Chemical Equilibrium: Principles, Calculations, and Applications

Ch. 15: Chemical Equilibrium

The Concept of Equilibrium

Chemical equilibrium is a fundamental concept in chemistry describing the state in which the concentrations of all reactants and products remain constant over time. This occurs because the forward and reverse reactions proceed at the same rate.

• Chemical equilibrium is reached when the rate of the forward reaction equals the rate of the reverse reaction.

• At equilibrium, the concentrations of reactants and products do not change, although both reactions continue to occur.

• Example: The decomposition of dinitrogen tetroxide to nitrogen dioxide:

$\mathrm{N_2O_4(g) \rightleftharpoons 2NO_2(g)}$

• Double arrows ($\rightleftharpoons$) indicate a reversible reaction at equilibrium.

Reversible Reactions

Most chemical reactions are reversible, meaning they can proceed in both the forward and reverse directions.

• Reversible reaction: A reaction that occurs simultaneously in both directions.

• Examples:

  • $\mathrm{N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)}$ (Haber process)

  • $\mathrm{CO(g) + 2H_2(g) \rightleftharpoons CH_3OH(g)}$

  • $\mathrm{N_2O_4(g) \rightleftharpoons 2NO_2(g)}$

Dynamic Nature of Equilibrium

At equilibrium, both the forward and reverse reactions continue to occur, but there is no net change in the concentrations of reactants and products.

• Equilibrium can be reached starting from either reactants or products.

• The relative amounts of substances at equilibrium depend on the stoichiometry of the reaction.

The Equilibrium Constant

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: $aA + bB \rightleftharpoons dD + eE$

• The equilibrium constant expression is:

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

• For reactions involving gases, the equilibrium constant can also be expressed in terms of partial pressures:

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

• 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)

Magnitude and Interpretation of Equilibrium Constants

The value of the equilibrium constant provides insight into the composition of the equilibrium mixture.

• If $K \gg 1$, products predominate at equilibrium (equilibrium lies to the right).

• If $K \ll 1$, reactants predominate at equilibrium (equilibrium lies to the left).

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

Direction of the Chemical Equation and $K$

The equilibrium constant for a reaction written in the reverse direction is the reciprocal of the equilibrium constant for the forward reaction.

• Example:

$\mathrm{N_2O_4(g) \rightleftharpoons 2NO_2(g)}$    $K_c = 0.212$ at $100^\circ$C $\mathrm{2NO_2(g) \rightleftharpoons N_2O_4(g)}$    $K_c = \frac{1}{0.212} = 4.72$ at $100^\circ$C

Stoichiometry and Equilibrium Constants

Changing the stoichiometric coefficients in a balanced equation affects the value of the equilibrium constant.

• If the equation is multiplied by $n$, raise $K$ to the $n$th power.

• If the equation is divided by $n$, take the $n$th root of $K$.

Combining Equilibrium Expressions

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

• Example:

$K_3 = K_1 \times K_2$

Homogeneous and Heterogeneous Equilibria

Equilibria are classified based on the phases of the reactants and products.

• Homogeneous equilibrium: All reactants and products are in the same phase.

• Heterogeneous equilibrium: Reactants and/or products are in different phases.

• Concentrations of pure solids and liquids are not included in the equilibrium constant expression.

• Example: $\mathrm{CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)}$ $K_c = [CO_2]$ and $K_p = P_{CO_2}$

Calculating Equilibrium Constants

To determine the equilibrium constant from experimental data:

1. Tabulate all known initial and equilibrium concentrations.

2. Calculate the change in concentration for each species.

3. Use stoichiometry to relate changes among all species.

4. Determine equilibrium concentrations.

5. Substitute values into the equilibrium constant expression.

Example Table: Initial and Equilibrium Concentrations

Experiment	Initial [N2O4]	Initial [NO2]	Equilibrium [N2O4]	Equilibrium [NO2]	Kc
1	0.0	0.0200	0.00140	0.0172	0.211
2	0.0	0.0300	0.00260	0.0243	0.211
3	0.0	0.0400	0.00452	0.0310	0.213
4	0.0200	0.0	0.00452	0.0310	0.213

Reaction Quotient ($Q$) and Predicting Direction

The reaction quotient ($Q$) is calculated using the same expression as $K$, but with current (not necessarily equilibrium) concentrations.

• If $Q < K$, the reaction proceeds forward (toward products).

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

• If $Q > K$, the reaction proceeds in reverse (toward reactants).

$Q = \frac{[P]^p[Q]^q}{[A]^a[B]^b}$

Calculating Equilibrium Concentrations

Given initial concentrations and $K$, equilibrium concentrations can be found by defining changes in terms of a variable (often $x$) and solving the resulting equation, sometimes using the quadratic formula.

• Set up an ICE (Initial, Change, Equilibrium) table.

• Write the equilibrium constant expression in terms of $x$.

• Solve for $x$ and substitute back to find equilibrium concentrations.

Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Le Chatelier’s Principle

Le Chatelier’s Principle predicts how a system at equilibrium responds to disturbances:

• If a system at equilibrium is disturbed by a change in concentration, pressure, or temperature, it will shift to counteract the disturbance and restore equilibrium.

• Changing concentration: Adding reactant or product shifts equilibrium to consume the added component.

• Changing pressure/volume: For gaseous equilibria, increasing pressure (decreasing volume) shifts equilibrium toward the side with fewer moles of gas.

• Changing temperature: For endothermic reactions, increasing temperature shifts equilibrium toward products (and increases $K$); for exothermic reactions, increasing temperature shifts equilibrium toward reactants (and decreases $K$).

• Catalysts: Speed up the attainment of equilibrium but do not affect the equilibrium position or $K$.

Summary Table: Effects of Disturbances on Equilibrium

Disturbance	Direction of Shift	Effect on $K$
Increase [Reactant]	Toward products	None
Decrease [Reactant]	Toward reactants	None
Increase pressure (decrease volume)	Toward fewer moles of gas	None
Increase temperature (endothermic)	Toward products	Increases $K$
Increase temperature (exothermic)	Toward reactants	Decreases $K$
Add catalyst	No shift	None

Example Application: The Haber process for ammonia synthesis ($\mathrm{N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)}$) is a classic industrial example of equilibrium manipulation using pressure, temperature, and catalysts to maximize yield.

Additional info: These notes are based on lecture slides and textbook content for a General Chemistry course, focusing on the principles and calculations of chemical equilibrium.