Question

Write the expressions for \(K_{c}\) for the following reactions. In each case indicate whether the reaction is homogeneous or heterogeneous. (a) \(2 \mathrm{O}_{3}(g) \rightleftharpoons 3 \mathrm{O}_{2}(g)\) (b) \(\mathrm{Ti}(s)+2 \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{TiCl}_{4}(l)\) (c) \(2 \mathrm{C}_{2} \mathrm{H}_{4}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+\mathrm{O}_{2}(g)\) (d) \(\mathrm{C}(s)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{4}(g)\) (e) \(4 \mathrm{HCl}(a q)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{H}_{2} \mathrm{O}(l)+2 \mathrm{Cl}_{2}(g)\)

Short answer

(a) \(K_c = \dfrac{[O_2]^3}{[O_3]^2}\) (homogeneous) (b) \(K_c = \dfrac{[\mathrm{TiCl}_{4}]}{[\mathrm{Cl}_{2}]^2}\) (heterogeneous) (c) \(K_c = \dfrac{[\mathrm{C}_{2} \mathrm{H}_{6}]^2 [O_2]}{[\mathrm{C}_{2} \mathrm{H}_{4}]^2 [H_2O]^2}\) (homogeneous) (d) \(K_c = \dfrac{[\mathrm{CH}_{4}]}{[\mathrm{H}_{2}]^2}\) (heterogeneous) (e) \(K_c = \dfrac{[\mathrm{H}_2 \mathrm{O}]^2 [\mathrm{Cl}_2]^2}{[\mathrm{HCl}]^4 [O_2]}\) (heterogeneous)

Step-by-step solution

Write the \(K_c\) expression

For a given balanced chemical equation, \(K_c\) can be expressed as the ratio of the product of the concentrations of the products raised to their stoichiometric coefficients, to the product of the concentrations of the reactants raised to their stoichiometric coefficients. So, for Reaction 1: \[K_c = \dfrac{[O_2]^3}{[O_3]^2}\]

Homogeneous or heterogeneous

Since all reactants and products are gases (indicated by the (g) subscript), Reaction 1 is homogeneous. (b) Reaction 2: \(\mathrm{Ti}(s)+2 \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{TiCl}_{4}(l)\)

Write the \(K_c\) expression

For Reaction 2: \[K_c = \dfrac{[\mathrm{TiCl}_{4}]}{[\mathrm{Cl}_{2}]^2}\] Note that the solid Ti does not appear in the \(K_c\) expression, as its concentration doesn't change during the reaction.

Homogeneous or heterogeneous

Since Reaction 2 has a solid, a gas, and a liquid, it is heterogeneous. (c) Reaction 3: \(2 \mathrm{C}_{2} \mathrm{H}_{4}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+\mathrm{O}_{2}(g)\)

Write the \(K_c\) expression

For Reaction 3: \[K_c = \dfrac{[\mathrm{C}_{2} \mathrm{H}_{6}]^2 [O_2]}{[\mathrm{C}_{2} \mathrm{H}_{4}]^2 [H_2O]^2}\]

Homogeneous or heterogeneous

Since all reactants and products are gases, Reaction 3 is homogeneous. (d) Reaction 4: \(\mathrm{C}(s)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{4}(g)\)

Write the \(K_c\) expression

For Reaction 4: \[K_c = \dfrac{[\mathrm{CH}_{4}]}{[\mathrm{H}_{2}]^2}\] Again, the solid C does not appear in the \(K_c\) expression as its concentration doesn't change during the reaction.

Homogeneous or heterogeneous

Since Reaction 4 has a solid and gases, it is heterogeneous. (e) Reaction 5: \(4 \mathrm{HCl}(a q)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{H}_{2} \mathrm{O}(l)+2 \mathrm{Cl}_{2}(g)\)

Write the \(K_c\) expression

For Reaction 5: \[K_c = \dfrac{[\mathrm{H}_2 \mathrm{O}]^2 [\mathrm{Cl}_2]^2}{[\mathrm{HCl}]^4 [O_2]}\]

Homogeneous or heterogeneous

Since Reaction 5 has aqueous, gas, and liquid components, it is heterogeneous.

Key concepts

Equilibrium Constant Expression

The equilibrium constant expression is a mathematical representation of a chemical equilibrium state. It quantifies the relative concentrations of reactants and products involved in a chemical reaction at equilibrium. In a chemical equation of the form \(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant (\(K_c\)) is given as \[K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}\] where \([A]\), \([B]\), \([C]\), and \([D]\) are the molar concentrations of the reactants and products, and \(a\), \(b\), \(c\), and \(d\) their respective stoichiometric coefficients in the balanced equation.
H4 Why Exclude Solids and Pure Liquids?Inhomogeneous reactions involving phases such as solids or pure liquids, their concentration remains constant throughout the reaction as their density does not change. These are excluded from the equilibrium constant expression because their concentrations are not part of the dynamic equilibrium between reactants and products in the reaction mixture.

For example, in the reaction between titanium and chlorine gas forming titanium tetrachloride, \(Ti(s) + 2Cl_2(g) \rightleftharpoons TiCl_4(l)\), the solid titanium (\(Ti\)) is excluded from the \(K_c\) expression, resulting in \[K_c = \frac{[TiCl_4]}{[Cl_2]^2}\]. This distinction is crucial for accurately calculating the equilibrium concentrations of the participating substances.

Homogeneous and Heterogeneous Reactions

Chemical reactions are classified as either homogeneous or heterogeneous, depending on whether the reactants and products exist in the same phase or in different phases.
Homogeneous reactions are those where all reactants and products are in the same phase, typically all gases or all liquids. This uniformity facilitates the expression of the equilibrium constant as a simple ratio of concentrations. For instance, the reaction involving ozone and oxygen, \(2O_3(g) \rightleftharpoons 3O_2(g)\), is a homogeneous reaction as both ozone and oxygen are in the gaseous phase.

On the other hand, heterogeneous reactions occur between reactants in different phases, like solids, liquids, gases, or even solutions. In our example of titanium and chlorine, we have a solid reactant and gaseous reactants combining to form a liquid product, defining it as a heterogeneous reaction. This multiphase nature of heterogeneous reactions introduces complexity in equilibrium expressions, as phase-bound substances like solids and pure liquids are not included in the equilibrium constant expression, while gases and species in solution are.

Stoichiometry in Chemical Equations

Stoichiometry plays a pivotal role in understanding chemical equations and the resulting equilibrium constant expressions. It refers to the quantitative relationship between reactants and products in a chemical reaction. Stoichiometric coefficients, the numbers before each species in a balanced chemical equation, indicate the proportionate amounts of each substance involved. These coefficients are used to derive the terms in the equilibrium constant expression.
In the equilibrium constant expression, the concentrations of products and reactants are raised to the power of their respective stoichiometric coefficients. In the chemical reaction \(2C_2H_4(g) + 2H_2O(g) \rightleftharpoons 2C_2H_6(g) + O_2(g)\), the stoichiometric coefficients are essential in assembling the equilibrium expression: \[K_c = \frac{[C_2H_6]^2 [O_2]}{[C_2H_4]^2 [H_2O]^2}\].
Careful balancing of the chemical equation is crucial, as any mistake in the coefficients would lead to an incorrect equilibrium expression, affecting the computation of the equilibrium constant and the understanding of the reaction's dynamics.