Chemistry

Problem

Ammonium nitrate decomposes explosively according to the overall reaction

\mathrm{NH_4NO_3(s) \to N_2O(g) + 2\,H_2O(g)}

. For this reaction,

\Delta H = +73.7\,\text{kJ mol}^{-1}

and

\Delta S = +259\,\text{J mol}^{-1}\,\text{K}^{-1}

(assumed constant with temperature). a. Part 1: Starting from the definition of the Gibbs free energy change, derive the general temperature condition that must be satisfied for the reaction to be at equilibrium (

\Delta G = 0

). b. Part 2: Compute the equilibrium temperature for the decomposition of

\mathrm{NH_4NO_3(s)}

. c. Part 3: Combine the result from part (a) with the relationship between

\Delta G^\circ

and the equilibrium constant

K

to obtain an expression for

\ln K

as a function of temperature for this reaction. d. Part 4: Evaluate

K

800\,\text{K}

using your expression from part (c). Clearly show your work. e. Part 5: Based on your calculations, discuss qualitatively whether increasing the temperature beyond the value found in part (b) will increase or decrease the extent of decomposition under constant-pressure conditions.

Intuitive explanation

This problem tests whether you understand when a reaction becomes spontaneous by examining the competition between enthalpy and entropy. Even though breaking bonds requires energy ( $\Delta H > 0$ ), the reaction can still proceed if the increase in disorder (measured by $\Delta S$ ) is large enough to overcome that cost. The key is using the Gibbs free energy equation $\Delta G = \Delta H - T\Delta S$ to find the temperature threshold where the entropy term finally dominates and makes the decomposition favorable.

Intuitive explanation

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