Chemical Equilibria: Principles, Calculations, and Applications

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Chapter 15: Chemical Equilibria

Introduction to Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry describing the state in which the concentrations of reactants and products remain constant over time. This chapter explores the dynamic nature of equilibrium, how to express and calculate equilibrium constants, and the factors that affect equilibrium systems.

Equilibrium processes: Reactions where both forward and reverse processes occur at equal rates.
Factors affecting equilibrium: Concentration, pressure, temperature, and the presence of catalysts.

Defining Equilibrium

Dynamic 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 dynamic: reactants are constantly forming products, and products are constantly reforming reactants at equal rates.

Dynamic process: Both forward and reverse reactions continue to occur.
Constant concentrations: The amounts of reactants and products remain unchanged at equilibrium.

The Equilibrium Process

At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.
Example: For the reaction $\mathrm{N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)}$, both the formation of ammonia and its decomposition occur at the same rate.

Equilibrium Notation: The Arrows

The double arrow ($\rightleftharpoons$) indicates that a reaction is reversible and both forward and reverse reactions are occurring.
Example: $\mathrm{N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)}$

Formation of Equilibrium

How Equilibrium is Established

When a reaction mixture is not at equilibrium, it will naturally shift towards equilibrium. This can occur whether only reactants or only products are initially present.

If only reactants are present, products will form until equilibrium is reached.
If only products are present, reactants will form until equilibrium is reached.
At equilibrium, both forward and reverse processes occur simultaneously.

Graphical Representation

Initial concentrations of reactants and products change over time until they reach constant values at equilibrium.
The relative concentrations at equilibrium depend on the reaction and its stoichiometry.

Equilibrium Constants

The Equilibrium Constant ($K$)

The equilibrium constant ($K$) quantifies the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of their stoichiometric coefficients.

For a general reaction: $aA + bB \rightleftharpoons xX + yY$

The equilibrium constant expression is:

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

Brackets [ ] denote molar concentrations at equilibrium.
Stoichiometric coefficients become exponents.
$K_c$ is unitless and temperature dependent.

Interpreting the Size of $K$

If $K > 1$, products are favored at equilibrium.
If $K < 1$, reactants are favored at equilibrium.
If $K \approx 1$, significant amounts of both reactants and products are present.

Example: Writing Equilibrium Constant Expressions

For $\mathrm{N_2O_4(g) \rightleftharpoons 2NO_2(g)}$:
$K_c = \frac{[NO_2]^2}{[N_2O_4]}$
For $\mathrm{N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)}$:
$K_c = \frac{[NH_3]^2}{[N_2][H_2]^3}$

Equilibrium with Gases: $K_p$ and $K_c$

Equilibrium Pressure

For reactions involving gases, equilibrium can be expressed in terms of partial pressures ($K_p$) instead of concentrations ($K_c$).

The ideal gas law: $PV = nRT$ or $P = \frac{n}{V}RT$
Partial pressure of gas A: $P_A = \frac{n_A}{V}RT$

Relationship Between $K_p$ and $K_c$

$K_p = K_c(RT)^{\Delta n}$

$\Delta n$ = moles of gaseous products - moles of gaseous reactants
$R = 0.08206\ \mathrm{L\ atm\ mol^{-1}\ K^{-1}}$

Example: Calculating $K_p$

Given $K_c$ and temperature, use the above formula to find $K_p$.

Heterogeneous Equilibria

Equilibria Involving Different Phases

Only aqueous ($aq$) and gaseous ($g$) species are included in equilibrium constant expressions. Pure solids ($s$) and pure liquids ($l$) are omitted because their concentrations are constant.

Heterogeneous equilibrium: Equilibrium involving more than one phase (e.g., solid and gas).

Example: Writing $K$ for Heterogeneous Equilibria

For $\mathrm{H_2O(g) + C(s) \rightleftharpoons H_2(g) + CO(g)}$:
$K = \frac{[H_2][CO]}{[H_2O]}$ (omit $C(s)$)

Manipulating Equilibrium Constants

$K$ for Reversed Reactions

Reversing a reaction inverts the equilibrium constant:
If $K$ for $A \rightleftharpoons B$ is $K$, then for $B \rightleftharpoons A$ it is $1/K$.

$K$ for Different Stoichiometry

If the coefficients in a balanced equation are multiplied by $n$, the new $K$ is the original $K$ raised to the $n$th power.
Example: If $K_1$ for $2A \rightleftharpoons B$ is $K_1$, then for $6A \rightleftharpoons 3B$, $K_2 = K_1^3$.

Combining Equilibria

When adding two or more reactions, multiply their $K$ values to get the overall $K$ for the net reaction.

Reaction Quotient ($Q$) and Predicting Shifts

Reaction Quotient ($Q$)

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

If $Q < K$, the reaction shifts right (toward products).
If $Q > K$, the reaction shifts left (toward reactants).
If $Q = K$, the system is at equilibrium.

ICE Tables: Calculating Equilibrium Concentrations

ICE Table Method

ICE stands for Initial, Change, and Equilibrium. ICE tables help organize data and solve for unknown concentrations at equilibrium.

Initial: Starting concentrations or pressures.
Change: Amounts by which concentrations change (often represented by $x$).
Equilibrium: Final concentrations, calculated using initial values and changes.

Example ICE Table

Species	Initial (M)	Change (M)	Equilibrium (M)
A	0.200	-x	0.200 - x
B	0	+x	x

Use the equilibrium constant expression and the ICE table to solve for $x$ and thus the equilibrium concentrations.

Le Châtelier's Principle

Predicting the Effect of Changes on Equilibrium

Le Châtelier's Principle states that if a system at equilibrium is disturbed, it will shift in the direction that counteracts the disturbance.

Concentration: Adding reactant shifts equilibrium toward products; removing reactant shifts toward reactants.
Pressure (for gases): Increasing pressure shifts equilibrium toward the side with fewer moles of gas; decreasing pressure shifts toward more moles of gas.
Temperature: For endothermic reactions, increasing temperature shifts equilibrium toward products; for exothermic, toward reactants.

Example: Le Châtelier's Principle and Pressure

For $\mathrm{N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)}$, increasing pressure shifts equilibrium toward ammonia (fewer moles of gas).

Example: Le Châtelier's Principle and Temperature

For an endothermic reaction: $\mathrm{Reactants + heat \rightleftharpoons Products}$, increasing temperature favors product formation.
For an exothermic reaction: $\mathrm{Reactants \rightleftharpoons Products + heat}$, increasing temperature favors reactant formation.

Summary Table: Factors Affecting Equilibrium

Change	System Response
Add reactant	Shifts toward products
Remove reactant	Shifts toward reactants
Add product	Shifts toward reactants
Remove product	Shifts toward products
Increase pressure (gases)	Shifts toward fewer moles of gas
Decrease pressure (gases)	Shifts toward more moles of gas
Increase temperature (endothermic)	Shifts toward products
Increase temperature (exothermic)	Shifts toward reactants

Key Concepts to Master

Definition and characteristics of equilibrium reactions
Writing equilibrium constant expressions for homogeneous and heterogeneous reactions
Calculating $K_c$ and $K_p$
Using ICE tables to solve for equilibrium concentrations
Predicting the effect of changes in concentration, pressure, and temperature using Le Châtelier's Principle
Understanding the relationship between $Q$ and $K$