Problem 3 - General Chemistry: Principles and Modern Applications

Question

Write equilibrium constant expressions, \(K_{\mathrm{c}},\) for the reactions (a) \(2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}_{2}(\mathrm{g})\) (b) \(\mathrm{Zn}(\mathrm{s})+2 \mathrm{Ag}^{+}(\mathrm{aq}) \rightleftharpoons \mathrm{Zn}^{2+}(\mathrm{aq})+2 \mathrm{Ag}(\mathrm{s})\) (c) \(\mathrm{Mg}(\mathrm{OH})_{2}(\mathrm{s})+\mathrm{CO}_{3}^{2-}(\mathrm{aq}) \rightleftharpoons\) \(\mathrm{MgCO}_{3}(\mathrm{s})+2 \mathrm{OH}^{-}(\mathrm{aq})\)

Step-by-Step Solution

Answer

The equilibrium constant expressions for the provided reactions are (a) \( K_c = \frac{{[\mathrm{NO_{2}}]^2}}{{[\mathrm{NO}]^2 [\mathrm{O_2}]}} \), (b) \( K_c = \frac{{[\mathrm{Zn}^{2+}]}}{{[\mathrm{Ag^{+}}]^2}} \), and (c) \( K_c = \frac{{[\mathrm{OH^{-}}]^2}}{{[\mathrm{CO_{3}^{2-}]}} \).

Step 1: Writing the Equilibrium Constant Expression for Reaction (a)

For the reaction \(2 \mathrm{NO(g)} + \mathrm{O_{2}(g)} \rightleftharpoons 2 \mathrm{NO_{2}(g)}\), the equilibrium constant expression would be \[ K_c = \frac{{[\mathrm{NO_{2}}]^2}}{{[\mathrm{NO}]^2 [\mathrm{O_2}]}} \] The concentrations of \(NO_{2}\), \(NO\) and \(O_{2}\) are raised to their stoichiometric coefficients.

Step 2: Writing the Equilibrium Constant Expression for Reaction (b)

For the reaction \(\mathrm{Zn(s)} + 2 \mathrm{Ag^{+}(aq)} \rightleftharpoons \mathrm{Zn^{2+}(aq)} + 2 \mathrm{Ag(s)}\), write the equilibrium constant expression as \[ K_c = \frac{{[\mathrm{Zn}^{2+}]}}{{[\mathrm{Ag^{+}}]^2}} \] In this reaction, \(Zn\) is a pure solid and \(Ag\) is a pure metal; therefore, they are not included in the equilibrium expression.

Step 3: Writing the Equilibrium Constant Expression for Reaction (c)

For the reaction \(\mathrm{Mg(OH)_{2}(s)} + \mathrm{CO_{3}^{2-}(aq)} \rightleftharpoons \mathrm{MgCO_{3}(s)} + 2 \mathrm{OH^{-}(aq)}\), the equilibrium constant expression would be \[ K_c = \frac{{[\mathrm{OH^{-}}]^2}}{{[\mathrm{CO_{3}^{2-}]}} \] Here, \(\mathrm{Mg(OH)_{2}}\) and \(\mathrm{MgCO_{3}}\) are both solids so they aren't included in the equilibrium expression.

Key Concepts

Equilibrium Constant Expressions

In any reversible chemical reaction, we often want to understand how far the reaction will proceed before reaching equilibrium. This is where the equilibrium constant expressions or \(K_c\) come into play. The equilibrium constant is a ratio that compares the concentration of products to the concentration of reactants when the reaction is at equilibrium.

For any given reaction, the concentrations in the expression are raised to the power of their stoichiometric coefficients. For example, in the reaction \(2 \mathrm{NO(g)} + \mathrm{O_{2}(g)} \rightleftharpoons 2 \mathrm{NO_{2}(g)}\), the expression for the equilibrium constant is defined as:

• \(K_c = \frac{[\mathrm{NO_{2}}]^2}{[\mathrm{NO}]^2 \cdot [\mathrm{O_{2}}]}\)

Raising the concentrations to the correct powers ensures that the equilibrium constant is directly tied to the stoichiometry and reaction coefficients of the chemical equation, helping predict the direction and extent of the reaction.

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that involves the calculation of reactants and products in chemical reactions. It is essentially about balancing the reactants and products based on the stoichiometric coefficients from the chemical equation.

In equilibrium constant expressions, stoichiometry guides how we determine the concentration terms to use. All reactants and products concentrations in the equilibrium expression are raised to the power of their stoichiometric coefficients. Consider the reaction \(\mathrm{Zn(s)} + 2 \mathrm{Ag^{+}(aq)} \rightleftharpoons \mathrm{Zn^{2+}(aq)} + 2 \mathrm{Ag(s)}\). For this reaction, due to stoichiometry, the concentration of \(\mathrm{Ag^+}\) in the expression for \(K_c\) is squared, reflecting its coefficient of 2 in the reaction:

• \(K_c = \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Ag^{+}}]^2}\)

Understanding stoichiometry allows for accurate calculation of equilibrium constant expressions.

Reaction Coefficients

Reaction coefficients are numbers that precede the chemical formula in a balanced chemical equation. They indicate the proportions in which reactants react and products form in a chemical reaction.

These coefficients are crucial when writing equilibrium constant expressions. The concentrations of reactants and products must reflect these ratios, as dictated by the balanced chemical equation. For example, a coefficient of 2 in front of \(\mathrm{NO_{2}}\) in \(2 \mathrm{NO(g)} + \mathrm{O_{2}(g)} \rightleftharpoons 2 \mathrm{NO_{2}(g)}\) requires \(\mathrm{NO_{2}}\) concentration to be squared:

• \(K_c = \frac{[\mathrm{NO_{2}}]^2}{[\mathrm{NO}]^2 \cdot [\mathrm{O_{2}}]}\)

This is true for any reactant or product, whether they appear on the reactant or the product side of the equilibrium. Correct use of reaction coefficients is key to properly representing the system at equilibrium.

Solid and Aqueous Phases

In chemical equilibrium, solid and aqueous phases of reactants and products behave differently, which must be considered in equilibrium constant expressions.

Solids and liquids have constant concentrations because their densities do not change during the reaction. Therefore, they are not included in equilibrium expressions. This is why in the equation \(\mathrm{Mg(OH)_{2}(s)} + \mathrm{CO_{3}^{2-}(aq)} \rightleftharpoons \mathrm{MgCO_{3}(s)} + 2 \mathrm{OH^{-}(aq)}\), \(\mathrm{Mg(OH)_{2}}\) and \(\mathrm{MgCO_{3}}\) do not appear in the \(K_c\) expression:

• \(K_c = \frac{[\mathrm{OH^{-}}]^2}{[\mathrm{CO_{3}^{2-}]}\)

In contrast, aqueous species are included because their concentrations change during the reaction. Correctly applying the rule for including solids and excluding aqueous solutions ensures accurate equilibrium expressions, reflecting the system's true behavior.