Chemical Equilibrium: Principles, Calculations, and Applications

Chemical Equilibrium

Introduction to Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry describing the state in which the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products. Many chemical reactions do not proceed to completion but instead reach a state of dynamic balance.

• Dynamic equilibrium: The condition in which the forward and reverse reactions occur at the same rate, so the concentrations of all species remain constant over time.

• Example: The synthesis of ammonia: $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$

The Equilibrium Constant, $K$, and the Reaction Quotient, $Q$

Reactions at Equilibrium

The equilibrium constant ($K$) expresses the ratio of product and reactant concentrations at equilibrium for a reversible reaction. The reaction quotient ($Q$) is calculated in the same way as $K$, but for any set of concentrations, not just equilibrium values.

• General equilibrium expression: For a reaction $aA + bB \rightleftharpoons cC + dD$:

$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

• Reaction quotient:

$Q_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

• If $Q_c < K_c$, the reaction proceeds forward (toward products).

• If $Q_c > K_c$, the reaction proceeds in reverse (toward reactants).

• If $Q_c = K_c$, the system is at equilibrium.

Equilibrium Calculations and Examples

To determine $K_c$, substitute equilibrium concentrations into the equilibrium expression. For example, for $N_2O_4(g) \rightleftharpoons 2NO_2(g)$:

$K_c = \frac{[NO_2]^2}{[N_2O_4]}$

• Given equilibrium concentrations, calculate $K_c$ directly.

• Given $K_c$ and initial concentrations, use an ICE (Initial, Change, Equilibrium) table to solve for unknowns.

Equilibria Involving Gases

For reactions involving gases, equilibrium can also be expressed in terms of partial pressures ($K_p$):

$K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b}$

The relationship between $K_p$ and $K_c$ is:

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

where $\Delta n$ is the change in moles of gas ($\text{moles of gaseous products} - \text{moles of gaseous reactants}$), $R$ is the gas constant, and $T$ is temperature in Kelvin.

Magnitude of the Equilibrium Constant

• If $K \gg 1$, products are favored at equilibrium (reaction goes nearly to completion).

• If $K \ll 1$, reactants are favored (very little product forms).

• If $K \approx 1$, significant amounts of both reactants and products are present.

Equilibrium Constant Expressions for Heterogeneous Systems

For reactions involving solids and liquids, their concentrations are omitted from the equilibrium expression because they are constant.

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

$K = [CO_2]$

Equilibrium and Gibbs Energy

The Relationship Between $\Delta G^\circ$ and $K$

The standard Gibbs free energy change ($\Delta G^\circ$) is related to the equilibrium constant by:

$\Delta G^\circ = -RT \ln K$

• If $\Delta G^\circ < 0$, $K > 1$ (products favored).

• If $\Delta G^\circ > 0$, $K < 1$ (reactants favored).

• If $\Delta G^\circ = 0$, $K = 1$.

How Systems at Equilibrium Respond to Change

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.

• Adding/removing reactants or products: The system shifts to consume the added component or replace the removed one.

• Changing pressure (gaseous reactions): Increasing pressure shifts equilibrium toward the side with fewer moles of gas; decreasing pressure shifts toward more moles of gas.

• Changing temperature: For endothermic reactions, increasing temperature shifts equilibrium toward products; for exothermic reactions, toward reactants.

• Adding a catalyst: Does not affect the position of equilibrium, only the rate at which equilibrium is achieved.

Calculating Equilibrium Concentrations

Using Initial Concentrations and $K$

To solve for equilibrium concentrations given initial amounts and $K$, use an ICE table:

• List initial concentrations.

• Define changes in terms of $x$ (the amount reacted or formed).

• Write expressions for equilibrium concentrations.

• Substitute into the equilibrium expression and solve for $x$.

Sample ICE Table Structure

	Initial	Change	Equilibrium
[A]	0.500	-x	0.500-x
[B]	0.500	-x	0.500-x
[C]	0	+x	x

Substitute equilibrium values into the $K$ expression and solve for $x$.

Summary Table: Key Equilibrium Concepts

Concept	Definition/Equation
Equilibrium Constant ($K_c$)	$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$
Reaction Quotient ($Q_c$)	Same form as $K_c$, but for any concentrations
Relationship between $K_p$ and $K_c$	$K_p = K_c(RT)^{\Delta n}$
Gibbs Free Energy and $K$	$\Delta G^\circ = -RT \ln K$
Le Châtelier’s Principle	System shifts to counteract disturbances

Additional info:

• Worked examples and practice exercises are included throughout the material to reinforce calculation methods and conceptual understanding.

• Tables and diagrams illustrate the effect of changing conditions on equilibrium systems.