Question

Write the expression for \(K_{c}\) for the following reactions. In each case indicate whether the reaction is homogeneous or heterogeneous. (a) \(3 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}(g)+\mathrm{NO}_{2}(g)\) (b) \(\mathrm{CH}_{4}(g)+2 \mathrm{H}_{2} \mathrm{~S}(g) \rightleftharpoons \mathrm{CS}_{2}(g)+4 \mathrm{H}_{2}(g)\) (c) \(\mathrm{Ni}(\mathrm{CO})_{4}(g) \rightleftharpoons \mathrm{Ni}(s)+4 \mathrm{CO}(g)\) (d) \(\mathrm{HF}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{F}^{-}(a q)\) (e) \(2 \mathrm{Ag}(s)+\mathrm{Zn}^{2+}(a q) \rightleftharpoons 2 \mathrm{Ag}^{+}(a q)+\mathrm{Zn}(s)\)

Short answer

(a) Homogeneous: \(K_c = \frac{[N_{2}O][NO_{2}]}{[NO]^3}\) (b) Homogeneous: \(K_c = \frac{[CS_{2}][H_{2}]^4}{[CH_{4}][H_{2}S]^2}\) (c) Heterogeneous: \(K_c = \frac{[CO]^4}{[Ni(CO)_{4}]}\) (d) Homogeneous: \(K_c = \frac{[H^{+}][F^{-}]}{[HF]}\) (e) Heterogeneous: \(K_c = \frac{[Ag^{+}]^2}{[Zn^{2+}]}\)

Step-by-step solution

Identify reaction types

All the reactants and products are in the gas phase, making this a homogeneous reaction.

Write the Kc expression

For a homogeneous reaction, the equilibrium constant expression (Kc) is the ratio of the product concentration raised to its stoichiometric coefficient divided by the reactant concentration raised to its stoichiometric coefficient. For this reaction: Kc = \(\frac{[N_{2}O][NO_{2}]}{[NO]^3}\) (b) CH4(g) + 2 H2S(g) ⇌ CS2(g) + 4 H2(g)

Identify reaction types

All the reactants and products are in the gas phase, making this a homogeneous reaction.

Write the Kc expression

For this reaction: Kc = \(\frac{[CS_{2}][H_{2}]^4}{[CH_{4}][H_{2}S]^2}\) (c) Ni(CO)4(g) ⇌ Ni(s) + 4 CO(g)

Identify reaction types

Since the reactants and products are not all in the gas phase, this is a heterogeneous reaction.

Write the Kc expression

For a heterogeneous reaction, we only include the concentrations of the gaseous components in the Kc expression. For this reaction: Kc = \(\frac{[CO]^4}{[Ni(CO)_{4}]}\) (d) HF(aq) ⇌ H+(aq) + F−(aq)

Identify reaction types

All the reactants and products are in the aqueous phase, making this a homogeneous reaction.

Write the Kc expression

For this reaction: Kc = \(\frac{[H^{+}][F^{-}]}{[HF]}\) (e) 2 Ag(s) + Zn2+(aq) ⇌ 2 Ag+(aq) + Zn(s)

Identify reaction types

Since the reactants and products are not all in the same phase, this is a heterogeneous reaction.

Write the Kc expression

For this heterogeneous reaction, we only include the concentrations of the aqueous components in the Kc expression. For this reaction: Kc = \(\frac{[Ag^{+}]^2}{[Zn^{2+}]}\)

Key concepts

Homogeneous and Heterogeneous Reactions

Understanding the distinction between homogeneous and heterogeneous reactions is foundational in studying chemical reactions and equilibrium.

Homogeneous reactions involve reactants and products that are in the same phase, typically all gases or all in a liquid solution. An example from our exercise is the reaction involving nitrogen oxide gases, where we see them all present as gases, making it a homogeneous reaction. On the other hand, heterogeneous reactions occur between substances in different phases. For example, in the reaction involving nickel tetracarbonyl gas and solid nickel, we have reactants and products in different states of matter: gas and solid, which makes it a heterogeneous reaction.

This distinction is crucial when deriving the equilibrium constant expression, as it dictates which concentrations we need to consider. For homogeneous reactions, concentrations of all reactants and products are included. In contrast, for heterogeneous reactions, the rule of thumb is to omit the concentrations of solids or pure liquids, as their concentration does not change during the reaction. Consequently, the equilibrium expression for heterogeneous reactions only includes the gaseous and aqueous components.

Chemical Equilibrium

When a chemical reaction reaches a state where the rates of the forward and reverse reactions are equal, it has achieved chemical equilibrium. At this point, the concentrations of all reactants and products remain constant over time, though they may not be equal to each other.

The equilibrium constant expression, denoted as \( K_c \), mathematically defines this balance. It is a ratio indicating the relationship between the concentrations of products and reactants raised to the power of their stoichiometric coefficients at equilibrium. The value of \( K_c \) helps predict the direction in which the reaction will proceed to reach equilibrium. If \( K_c \) is large, the reaction tends to favor the formation of products, while a small \( K_c \) suggests a preference for reactants.

The reactions given in the exercise demonstrate several scenarios of equilibrium. Each equilibrium constant expression we derive from these reactions is a snapshot of this balance, reflecting how reactant and product concentrations relate to each other in a system that has reached a state of dynamic equilibrium.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers before the chemical formulas in a balanced chemical equation. They represent the relative quantities of reactants and products involved in a reaction. These coefficients play a fundamental role in calculating the equilibrium constant expression.

In the equilibrium constant expression, the concentrations of the products and reactants are raised to the power of their respective stoichiometric coefficients. This aspect highlights the importance of having a balanced equation before attempting to write the expression for \( K_c \). For instance, in the reaction between methane and hydrogen sulfide, which forms carbon disulfide and hydrogen gas, the stoichiometric coefficients tell us how to form the ratio in the \( K_c \) expression: \( K_c = \frac{[CS_{2}][H_{2}]^4}{[CH_{4}][H_{2}S]^2} \). The coefficients ensure the expression accurately reflects the proportions in which the reacting molecules interact.

Each time we solve for \( K_c \), we're applying these coefficients to understand the chemical system's behavior. Grasping this numerically quantitative relationship helps us predict how changes in concentration, temperature, or pressure may shift the equilibrium position.