Chemical Equilibrium Principles and Calculations

Dynamic Chemical Equilibrium

• Reversible Reactions: Chemical reactions in which products react to regenerate the original reactants in opposition to the forward process.

• Chemical Equilibrium: The state reached by a reaction mixture when the rates of the forward and reverse reactions become equal ($r_f = r_r$).

• Dynamic Process: At equilibrium, forward and reverse reactions continue at identical rates, resulting in constant concentrations of reactants and products over time.

The Equilibrium Constant

• Law of Mass Action: For a general reaction $a\text{A}(g) + b\text{B}(g) \rightleftharpoons c\text{C}(g) + d\text{D}(g)$, the concentration equilibrium constant $K_c$ is defined as:   $K_c = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b}$

• Kinetics Relation: At dynamic equilibrium, equal rates ($k_f [\text{A}]^a [\text{B}]^b = k_r [\text{C}]^c [\text{D}]^d$) yield the relationship:   $K_c = \frac{k_f}{k_r}$

• Pressure Equilibrium Constant ($K_p$): Formulated for gas-phase reactions using partial pressures:   $K_p = \frac{P_{\text{C}}^c P_{\text{D}}^d}{P_{\text{A}}^a P_{\text{B}}^b}$

• Relation Between $K_p$ and $K_c$:   $K_p = K_c (RT)^{\Delta n}$

  • $\Delta n = (c + d) - (a + b)$ (moles of gaseous products minus moles of gaseous reactants).

  • $R = 0.0821\,\text{L}\,\text{atm}\,\text{K}^{-1}\,\text{mol}^{-1}$.

  • $T$ is the temperature in Kelvin ($\text{K}$).

• Rules for Manipulating Equations and $K_c$:

  • Reversing a chemical equation inverts its equilibrium constant: $K_{\text{rev}} = \frac{1}{K_c}$.

  • Multiplying coefficients by a factor $n$ raises $K_c$ to the power $n$: K_{\text{new}} = K_c^n$.\n - Summing chemical equations multiplies their equilibrium constants: K_3 = K_1 \times K_2$.

Heterogeneous Equilibria

• Heterogeneous Equilibrium: An equilibrium involving reactants and products present in more than one phase.

• Omission of Pure Phases: Concentrations of pure solids $(s)$, pure liquids $(l)$, and solvents in dilute solutions remain constant and are omitted from equilibrium-constant expressions.

• Example: For $3\text{Fe}(s) + 4\text{H}_2\text{O}(g) \rightleftharpoons \text{Fe}_3\text{O}_4(s) + 4\text{H}_2(g)$, the equilibrium expression is:   $K_c = \frac{[\text{H}_2]^4}{[\text{H}_2\text{O}]^4}$

Using the Equilibrium Constant

• Qualitative Interpretation of $K_c$:

  • $K_c \gg 1$: Products strongly favored at equilibrium.

  • $K_c \ll 1$: Reactants strongly favored at equilibrium.

  • $K_c \approx 1$: Substantial amounts of both reactants and products exist at equilibrium.

• Reaction Quotient ($Q_c$): Calculated using non-equilibrium initial concentrations in the equilibrium expression:   $Q_c = \frac{[\text{C}]_i^c [\text{D}]_i^d}{[\text{A}]_i^a [\text{B}]_i^b}$

  • $Q_c < K_c$: Reaction shifts right (forward) toward products.

  • $Q_c > K_c$: Reaction shifts left (reverse) toward reactants.

  • $Q_c = K_c$: System is at equilibrium.

• Equilibrium Calculations: Set up an ICE (Initial, Change, Equilibrium) table in terms of unknown variable $x$, substitute into the $K_c$ expression, and solve algebraically or via the quadratic formula:   $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Le Châtelier's Principle

• Le Châtelier's Principle: When a system at chemical equilibrium is disturbed by a change in concentration, pressure, or temperature, it shifts to counter the applied change.

• Concentration Disturbance:

  • Adding reactant or removing product shifts equilibrium right (forward).

  • Adding product or removing reactant shifts equilibrium left (reverse).

• Pressure / Volume Disturbance (Gas Phase):

  • 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 partial pressures or equilibrium position.

• Temperature Disturbance:

  • Exothermic Reaction ($\Delta H^\circ < 0$): Heat is a product. Raising temperature shifts equilibrium left (lowering $K_c$); lowering temperature shifts right (raising $K_c$).

  • Endothermic Reaction ($\Delta H^\circ > 0$): Heat is a reactant. Raising temperature shifts equilibrium right (raising $K_c$); lowering temperature shifts left (lowering $K_c$).

• Effect of a Catalyst:

  • Increases both forward and reverse rates equally by lowering activation energy.

  • Speeds up attainment of equilibrium without altering $K_c$ or equilibrium composition.