Question

For each of the following gas-phase reactions, indicate how the rate of disappearance of each reactant is related to the rate of appearance of each product: (a) ${\mathrm{H}}_2{\mathrm{O}}_2(g)⟶{\mathrm{H}}_2(g)+{\mathrm{O}}_2(g)$ (b) $2{\mathrm{N}}_2\mathrm{O}(g)⟶2{\mathrm{N}}_2(g)+{\mathrm{O}}_2(g)$ (c) ${\mathrm{N}}_2(g)+3{\mathrm{H}}_2(g)⟶2{\mathrm{NH}}_3(g)$ (d) ${\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2(g)⟶{\mathrm{C}}_2{\mathrm{H}}_4(g)+{\mathrm{NH}}_3(g)$

Short answer

The rate relationships for each gas-phase reaction are: (a) $\text{Rate}({\mathrm{H}}_2)=-\text{Rate}({\mathrm{H}}_2{\mathrm{O}}_2)$, $\text{Rate}({\mathrm{O}}_2)=-\text{Rate}({\mathrm{H}}_2{\mathrm{O}}_2)$ (b) $\text{Rate}({\mathrm{N}}_2)=\frac{1}{2}\cdot \text{-Rate}({\mathrm{N}}_2\mathrm{O})$, $\text{Rate}({\mathrm{O}}_2)=\frac{1}{2}\cdot \text{-Rate}({\mathrm{N}}_2\mathrm{O})$ (c) $\text{Rate}({\mathrm{H}}_2)=\frac{1}{3}\cdot \text{-Rate}({\mathrm{N}}_2)$, $\text{Rate}({\mathrm{NH}}_3)=\frac{2}{1}\cdot \text{-Rate}({\mathrm{N}}_2)$ (d) $\text{Rate}({\mathrm{C}}_2{\mathrm{H}}_4)=-\text{Rate}({\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2)$, $\text{Rate}({\mathrm{NH}}_3)=-\text{Rate}({\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2)$

Step-by-step solution

Find rates in terms of reactants and products

Let's denote the rate of disappearance of ${\mathrm{H}}_2{\mathrm{O}}_2$ as $-\text{Rate}({\mathrm{H}}_2{\mathrm{O}}_2)$. Since there is a 1:1 stoichiometry between ${\mathrm{H}}_2{\mathrm{O}}_2$, ${\mathrm{H}}_2$, and ${\mathrm{O}}_2$, their rates of appearance have the same magnitude but opposite sign. The rate relationships for this reaction are: - $\text{Rate}({\mathrm{H}}_2)=-\text{Rate}({\mathrm{H}}_2{\mathrm{O}}_2)$ - $\text{Rate}({\mathrm{O}}_2)=-\text{Rate}({\mathrm{H}}_2{\mathrm{O}}_2)$ (b) Reaction: $2{\mathrm{N}}_2\mathrm{O}(g)⟶2{\mathrm{N}}_2(g)+{\mathrm{O}}_2(g)$

Find rates in terms of reactants and products

Let's denote the rate of disappearance of ${\mathrm{N}}_2\mathrm{O}$ as $-\text{Rate}({\mathrm{N}}_2\mathrm{O})$. Since there is a 2:2:1 stoichiometry between ${\mathrm{N}}_2\mathrm{O}$, ${\mathrm{N}}_2$, and ${\mathrm{O}}_2$, their rates need to be adjusted by their respective stoichiometric coefficients. The rate relationships for this reaction are: - $\text{Rate}({\mathrm{N}}_2)=\frac{1}{2}\cdot \text{-Rate}({\mathrm{N}}_2\mathrm{O})$ - $\text{Rate}({\mathrm{O}}_2)=\frac{1}{2}\cdot \text{-Rate}({\mathrm{N}}_2\mathrm{O})$ (c) Reaction: ${\mathrm{N}}_2(g)+3{\mathrm{H}}_2(g)⟶2{\mathrm{NH}}_3(g)$

Find rates in terms of reactants and products

Let's denote the rate of disappearance of ${\mathrm{N}}_2$ as $-\text{Rate}({\mathrm{N}}_2)$. Since there is a 1:3:2 stoichiometry between ${\mathrm{N}}_2$, ${\mathrm{H}}_2$, and ${\mathrm{NH}}_3$, their rates need to be adjusted by their respective stoichiometric coefficients. The rate relationships for this reaction are: - $\text{Rate}({\mathrm{H}}_2)=\frac{1}{3}\cdot \text{-Rate}({\mathrm{N}}_2)$ - $\text{Rate}({\mathrm{NH}}_3)=\frac{2}{1}\cdot \text{-Rate}({\mathrm{N}}_2)$ (d) Reaction: ${\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2(g)⟶{\mathrm{C}}_2{\mathrm{H}}_4(g)+{\mathrm{NH}}_3(g)$

Find rates in terms of reactants and products

Let's denote the rate of disappearance of ${\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2$ as $-\text{Rate}({\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2)$. Since there is a 1:1:1 stoichiometry between ${\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2$, ${\mathrm{C}}_2{\mathrm{H}}_4$, and ${\mathrm{NH}}_3$, their rates of appearance have the same magnitude but opposite sign. The rate relationships for this reaction are: - $\text{Rate}({\mathrm{C}}_2{\mathrm{H}}_4)=-\text{Rate}({\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2)$ - $\text{Rate}({\mathrm{NH}}_3)=-\text{Rate}({\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{NH}}_2)$

Key concepts

Rate of Disappearance

Understanding the rate of disappearance in chemical reactions is crucial as it quantitatively describes how quickly reactants are consumed. In a chemical equation, reactants are the substances that undergo a transformation to produce new substances, or products. The rate of disappearance is expressed as the change in concentration of a reactant over time, typically noted with a negative sign to indicate a decrease.

For instance, if we consider reaction (a) from the exercise, ${\mathrm{H}}_2{\mathrm{O}}_2(g)⟶{\mathrm{H}}_2(g)+{\mathrm{O}}_2(g)$, the rate at which hydrogen peroxide ${\mathrm{H}}_2{\mathrm{O}}_2$ is vanishing from the reaction is referred to as its rate of disappearance. Since the reaction is direct and with a 1:1:1 stoichiometry, we know that for each molecule of ${\mathrm{H}}_2{\mathrm{O}}_2$ that disappears, one molecule of ${\mathrm{H}}_2$ and one molecule of ${\mathrm{O}}_2$ appear. This allows us to equate the rate of disappearance of ${\mathrm{H}}_2{\mathrm{O}}_2$ with the rate of appearance of the products.

Stoichiometry

Moving to the concept of stoichiometry, it is the quantitative relationship between the amounts of reactants and products in a chemical reaction. It allows scientists to predict the amount of products that will be formed from a given quantity of reactants, based on the balanced chemical equation.

The coefficients present in a balanced chemical equation provide the ratio of molecules required for the reaction to proceed and the ratio of molecules produced. For example, in reaction (b) from the original problem, $2{\mathrm{N}}_2\mathrm{O}(g)⟶2{\mathrm{N}}_2(g)+{\mathrm{O}}_2(g)$, the coefficients initial '2' for ${\mathrm{N}}_2\mathrm{O}$ and ${\mathrm{N}}_2$ indicate that two molecules of nitrous oxide are needed to produce two molecules of nitrogen. The coefficient '1' for ${\mathrm{O}}_2$ indicates that these two molecules of ${\mathrm{N}}_2\mathrm{O}$ will also produce only one molecule of oxygen. Hence, the rates of disappearance and appearance are directly connected to these stoichiometric coefficients. Applying stoichiometry ensures the conservation of mass and atoms during the chemical reaction.

Rate of Appearance

Finally, we examine the rate of appearance, which is the flip side of the coin to the rate of disappearance. It defines how quickly a product forms in a chemical reaction. Just like the rate of disappearance, the rate of appearance is based on the change in concentration of a product over time, and it is generally expressed with a positive sign.

Taking reaction (c), ${\mathrm{N}}_2(g)+3{\mathrm{H}}_2(g)⟶2{\mathrm{NH}}_3(g)$, we see an example with differing stoichiometric coefficients. These coefficients imply that for every molecule of ${\mathrm{N}}_2$ that reacts, two molecules of ammonia ${\mathrm{NH}}_3$ are formed. To accurately express the rate of appearance, we use the stoichiometry of the reaction to find that the rate at which ${\mathrm{NH}}_3$ forms is twice the magnitude of the rate of disappearance of ${\mathrm{N}}_2$. Understanding the rate of appearance is important for predicting the yield of a product and for determining how long a reaction needs to be run to produce a desired amount of product.