Chemical Equilibria: Principles, Calculations, and Applications

Chapter 15: Chemical Equilibria

Introduction to Chemical Equilibrium

Chemical equilibrium is a fundamental concept in general chemistry, describing the state in which the concentrations of reactants and products remain constant over time. This chapter covers the dynamic nature of equilibrium, how to express equilibrium mathematically, and how changes in conditions affect equilibrium systems.

• Equilibrium processes are dynamic, with forward and reverse reactions occurring at equal rates.

• Factors affecting equilibrium include concentration, pressure, and temperature.

Defining Equilibrium

An equilibrium reaction is one where both reactants and products are present, and their concentrations do not change over time. Although it appears static, equilibrium is actually dynamic, with reactants constantly forming products and vice versa.

• Dynamic equilibrium: The forward and reverse reactions occur at the same rate.

• Equilibrium process: Reactants form products and products form reactants simultaneously.

• Equilibrium notation: Double arrows (⇌) indicate a reversible reaction.

Phase Equilibrium and Reaction Examples

Equilibrium can occur in different phases, such as between liquid and gas. For example, the rate of evaporation equals the rate of condensation at equilibrium:

• $k_{(l \rightarrow g)} = k_{(g \rightarrow l)}$

• For chemical reactions: $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$

Formation of Equilibrium

When a system is not at equilibrium, it will naturally shift towards equilibrium. The direction depends on the initial presence of reactants or products.

• If only reactants are present, products will form.

• If only products are present, reactants will form.

• At equilibrium, both processes occur simultaneously.

Example: Sealed water container:

• Initially only $H_2O(l)$ is present.

• Over time, $H_2O(l)$ evaporates to $H_2O(g)$.

• At equilibrium: $H_2O(l) \rightleftharpoons H_2O(g)$

Graphical Representation of Equilibrium

When graphing the formation of equilibrium, initial concentrations of reactants and products change until equilibrium is reached. The final concentrations depend on the reaction's stoichiometry.

• Example: $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$

• Concentrations at equilibrium are unique for each reaction.

Equilibrium Constant ($K$)

The equilibrium constant ($K$) quantifies the ratio of products to reactants at equilibrium. It is specific to each reaction and depends on temperature.

• $K$ can be expressed in terms of concentration ($K_c$) or pressure ($K_p$).

• General form for a reaction: $aA + bB \rightleftharpoons xX + yY$

Equilibrium constant expression:

• $K_c = \frac{[X]^x[Y]^y}{[A]^a[B]^b}$

• Brackets $[ ]$ denote concentration in molarity (M).

• Stoichiometric coefficients become exponents.

• $K$ is unitless and temperature dependent.

Interpreting the Size of $K$

• If $K = 1$, products and reactants are present in similar amounts.

• If $K > 1$, products predominate.

• If $K < 1$, reactants predominate.

Equilibrium Pressure ($K_p$)

For reactions involving gases, equilibrium can be expressed in terms of partial pressures.

• Ideal gas law: $PV = nRT$ or $P = \frac{n}{V}RT$

• Partial pressure: $P_A = \frac{n_A}{V}RT$

• Equilibrium constant in pressure: $K_p = \frac{P_X^x P_Y^y}{P_A^a P_B^b}$

• Relationship between $K_p$ and $K_c$: $K_p = K_c(RT)^{\Delta n}$, where $\Delta n =$ moles of gaseous products $-$ moles of gaseous reactants.

Calculating Equilibrium Constants

To calculate $K_c$ or $K_p$, use the equilibrium concentrations or pressures of all reactants and products.

• Set up the equilibrium constant expression.

• Plug in the values and solve for $K$.

Heterogeneous Equilibria

When a reaction involves solids or liquids, only the concentrations of aqueous ($aq$) and gaseous ($g$) species are included in the equilibrium expression. Solids and pure liquids are omitted because their concentrations are constant.

• Example: $H_2O(g) + C(s) \rightleftharpoons H_2(g) + CO(g)$

• Equilibrium expression: $K = \frac{[H_2][CO]}{[H_2O]}$ (ignore $C(s)$)

Manipulating Equilibrium Constants

K for Reversed Reactions

If a reaction is reversed, the equilibrium constant becomes its reciprocal:

• Original: $A \rightleftharpoons B$, $K = \frac{[B]}{[A]}$

• Reversed: $B \rightleftharpoons A$, $K' = \frac{[A]}{[B]} = \frac{1}{K}$

K for Different Stoichiometry

If the stoichiometry of a reaction is changed (e.g., multiplied by a factor), the equilibrium constant is raised to that power.

• For $2A \rightleftharpoons B$, $K_1 = \frac{[B]}{[A]^2}$

• For $6A \rightleftharpoons 3B$, $K_2 = \frac{[B]^3}{[A]^6} = (K_1)^3$

K for Added Reactions

When reactions are added together, the overall equilibrium constant is the product of the individual constants.

• If $K_1$ and $K_2$ are for two reactions, then $K_{total} = K_1 \times K_2$

Reaction Quotient ($Q$) and Shifts

The reaction quotient ($Q$) is calculated like $K$, but with current (not necessarily equilibrium) concentrations. Comparing $Q$ to $K$ predicts the direction the reaction will shift:

• If $Q < K$, reaction shifts forward (towards products).

• If $Q > K$, reaction shifts backward (towards reactants).

• If $Q = K$, system is at equilibrium.

ICE Tables (Initial, Change, Equilibrium)

ICE tables are used to organize and solve equilibrium problems involving concentrations or pressures.

Step	Description
Initial	Initial concentrations or pressures
Change	Change in concentration/pressure as reaction proceeds
Equilibrium	Final concentrations/pressures at equilibrium

• Use stoichiometry to relate changes in reactants and products.

• Set up algebraic expressions to solve for unknowns.

Le Châtelier's Principle

Le Châtelier's Principle states that if a system at equilibrium is disturbed, it will shift to counteract the disturbance and restore equilibrium.

• Change in concentration: Adding reactant shifts equilibrium towards products; adding product shifts towards reactants.

• Change in pressure: Increasing pressure shifts equilibrium towards the side with fewer moles of gas; decreasing pressure shifts towards more moles.

• Change in temperature: For endothermic reactions, increasing temperature shifts towards products; for exothermic, towards reactants.

Summary Table: Effects on Equilibrium

Disturbance	Shift Direction	Reason
Add reactant	Towards products	System uses up added reactant
Add product	Towards reactants	System uses up added product
Increase pressure	Towards fewer gas moles	Reduces total pressure
Increase temperature (endothermic)	Towards products	Absorbs added heat
Increase temperature (exothermic)	Towards reactants	Removes excess heat

Key Concepts to Master

• Definition and meaning of equilibrium reactions

• Writing equilibrium expressions for homogeneous and heterogeneous reactions

• Predicting shifts in equilibrium due to changes in concentration, pressure, or temperature

• Calculating $K_c$, $K_p$, and using ICE tables

• Comparing $Q$ and $K$ to predict reaction direction

Additional info: Some examples and tables have been expanded for clarity and completeness. All equations are provided in LaTeX format for academic accuracy.