Problem 2 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The oxidation of ammonia in air is catalysed by platinum metal \\[ 4 \mathrm{NH}_{3}(\mathrm{g})+5 \mathrm{O}_{2}(\mathrm{g}) \rightarrow 4 \mathrm{NO}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\] Write an expression for the rate of reaction in terms of differentials for the consumption of the reactants and formation of the products. (Sections \(9.2 \text { and } 9.4)\)

Step-by-Step Solution

Answer

Rate of reaction: \(-\frac{1}{4}\frac{d[\mathrm{NH}_3]}{dt} = -\frac{1}{5}\frac{d[\mathrm{O}_2]}{dt} = \frac{1}{4}\frac{d[\mathrm{NO}]}{dt} = \frac{1}{6}\frac{d[\mathrm{H}_2\mathrm{O}]}{dt}\).

Step 1: Understanding the Rate of Reaction

The rate of a chemical reaction is defined as the change in concentration of a reactant or product per unit time. The rate can be expressed in terms of the consumption of reactants or the formation of products.

Step 2: Write Rate Expression for Each Substance

The balanced equation is \(4 \mathrm{NH}_{3}(\mathrm{g})+5 \mathrm{O}_{2}(\mathrm{g}) \rightarrow 4 \mathrm{NO}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\). We can express the rate of reaction in terms of each component: \(-\frac{1}{4}\frac{d[\mathrm{NH}_3]}{dt}\), \(-\frac{1}{5}\frac{d[\mathrm{O}_2]}{dt}\), \(\frac{1}{4}\frac{d[\mathrm{NO}]}{dt}\), \(\frac{1}{6}\frac{d[\mathrm{H}_2\mathrm{O}]}{dt}\).

Step 3: Consider Stoichiometry in the Rate Expression

The stoichiometric coefficients from the balanced equation determine the relationship between the rates of consumption and formation. Therefore, the rate expressions are made consistent by dividing the change in concentration by the coefficient: \(-\frac{1}{4}\frac{d[\mathrm{NH}_3]}{dt} = -\frac{1}{5}\frac{d[\mathrm{O}_2]}{dt} = \frac{1}{4}\frac{d[\mathrm{NO}]}{dt} = \frac{1}{6}\frac{d[\mathrm{H}_2\mathrm{O}]}{dt}\).

Step 4: Combine Rate Expressions

The overall rate of the reaction can be represented by any of the rate expressions derived. Therefore, each expression for rate is equivalent. Typically, the expression is given based on one of the components, which is often a reactant. Here, one could express it simply as \(-\frac{1}{4}\frac{d[\mathrm{NH}_3]}{dt}\), which equals the others.

Key Concepts

Rate of Reaction

The rate of reaction is a fundamental concept in chemical kinetics, representing how quickly a reactant is consumed or a product is formed in a chemical process. It is expressed as the change in concentration of a specific species over time. For the reaction involving ammonia oxidation, we are interested in understanding how the concentration of ammonia and other species change as the reaction proceeds. In simple terms, it’s like tracking how fast soda in a bottle decreases as you drink it. By studying rates, chemists can predict how reactions proceed and determine the conditions needed to effectively control them. Breaking down into a more technical explanation, the rate can be represented mathematically. For reactants, it's shown with a negative sign indicating consumption, for example, \(-\frac{d[\mathrm{NH}_3]}{dt}\). For products, there’s no negative sign, such as \(\frac{d[\mathrm{NO}]}{dt}\). This basic technique provides insight into how varying concentrations influence reaction speed.

Stoichiometry

Stoichiometry provides a quantitative relationship between reactants and products in a balanced chemical equation. Imagine stoichiometry as a recipe: if you know how much ingredients you need to bake a cake, you can determine how much cake you'll make or how much of each ingredient is necessary based on the reaction. In the ammonia oxidation reaction, the stoichiometric coefficients (4 for \(\mathrm{NH}_3\), 5 for \(\mathrm{O}_2\), 4 for \(\mathrm{NO}\), and 6 for \(\mathrm{H}_2\mathrm{O}\)) help chemists relate the rates of disappearance of reactants to the rates of formation of products. This ensures everyone "cooks" with the same quantities, maintaining consistency. The stoichiometric coefficients provide a **magnifying glass** on how the ingredients (reactants) turn into the results (products) in exact proportions.

Differential Rate Expressions

Differential rate expressions provide a mathematical description of how reaction rates relate to concentrations at any given moment. They are derived from stoichiometric relations in the balanced equation. By using differentials, we express these rates with respect to time. For ammonia oxidation, these expressions are crafted using changes in concentrations divided by their stoichiometric coefficients. For instance, \(-\frac{1}{4}\frac{d[\mathrm{NH}_3]}{dt}\) suggests that for every unit of time, \(\mathrm{NH}_3\) decreases based on the stoichiometry of the process. It's like having a stopwatch that tracks how fast each element in your chemical "race" is moving according to its weight. Thus, the differentials tell us the real-time story of the reaction's pace and provide a uniform method to compare all species involved.

Catalysis

Catalysis involves a catalyst, a substance that speeds up a chemical reaction without being consumed in the process. In the case of ammonia oxidation, platinum acts as a catalyst. Imagine a catalyst as a friendly guide helping molecules find and react with each other more efficiently, just like a coach guiding athletes to improve performance without running the race themselves. Catalysts operate by lowering the activation energy—the hill molecules must climb to react—thus increasing the reaction rate. They make reactions more efficient and are pivotal in many industrial processes, including the Haber process for ammonia production. Catalysts ensure that reactions happen promptly, paving the way for numerous advancements in science and sustainability.