Question

For each of the following gas-phase reactions, write the rate expression in terms of the appearance of each product and disappearance of each reactant: (a) \(2 \mathrm{H}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)\) (b) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (c) \(2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) (d) \(\mathrm{N}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{H}_{4}(g)\)

Short answer

(a) Rate = -\(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt}\) = \(\frac{d[\mathrm{H}_{2}]}{dt}\) = \(\frac{1}{2}\) \(\frac{d[\mathrm{O}_{2}]}{dt}\) (b) Rate = -\(\frac{d[\mathrm{SO}_{2}]}{dt}\) = \(\frac{1}{2}\) \(\frac{-d[\mathrm{O}_{2}]}{dt}\) = \(\frac{d[\mathrm{SO}_{3}]}{dt}\) (c) Rate = -\(\frac{1}{2}\) \(\frac{d[\mathrm{NO}]}{dt}\) = -\(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2}]}{dt}\) = \(\frac{d[\mathrm{N}_{2}]}{dt}\) = \(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt}\) (d) Rate = -\(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2}]}{dt}\) = -\(\frac{d[\mathrm{N}_{2}]}{dt}\) = \(\frac{d[\mathrm{N}_{2} \mathrm{H}_{4}]}{dt}\)

Step-by-step solution

Identify the rate of the formation of products and the consumption of reactants

Using the coefficients of the balanced chemical equation, we can determine the relationship between the rate of formation of products and the rate of consumption of reactants: Rate of formation of \(\mathrm{H}_{2}\) = 2 times the rate of consumption of \(\mathrm{H}_{2} \mathrm{O}\) Rate of formation of \(\mathrm{O}_{2}\) = 1 times the rate of consumption of \(\mathrm{H}_{2} \mathrm{O}\)

Determine the rate expressions for reactants and products

Now that we know the relationship between the rates, we can write the rate expressions in terms of each reactant and product: Rate = -\(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt}\) = \(\frac{d[\mathrm{H}_{2}]}{dt}\) = \(\frac{1}{2}\) \(\frac{d[\mathrm{O}_{2}]}{dt}\) (b) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\)

Identify the rate of the formation of products and the consumption of reactants

Using the coefficients of the balanced chemical equation, we can determine the relationship between the rate of formation of products and the rate of consumption of reactants: Rate of formation of \(\mathrm{SO}_{3}\) = 1 times the rate of consumption of \(\mathrm{SO}_{2}\) and Rate of formation of \(\mathrm{SO}_{3}\) = \(\frac{1}{2}\) times the rate of consumption of \(\mathrm{O}_{2}\)

Determine the rate expressions for reactants and products

Now that we know the relationship between the rates, we can write the rate expressions in terms of each reactant and product: Rate = -\(\frac{d[\mathrm{SO}_{2}]}{dt}\) = \(\frac{1}{2}\) \(\frac{-d[\mathrm{O}_{2}]}{dt}\) = \(\frac{d[\mathrm{SO}_{3}]}{dt}\) (c) \(2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}(g)+2\mathrm{H}_{2} \mathrm{O}(g)\)

Identify the rate of the formation of products and the consumption of reactants

Using the coefficients of the balanced chemical equation, we can determine the relationship between the rate of formation of products and the rate of consumption of reactants: Rate of formation of \(\mathrm{N}_{2}\) = 1 times the rate of consumption of \(\mathrm{NO}\) Rate of formation of \(\mathrm{H}_{2} \mathrm{O}\) = \(\frac{1}{2}\) times the rate of consumption of \(\mathrm{H}_{2}\)

Determine the rate expressions for reactants and products

Now that we know the relationship between the rates, we can write the rate expressions in terms of each reactant and product: Rate = -\(\frac{1}{2}\) \(\frac{d[\mathrm{NO}]}{dt}\) = -\(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2}]}{dt}\) = \(\frac{d[\mathrm{N}_{2}]}{dt}\) = \(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt}\) (d) \(\mathrm{N}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{N}_{2}\mathrm{H}_{4}(g)\)

Identify the rate of the formation of products and the consumption of reactants

Using the coefficients of the balanced chemical equation, we can determine the relationship between the rate of formation of products and the rate of consumption of reactants: Rate of formation of \(\mathrm{N}_{2} \mathrm{H}_{4}\) = 2 times the rate of consumption of \(\mathrm{H}_{2}\) Rate of formation of \(\mathrm{N}_{2} \mathrm{H}_{4}\) = 1 times the rate of consumption of \(\mathrm{N}_{2}\)

Determine the rate expressions for reactants and products

Now that we know the relationship between the rates, we can write the rate expressions in terms of each reactant and product: Rate = -\(\frac{1}{2}\) \(\frac{d[\mathrm{H}_{2}]}{dt}\) = -\(\frac{d[\mathrm{N}_{2}]}{dt}\) = \(\frac{d[\mathrm{N}_{2} \mathrm{H}_{4}]}{dt}\)