Chemical Equilibrium: Principles, Calculations, and Le Châtelier’s Principle

Chapter 15: Chemical Equilibrium

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 in a reversible reaction. This chapter explores the nature of equilibrium, the equilibrium constant, and how systems respond to changes according to Le Châtelier’s Principle.

What is Equilibrium?

• Definition: Equilibrium is the state in a reversible chemical reaction where the concentrations of reactants and products remain unchanged over time.

• At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.

• Equilibrium does not mean the concentrations of reactants and products are equal; rather, their rates are equal.

• Reactions that can proceed in both directions are called reversible reactions and are represented by a double arrow (\( \rightleftharpoons \)).

• Kinetics studies how fast a reaction proceeds, while equilibrium concerns the final state where rates are balanced.

• At equilibrium, both reactants and products are present, and their concentrations remain constant (dynamic equilibrium).

Example: For the reaction 2 Red \( \rightleftharpoons \) Blue, the concentration of Red decreases and Blue increases until both stabilize, indicating equilibrium.

Position of Equilibrium

• The position of equilibrium describes the relative amounts of reactants and products at equilibrium.

• If almost all reactants are converted to products, equilibrium favors the products.

• If little reactant is converted, equilibrium favors the reactants.

The Equilibrium Constant (K)

• The equilibrium constant (K) quantifies the ratio of product and reactant concentrations at equilibrium.

• For a general reaction: \( aA + bB \rightleftharpoons cC + dD \), the Law of Mass Action gives:

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

• K is unitless and always written as products over reactants, each raised to the power of their coefficients.

• Kc uses concentrations (mol/L), Kp uses partial pressures (atm).

Example: For 2 N2O5 \( \rightleftharpoons \) 4 NO2 + O2:

$K = \frac{[NO_2]^4 [O_2]}{[N_2O_5]^2}$

Interpreting the Value of K

• If \( K \gg 1 \): Products are favored at equilibrium (more products than reactants).

• If \( K \ll 1 \): Reactants are favored at equilibrium (more reactants than products).

Relationships Between K and Chemical Equations

• Reversing the reaction inverts K: \( K_{reverse} = \frac{1}{K_{forward}} \)

• Multiplying coefficients by n raises K to the nth power: \( K_{new} = (K_{original})^n \)

• Adding equations multiplies their K values: \( K_{overall} = K_1 \times K_2 \)

Example Table: Effects of Manipulating Chemical Equations on K

Operation	Effect on K
Reverse equation	\( K_{new} = 1/K_{original} \)
Multiply equation by n	\( K_{new} = (K_{original})^n \)
Add equations	\( K_{new} = K_1 \times K_2 \)

Kp and Kc: Gaseous Equilibria

• For gas-phase reactions, K can be expressed in terms of partial pressures (Kp):

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

• Kp and Kc are related by:

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

• \( \Delta n = \) moles of gaseous products minus moles of gaseous reactants.

• Kp = Kc when \( \Delta n = 0 \).

Solids and Liquids in Equilibrium Expressions

• Pure solids and liquids are not included in equilibrium constant expressions.

• Only concentrations of gases and aqueous species appear in K expressions.

Example: For aA(s) + bB(aq) \( \rightleftharpoons \) cC(l) + dD(aq):

$K = \frac{[D]^d}{[B]^b}$

Calculating Equilibrium Constants from Measured Concentrations

• Measure equilibrium concentrations of all species and substitute into the K expression.

• K is constant at a given temperature, regardless of initial concentrations.

Example Table: Calculating Kc for H2(g) + I2(g) \( \rightleftharpoons \) 2HI(g)

Initial [H2]	Initial [I2]	Initial [HI]	Equilibrium [H2]	Equilibrium [I2]	Equilibrium [HI]
0.50	0.50	0.0	0.11	0.11	0.78
0.0	0.0	0.50	0.055	0.055	0.39
0.50	0.50	0.50	0.165	0.165	1.17
1.0	0.5	0.0	0.53	0.033	0.934

ICE Tables and Equilibrium Calculations

• Use ICE tables (Initial, Change, Equilibrium) to organize data and solve for unknown concentrations.

• Type 1: Given initial concentrations and one equilibrium concentration, calculate all others.

• Type 2: Given K and initial concentrations, solve for equilibrium concentrations (may require solving quadratic equations).

Example: For 2A + B \( \rightleftharpoons \) 4C, if [A]initial = 1.00 M, [B]initial = 1.00 M, [C]initial = 0, and [C]eq = 0.50 M, use stoichiometry to find [A]eq and [B]eq.

Approximations in Equilibrium Calculations

• If K is very small and initial concentrations are large, the change in concentration (x) may be negligible.

• Approximation is valid if x is less than 5% of the initial concentration.

$\text{If } \frac{x}{[X]_{initial}} \times 100\% < 5\%, \text{ the approximation is valid.}$

The Reaction Quotient (Q)

• Reaction quotient (Q): Calculated like K, but with current (not necessarily equilibrium) concentrations.

• For aA + bB \( \rightleftharpoons \) cC + dD:

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

• Compare Q to K to predict the direction of reaction:

  • If Q > K: Reaction proceeds in the reverse direction (products to reactants).

  • If Q < K: Reaction proceeds in the forward direction (reactants to products).

  • If Q = K: System is at equilibrium.

  • If only reactants are present, Q = 0; if only products, Q = ∞.

Le Châtelier’s Principle

• Le Châtelier’s Principle: If a system at equilibrium is disturbed, it will shift to counteract the disturbance and re-establish equilibrium (with the same K, unless temperature changes).

• Disturbances include changes in concentration, pressure/volume, or temperature.

Effect of Concentration Changes

• Adding reactant: Shifts equilibrium toward products.

• Removing product: Shifts equilibrium toward products.

• Removing reactant or adding product: Shifts equilibrium toward reactants.

Effect of Pressure and Volume Changes (Gases Only)

• Decreasing volume (increasing pressure): Shifts equilibrium toward the side with fewer moles of gas.

• Increasing volume (decreasing pressure): Shifts equilibrium toward the side with more moles of gas.

• Adding an inert gas at constant volume does not affect equilibrium position.

Effect of Temperature Changes

• Treat heat as a product (exothermic) or reactant (endothermic).

• Increasing temperature:

  • Exothermic: Shifts equilibrium toward reactants; K decreases.

  • Endothermic: Shifts equilibrium toward products; K increases.

• Decreasing temperature has the opposite effect.

Effect of Catalysts

• Catalysts lower the activation energy for both forward and reverse reactions equally.

• Catalysts do not affect the position of equilibrium or the value of K; they only help the system reach equilibrium faster.

Summary Table: Le Châtelier’s Principle Effects

Change	Effect on Equilibrium
Add reactant	Shifts toward products
Remove reactant	Shifts toward reactants
Add product	Shifts toward reactants
Remove product	Shifts toward products
Decrease volume (gases)	Shifts toward fewer moles of gas
Increase volume (gases)	Shifts toward more moles of gas
Increase temperature (exothermic)	Shifts toward reactants; K decreases
Increase temperature (endothermic)	Shifts toward products; K increases
Add catalyst	No effect on equilibrium position

Key Equations

• General equilibrium constant (concentration): $K = \frac{[C]^c [D]^d}{[A]^a [B]^b}$

• General equilibrium constant (partial pressure): $K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b}$

• Relationship between Kp and Kc: $K_p = K_c (RT)^{\Delta n}$

• Reaction quotient: $Q = \frac{[C]^c [D]^d}{[A]^a [B]^b}$

Additional info: For more complex equilibrium calculations, quadratic equations or approximations may be required. Always check the validity of approximations using the 5% rule.