Question

Write the reaction rate expressions for the following reactions in terms of the disappearance of the reactants and the appearance of products: (a) \(2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)\) (b) \(4 \mathrm{NH}_{3}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)\)

Short answer

(a) \(-\frac{1}{2}\frac{d[\mathrm{H}_2]}{dt} = -\frac{d[\mathrm{O}_2]}{dt} = \frac{1}{2}\frac{d[\mathrm{H}_2\mathrm{O}]}{dt}\). (b) \(-\frac{1}{4}\frac{d[\mathrm{NH}_3]}{dt} = -\frac{1}{5}\frac{d[\mathrm{O}_2]}{dt} = \frac{1}{4}\frac{d[\mathrm{NO}]}{dt} = \frac{1}{6}\frac{d[\mathrm{H}_2\mathrm{O}]}{dt}\).

Step-by-step solution

Understand the Reaction Rate Expressions

The reaction rate is expressed as the change in concentration over time. For a reaction, it is defined using the disappearance of reactants and the appearance of products. The rate for reactants is often negative, as they are being consumed, while for products, it is positive, as they are being formed.

Write Rate Expression for Reaction (a)

The reaction is \(2 \mathrm{H}_{2}(g) + \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)\). The rate of reaction is expressed as:\[\text{Rate} = -\frac{1}{2}\frac{d[\mathrm{H}_{2}]}{dt} = -\frac{d[\mathrm{O}_{2}]}{dt} = \frac{1}{2}\frac{d[\mathrm{H}_{2}\mathrm{O}]}{dt}\]This equation shows the rates in terms of stoichiometry, indicating the disappearance of \(\mathrm{H}_{2}\) and \(\mathrm{O}_{2}\), and the appearance of \(\mathrm{H}_{2}\mathrm{O}\).

Write Rate Expression for Reaction (b)

The reaction is \(4 \mathrm{NH}_{3}(g) + 5 \mathrm{O}_{2}(g) \rightarrow 4 \mathrm{NO}(g) + 6 \mathrm{H}_{2} \mathrm{O}(g)\). The rate of reaction is expressed as:\[\text{Rate} = -\frac{1}{4}\frac{d[\mathrm{NH}_{3}]}{dt} = -\frac{1}{5}\frac{d[\mathrm{O}_{2}]}{dt} = \frac{1}{4}\frac{d[\mathrm{NO}]}{dt} = \frac{1}{6}\frac{d[\mathrm{H}_{2}\mathrm{O}]}{dt}\]These expressions show the rates are proportional to the coefficients from the balanced equation.

Key concepts

Stoichiometry

Stoichiometry refers to the relationship between the quantities of reactants and products in a chemical reaction. It is based on the balanced chemical equation which indicates the exact proportions of substances involved. Think of stoichiometry as a recipe that tells you the exact amounts of each ingredient needed. In a reaction, the coefficients in the balanced equation are essential:

• They tell you how many moles of each substance are involved.
• They help in determining the reaction rate, as they dictate how much of a reactant is consumed or product is produced over time.

For example, in the reaction \(2 \mathrm{H}_{2}(g) + \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)\), the stoichiometric coefficients, 2 for \(\mathrm{H}_{2}\) and 1 for \(\mathrm{O}_{2}\), show that two moles of hydrogen react with one mole of oxygen. A solid grasp of stoichiometry allows one to beautifully predict the amount of products formed or reactants used in a given reaction.

Disappearance of Reactants

The disappearance of reactants during a chemical reaction is a crucial aspect of reaction rates. It describes how fast the starting materials are used up as the reaction proceeds:

• The rate of disappearance is often measured in terms of concentration change over time \(-\frac{d[\text{Reactant}]}{dt}\).
• The negative sign indicates a decrease in concentration since the reactant is being consumed.

In reaction \(2\mathrm{H}_{2}(g) + \mathrm{O}_{2}(g) \rightarrow 2\mathrm{H}_{2}\mathrm{O}(g)\), the rate of disappearance for \(\mathrm{H}_{2}\) is \(-\frac{1}{2}\frac{d[\mathrm{H}_{2}]}{dt}\), illustrating that hydrogen is being used up at a rate proportional to its stoichiometric coefficient. Similarly, \(\mathrm{O}_{2}\) vanishes at a rate \(-\frac{d[\mathrm{O}_{2}]}{dt}\), matching its coefficient.

Appearance of Products

The appearance of products in a chemical reaction illustrates how quickly new substances are being formed. This is an integral part of understanding reaction rates:

• The rate of appearance is expressed as \(\frac{d[\text{Product}]}{dt}\).
• It is positive because products' concentrations increase as the reaction progresses.

In our example with \(4 \mathrm{NH}_{3}(g) + 5 \mathrm{O}_{2}(g) \rightarrow 4 \mathrm{NO}(g) + 6 \mathrm{H}_{2} \mathrm{O}(g)\), the rate at which \(\mathrm{NO}\) appears is \(\frac{1}{4}\frac{d[\mathrm{NO}]}{dt}\), reflecting its stoichiometric coefficient. Likewise, the formation of \(\mathrm{H}_{2}\mathrm{O}\) is described by \(\frac{1}{6}\frac{d[\mathrm{H}_{2}\mathrm{O}]}{dt}\). These expressions align with the concept that chemical equations not only predict quantities but also rate changes, balancing the disappearance of reactants and the emergence of products.