standard free energy of formation of \(\mathrm{NO},\) and (b) calculate \(K_{P}\) of the reaction. (c) One of the starting substances in smog formation is NO. Assuming that the temperature in a running automobile engine is \(1100^{\circ} \mathrm{C},\) estimate \(K_{P}\) for the above reaction. (d) As farmers know, lightning helps to produce a better crop. Why?

Short Answer

Expert verified
a) The standard free energy of formation for NO is 86.7 kJ/mol. b) The equilibrium constant \(K_{P}\) for the reaction at 25°C is \(4.15*10^{-28}\). c) At 1100°C, \(K_{P}\) is about 179.3. d) Lightning helps produce better crops because it enables the formation of nitrates, a kind of fertilizer.

Step by step solution

01

Calculate the standard free energy of formation for NO

The given balance equation is \(N_{2}(g)+O_{2}(g) \rightleftharpoons 2 NO(g)\). The standard free energy of formation of elemental species in their standard states is zero. So, for \(N_{2}(g)\) and \(O_{2}(g)\), \(\Delta G_f^{\circ}\) is 0. The standard free energy change for the reaction \(\Delta G^{\circ}\) is 173.4 kJ/mol. Using the equation \(\Delta G^{\circ} = \Delta G_f^{\circ} (products) - \Delta G_f^{\circ} (reactants)\), we get \(\Delta G^{\circ} = 2 \Delta G_f^{\circ} (NO(g)) - 0\). Solving this gives \(\Delta G_f^{\circ} (NO(g))\) = 173.4/2 = 86.7 kJ/mol.
02

Calculate \(K_{P}\) for the reaction

The equation relating standard free energy change and equilibrium constant is given by \(\Delta G^{\circ} = -RT ln(K_{P})\), where R is the gas constant (8.314 J/K*mol) and T is the temperature in Kelvin (25°C = 298 K). Substituting the values, we get \(-173.4*10^3 J = -(8.314 J/K*mol * 298 K) ln(K_{P})\) , solving this yields \(K_{P} = e^{-173.4*10^3/(8.314*298)} = 4.15*10^{-28}\).
03

Estimate \(K_{P}\) at a different temperature

In order to calculate the equilibrium constant at a different temperature we use the Van't Hoff equation. We can rearrange the equation to: \(ln(K_2/K_1) = - \Delta H_m / R * (1/T_2 - 1/T_1)\), which gives us: \(ln(K_{P2}/K_{P1}) = (-\Delta H_m / 8.314) * (1/1373 - 1/298)\). DeltaH_m is the enthalpy change for the reaction, which can be calculated from DeltaG_m and T: \(\Delta H_m = \Delta G_m + T\Delta S_m = \Delta G_m + T(\Delta G_m / T) = 2*\Delta G_m = 2*173.4 kJ/mol\). Solving the Van't Hoff equation gives: \(K_{P2} = K_{P1} * exp[(-2*173.4*10^3 / 8.314) * (1/1373 - 1/298)] = 4.15 * 10^{-28} * exp[(-2*173.4*10^3 / 8.314) * (1/1373 - 1/298)] = 179.3\).
04

Explain why lightning helps produce better crops

Lightning helps produce better crops because the electrical energy from lightning can cause nitrogen and oxygen in the atmosphere to react and form nitrogen oxides, including NO and NO2. These nitrogen oxides dissolve in rain to form nitric acid, which is a form of nitrogen that plants can absorb and use to build proteins and other complex molecules. When this 'nitrate' falls to the ground in rain, it gets into the soil and provides plants with necessary nutrients.

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