Chapter 14, Problem 79GQ

Nitramide, NO₂NH₂, decomposes slowly in aqueous solution according to the following reaction:

NO₂NH₂(aq) → N₂O(g) + H₂O(ℓ)

The reaction follows the experimental rate law

$\text{Rate=}\frac{{\text{k[NO}}_{\text{2}}\text{NH}{\text{2}}_{}\text{]}}{{\text{[H}}_{\text{3}}{\text{O}}^{\text{+}}\text{]}}$

1. (a) What is the apparent order of the reaction in a pH buffered solution? (In a pH buffered solution, the concentration of H₃O⁺ is a constant.)
2. (b) Which of the following mechanisms is the most appropriate for the interpretation of this rate law? Explain. (Note that when writing the expression for K, the equilibrium constant, [H₂O] is not involved. See Chapter 15.)

   Mechanism 1

   ${\text{NO}}_{\text{2}}{\text{NH}}_{\text{2}}\stackrel{K_1}{\to }{\text{ N}}_{\text{2}}{\text{O + H}}_{\text{2}}\text{O}$

   Mechanism 2

   ${\text{NO}}_{\text{2}}{\text{NH}}_{\text{2}}{\text{+H}}_{\text{3}}{\text{O}}^{\text{+}}\underset{k_{\text{2}}\prime }{\overset{k_{\text{2}}}{\rightleftarrows }}{\text{ NO}}_{\text{2}}{\text{NH}}_{\text{3}}^{\text{+}}{\text{ + H}}_{\text{2}}\text{O (rapid equilibrium)}$

   ${\text{NO}}_{\text{2}}{\text{NH}}_{\text{3}}^{\text{+}}\stackrel{k_{\text{3}}}{\to }{\text{N}}_{\text{2}}\text{O}+{\text{H}}_{\text{3}}{\text{O}}^{\text{+}}\text{(rate-limiting step)}$

   Mechanism 3

   $\begin{array}{l}{\text{NO}}_{\text{2}}{\text{NH}}_{\text{2}}{\text{+H}}_{\text{2}}\text{O}\underset{k_{\text{4}}{}^{\prime }}{\overset{k_{\text{4}}}{\rightleftarrows }}{\text{ NO}}_{\text{2}}{\text{NH}}^{-}{\text{+ H}}_{\text{3}}{\text{O}}^{\text{+}}\text{(rapid equilibrium)}\\ {\text{NO}}_{\text{2}}{\text{NH}}^{-}\stackrel{k_{\text{5}}}{\to }{\text{N}}_{\text{2}}{\text{O+OH}}^{-}\text{(rate-limiting step)}\\ {\text{H}}_{\text{3}}{\text{O}}^{\text{+}}{\text{+OH}}^{-}\stackrel{k_{\text{6}}}{\to }{\text{2H}}_{\text{2}}\text{O (very fast reaction)}\end{array}$

3. (c) Show the relationship between the experimentally observed rate constant, k, and the rate constants in the selected mechanism.
4. (d) Based on the experimental rate law, will the reaction rate increase or decrease if the pH of the solution is increased?