Equilibrium Constants and Heterogeneous Equilibria

Equilibrium Constants and Their Applications

Understanding the Magnitude and Direction of K

The equilibrium constant, K, is a fundamental concept in chemical equilibrium that quantifies the ratio of product concentrations to reactant concentrations at equilibrium. The value of K provides insight into the preferred direction of a chemical reaction.

• Definition of K: For a general reaction, the equilibrium constant is given by:

$K = \frac{[\text{products}]}{[\text{reactants}]}$

• If K > 1: The equilibrium favors the products. The reaction proceeds predominantly to the right, and the equilibrium "lies to the right."

• If K < 1: The equilibrium favors the reactants. The reaction proceeds predominantly to the left, and the equilibrium "lies to the left."

Mathematical Properties of the Equilibrium Constant

The equilibrium constant can be manipulated mathematically when reactions are reversed, multiplied, or combined. These properties are essential for solving complex equilibrium problems.

• Reversing a Reaction: When the direction of a reaction is reversed, the new equilibrium constant is the reciprocal of the original.

$K_2 = \frac{1}{K_1}$

• Multiplying a Reaction by a Factor: When a reaction is multiplied by a coefficient n, the equilibrium constant is raised to the nth power.

$K_2 = (K_1)^n$

• Combining Reactions: When two or more reactions are added, the overall equilibrium constant is the product of the individual constants.

$K_3 = K_1 \times K_2$

• Example: For the reactions:

Equation	K Value
2 NOBr → 2 NO + Br2	0.014
Br2 + Cl2 → 2 BrCl	7.2
2 NOBr + Cl2 → 2 NO + 2 BrCl	0.014 × 7.2 = 0.10

Combining Equilibrium Constants: Worked Example

Given the following reactions and their equilibrium constants:

• $2 NH_3(g) + 3 I_2(g) \rightleftharpoons 6 HI(g) + N_2(g)$, $K_1 = 54.0$

• $2 NH_3(g) \rightleftharpoons N_2(g) + 3 H_2(g)$, $K_2 = 1.04 \times 10^{-2}$

To find the equilibrium constant for:

$6 HI(g) \rightleftharpoons 3 H_2(g) + 3 I_2(g)$

Combine the reactions appropriately and multiply/divide the K values as needed:

$K = \frac{K_1}{K_2} = 5.1 \times 10^3$

Heterogeneous Equilibria

Law of Mass Action for Heterogeneous Systems

Heterogeneous equilibria involve reactions with more than one phase (solid, liquid, gas). The law of mass action applies, but with an important modification: the concentrations (or partial pressures) of pure solids and pure liquids are omitted from the equilibrium expression.

• Example: For the reaction:

$CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$

The equilibrium constant expression is:

$K = [CO_2]$

• Only the gaseous component is included; the solids are omitted.

Writing Equilibrium Expressions

• General Rule: Only include concentrations (or partial pressures) of gases and aqueous species in the equilibrium expression.

• Example: For $CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$:

$K = [CO_2]$

• For a reaction involving only gases:

$K_p = \frac{(P_{HI})^2}{P_{H_2} (P_{I_2})}$

Assessing Heterogeneous Equilibrium

To establish equilibrium in a heterogeneous system, all necessary reactants and products must be present in the correct phases. For example, in the reaction:

$3 Fe(s) + 4 H_2O(g) \rightleftharpoons Fe_3O_4(s) + 4 H_2(g)$

• Adding $H_2(g)$ to a closed container with $Fe_3O_4(s)$ and $H_2O(g)$ allows equilibrium to be established.

Summary Table: Key Properties of Equilibrium Constants

Operation	Effect on K
Reverse reaction	$K_{new} = 1/K_{original}$
Multiply reaction by n	$K_{new} = (K_{original})^n$
Add reactions	$K_{overall} = K_1 \times K_2$
Omit solids/liquids	Do not include in K expression

Additional info: These principles are foundational for understanding chemical equilibrium, predicting reaction direction, and solving equilibrium problems in both homogeneous and heterogeneous systems.