Problem 23 - Chemistry: The Central Science

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) \(\mathrm{H}_{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)\) (b) \(2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g)\) (c) \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)\) (d) \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{NH}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{NH}_{3}(g)\)

Step-by-Step Solution

Answer

(a) Rate of disappearance of \(\mathrm{H}_{2} \mathrm{O}_{2} = -\)rate of appearance of \(\mathrm{H}_{2} = -\)rate of appearance of \(\mathrm{O}_{2}\) (b) Rate of disappearance of \(\mathrm{N}_{2}\mathrm{O} = -\cfrac{1}{2}\) rate of appearance of \(\mathrm{N}_{2}=-\cfrac{1}{2}\) rate of appearance of \(\mathrm{O}_{2}\) (c) Rate of disappearance of \(\mathrm{N}_{2} = -\cfrac{1}{3}\) rate of disappearance of \(\mathrm{H}_{2}=-\cfrac{1}{2}\) rate of appearance of \(\mathrm{NH}_{3}\) (d) Rate of disappearance of \(\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{NH}_{2} = -\)rate of appearance of \(\mathrm{C}_{2}\mathrm{H}_{4}=-\)rate of appearance of \(\mathrm{NH}_{3}\)

Step 1: Analyze the coefficients

The coefficients for this reaction are all 1, which means the molar ratios between all elements are equal.

Step 2: Write the rate relationships

Since the coefficients are all 1, the rate of disappearance of \(\mathrm{H}_{2} \mathrm{O}_{2}\) is equal to the rate of appearance of both \(\mathrm{H}_{2}\) and \(\mathrm{O}_{2}\). Thus, we have: Rate of disappearance of \(\mathrm{H}_{2} \mathrm{O}_{2} = -\)rate of appearance of \(\mathrm{H}_{2} = -\)rate of appearance of \(\mathrm{O}_{2}\) (b) Reaction: \(2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g)\)

Step 1: Analyze the coefficients

The coefficients in this reaction are 2 for \(\mathrm{N}_{2}\mathrm{O}\), \(\mathrm{N}_{2}\) and 1 for \(\mathrm{O}_{2}\).

Step 2: Write the rate relationships

Given the coefficients, the rate of disappearance of \(2\,\mathrm{N}_{2}\mathrm{O}\) is twice the rate of appearance of both \(2\,\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\). Thus: Rate of disappearance of \(\mathrm{N}_{2}\mathrm{O} = -\cfrac{1}{2}\) rate of appearance of \(\mathrm{N}_{2}=-\cfrac{1}{2}\) rate of appearance of \(\mathrm{O}_{2}\) (c) Reaction: \(\mathrm{N}_{2}(g) + 3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)\)

Step 1: Analyze the coefficients

The coefficients in this reaction are 1 for \(\mathrm{N}_{2}\), 3 for \(\mathrm{H}_{2}\) and 2 for \(\mathrm{NH}_{3}\).

Step 2: Write the rate relationships

With the given coefficients, the rate of disappearance of \(\mathrm{N}_{2}\) is \(\cfrac{1}{3}\) of the rate of disappearance of \(\mathrm{H}_{2}\) and \(\cfrac{1}{2}\) of the rate of appearance of \(\mathrm{NH}_{3}\): Rate of disappearance of \(\mathrm{N}_{2} = -\cfrac{1}{3}\) rate of disappearance of \(\mathrm{H}_{2}=-\cfrac{1}{2}\) rate of appearance of \(\mathrm{NH}_{3}\) (d) Reaction: \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{NH}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{NH}_{3}(g)\)

Step 1: Analyze the coefficients

The coefficients for this reaction are all 1, which means the molar ratios between all elements are equal.

Step 2: Write the rate relationships

Since the coefficients are all 1, the rate of disappearance of \(\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{NH}_{2}\) is equal to the rate of appearance of both \(\mathrm{C}_{2}\mathrm{H}_{4}\) and \(\mathrm{NH}_{3}\). Thus: Rate of disappearance of \(\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{NH}_{2} = -\)rate of appearance of \(\mathrm{C}_{2}\mathrm{H}_{4}=-\)rate of appearance of \(\mathrm{NH}_{3}\)

Key Concepts

Rate of Disappearance

In chemical reactions, substances are transformed as reactants turn into products. Naturally, this means that as a reaction progresses, the quantities of reactants decrease. The speed at which a reactant is consumed in a reaction is known as the rate of disappearance. This rate is crucial because it helps scientists and engineers understand how quickly a reactant is being used up, which influences how they might conduct or control a reaction. The rate of disappearance is typically expressed as a negative value. This negative sign is important because concentrations of reactants decrease over time. Mathematically, for a reactant \( A \), the rate of disappearance is represented as:\[ -\frac{d[A]}{dt} \]where \([A]\) is the concentration of reactant \( A \) and \( t \) is time. In the context of balanced chemical equations, the rate of disappearance can help determine how fast each reactant is being consumed relative to the creation of products by comparing their molar relationships.

Rate of Appearance

Conversely to disappearance, the rate of appearance refers to how quickly products form in a chemical reaction. As reactants decrease, products increase, thus scientists are just as interested in the rate at which products appear. The rate of appearance is expressed as a positive value, indicating an increase in product concentration over time. For a product \( B \), this rate can be described as:\[ \frac{d[B]}{dt} \]which considers the concentration of product \( B \) over time \( t \). These rates can vary depending on the conditions of the reaction and the stoichiometry of the balanced equation. Understanding the rate of appearance is essential for optimizing conditions in industrial and laboratory settings, ensuring products are formed at desired rates.

Molar Coefficients

Molar coefficients are integral parts of balanced chemical equations. They indicate the relative amounts of each reactant and product involved in a reaction. These coefficients are essential for understanding how the rate of disappearance of reactants relates to the rate of appearance of products.In chemical equations, molar coefficients are the numbers placed before compounds or elements. For instance, in the reaction \( 2 \mathrm{N}_2\mathrm{O}(g) \rightarrow 2 \mathrm{N}_2(g) + \mathrm{O}_2(g) \), the molar coefficients are 2 for \( \mathrm{N}_2\mathrm{O} \) and \( \mathrm{N}_2 \), and 1 for \( \mathrm{O}_2 \).Using these coefficients, we can deduce the stoichiometric relations, which allow us to calculate the rates at which different substances decrease or increase during the reaction. The molar ratio given by these coefficients helps in setting up proportional relationships between rates of disappearing reactants and appearing products.

Chemical Reactions

Chemical reactions are processes where reactants are converted into products through the breaking and forming of chemical bonds. These reactions are fundamental to chemistry, underlying everything from biological functions to industrial processes. Understanding a chemical reaction involves looking at the equation, which reveals vital details like reactants, products, and stoichiometric relationships. For example, the reaction \( \mathrm{N}_2 + 3\mathrm{H}_2 \rightarrow 2\mathrm{NH}_3 \) illustrates how nitrogen and hydrogen combine to form ammonia. Reactions have certain characteristics, such as exothermic or endothermic nature, which describe whether they release or absorb heat. Understanding rates and stoichiometry enables chemists to control reactions for various applications, including manufacturing, pharmaceuticals, and energy production. By evaluating these factors, scientists can predict and manipulate the outcomes of reactions to align with desired goals, making the study of reaction rates crucial in the field of chemistry.