Problem 44 - Chemistry Matter and Change

Question

Write equilibrium constant expressions for these homogeneous equilibria. \begin{equation} \begin{array}{l}{\text { a. } 2 \mathrm{N}_{2} \mathrm{H}_{4}(\mathrm{g})+2 \mathrm{NO}_{2}(\mathrm{g}) \rightleftharpoons 3 \mathrm{N}_{2}(\mathrm{g})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})} \\ {\text { b. } 2 \mathrm{NbCl}_{4}(\mathrm{g}) \rightleftharpoons \mathrm{NbCl}_{3}(\mathrm{g})+\mathrm{NbCl}_{5}(\mathrm{g})}\end{array} \end{equation}

Step-by-Step Solution

Answer

a) \( K_c = \frac{[\text{N}_2]^3[\text{H}_2\text{O}]^4}{[\text{N}_2\text{H}_4]^2[\text{NO}_2]^2} \); b) \( K_c = \frac{[\text{NbCl}_3][\text{NbCl}_5]}{[\text{NbCl}_4]^2} \).

Step 1: Understand the Concept of Equilibrium Constant

The equilibrium constant, denoted as \( K_c \), is a number that expresses the ratio of the concentrations of products to reactants, each raised to the power of their coefficients in the balanced equation, at equilibrium for a reaction at a given temperature.

Step 2: Identify Components of the Reaction

For each given reaction, identify the reactants and products. Ensure the reaction is balanced to accurately assign stoichiometric coefficients, which will be used as powers in the equilibrium expression.

Step 3: Write Equilibrium Expression for Reaction a

Given reaction a: \( 2 \text{N}_2\text{H}_4(g) + 2 \text{NO}_2(g) \rightleftharpoons 3 \text{N}_2(g) + 4 \text{H}_2\text{O}(g) \). For this reaction, the equilibrium expression is:\[K_c = \frac{[\text{N}_2]^3[\text{H}_2\text{O}]^4}{[\text{N}_2\text{H}_4]^2[\text{NO}_2]^2}\]

Step 4: Write Equilibrium Expression for Reaction b

Given reaction b: \( 2 \text{NbCl}_4(g) \rightleftharpoons \text{NbCl}_3(g) + \text{NbCl}_5(g) \). For this reaction, the equilibrium expression is:\[K_c = \frac{[\text{NbCl}_3][\text{NbCl}_5]}{[\text{NbCl}_4]^2}\]

Step 5: Review and Verify Expressions

Ensure that each equilibrium expression is correctly set up by checking the powers of concentration terms match the coefficients from the balanced chemical equations and that the form \( \frac{\text{Products}}{\text{Reactants}} \) is consistently used.

Key Concepts

Equilibrium Constant

The equilibrium constant, commonly denoted as \( K_c \), plays a crucial role in understanding chemical reactions at equilibrium. It describes the balance between products and reactants in a chemical reaction. It's essentially a fixed numerical value that reflects the ratio of concentrations of the products to reactants when a reaction is at equilibrium in accordance with temperature. Each concentration is raised to the power indicated by the coefficients in the balanced chemical equation. For example, in the reaction \( 2 \text{A} + 3 \text{B} \rightleftharpoons 2 \text{C} + \text{D} \), the equilibrium constant expression will be:\[K_c = \frac{[C]^2[D]}{[A]^2[B]^3}\]Here, square brackets denote the concentration of each species, and coefficients are exponents in the expression.

• If \( K_c \) is high, the reaction favors products at equilibrium.
• A low \( K_c \) indicates an equilibrium that favors reactants.

This way, \( K_c \) provides insights into the reaction's dynamics and helps predict the concentration of substances at equilibrium.

Homogeneous Equilibria

Homogeneous equilibria refer to reactions in which all reactants and products are in the same phase, typically as gases or dissolved in a solution. This concept is useful when writing equilibrium constant expressions because it simplifies the way we express the equilibrium state.In a homogeneous gaseous equilibrium, like the ones from the original exercise, all reactants and products are gases:

• Example a: \( 2 \text{N}_2\text{H}_4(g) + 2 \text{NO}_2(g) \rightleftharpoons 3 \text{N}_2(g) + 4 \text{H}_2\text{O}(g) \)
• Example b: \( 2 \text{NbCl}_4(g) \rightleftharpoons \text{NbCl}_3(g) + \text{NbCl}_5(g) \)

For both reactions, the equilibrium constant expression only involves the concentrations of these gaseous substances. This homogeneity makes it easier to calculate and compare \( K_c \) values, without needing to account for different phases.

Stoichiometry

Stoichiometry involves calculating the relative quantities of reactants and products in a chemical reaction. It is fundamentally based on the conservation of mass, ensuring that atoms are neither created nor destroyed. Properly understanding stoichiometry is essential when constructing equilibrium constant expressions.In the context of equilibrium expressions, stoichiometric coefficients become the exponents. For instance, in reaction a of our exercise, the expression for \( K_c \) is:\[K_c = \frac{[\text{N}_2]^3[\text{H}_2\text{O}]^4}{[\text{N}_2\text{H}_4]^2[\text{NO}_2]^2}\]Notice how the coefficients from the balanced chemical equation directly affect the powers to which product and reactant concentrations are raised. For any student dealing with chemical equilibria, getting comfortable with these coefficients is indispensable.

Balanced Chemical Equation

A balanced chemical equation is the foundation of calculating an equilibrium constant expression. It ensures that the same number of each type of atom is present on both sides of the equation, adhering to the law of conservation of mass. Without balancing, stoichiometric relationships and subsequent calculations, such as equilibrium constant expressions, wouldn't be accurate.To balance a chemical equation:

• Identify the reactants and products involved.
• Count the number of atoms of each element on both sides.
• Adjust the coefficients to ensure equal numbers of atoms for each element on both sides.

For example, in reaction b of our exercise, \( 2 \text{NbCl}_4(g) = \text{NbCl}_3(g) + \text{NbCl}_5(g) \), the coefficients of \( 2, 1, \) and \( 1 \) allow us to maintain atom conservation across the reaction.
Balancing serves not just as a manual check, but also guides us in determining the powers used in the expression for \( K_c \). This emphasizes how crucial a balanced equation is in all stoichiometric calculations.