Chemical Equilibrium: Comprehensive Guide

Chemical equilibrium occurs when reversible reactions balance, with forward and reverse rates equal, stabilizing reactant and product concentrations. For example, in the Haber-Bosch process (\( \ce{N2 + 3H2 <=> 2NH3} \)), ammonia forms and decomposes simultaneously. This MathMultiverse guide explores dynamic equilibrium, equilibrium constants (\( K_c \), \( K_p \)), Le Chatelier’s principle, detailed calculations, and applications in industry, biology, and environmental science, enhanced with interactive visualizations.

Rooted in thermodynamics and pioneered by scientists like Le Chatelier, equilibrium governs reactions like esterification (\( \ce{CH3COOH + C2H5OH <=> CH3COOC2H5 + H2O} \)) or gas dissociation (\( \ce{N2O4 <=> 2NO2} \)). Concentrations remain constant macroscopically, but molecular exchange persists. This guide blends theory, mathematics, and practical relevance for a thorough understanding.

Dynamic Equilibrium

Dynamic equilibrium is achieved when forward and reverse reaction rates equalize in a reversible system, maintaining constant concentrations.

Rate Balance

For \( \ce{2A + B <=> C + D} \):

\[ \text{Rate}_{\text{forward}} = k_f [\ce{A}]^2 [\ce{B}] \] \[ \text{Rate}_{\text{reverse}} = k_r [\ce{C}] [\ce{D}] \] \[ k_f [\ce{A}]^2 [\ce{B}] = k_r [\ce{C}] [\ce{D}] \]

Example: \( \ce{H2 + I2 <=> 2HI} \) stabilizes at specific concentrations.

Concentration vs. Time

Concentration changes for \( \ce{H2 + I2 <=> 2HI} \).

Equilibrium Constants

The equilibrium constant (\( K \)) quantifies the reaction’s extent.

Concentration-Based (\( K_c \))

For \( \ce{aA + bB <=> cC + dD} \):

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

Example: \( \ce{2SO2 + O2 <=> 2SO3} \), at equilibrium:

\[ [\ce{SO2}] = 0.05 \, \text{M}, [\ce{O2}] = 0.02 \, \text{M}, [\ce{SO3}] = 0.90 \, \text{M} \] \[ K_c = \frac{(0.90)^2}{(0.05)^2 (0.02)} = 16200 \]

Pressure-Based (\( K_p \))

For gases:

\[ K_p = \frac{(P_{\ce{C}})^c (P_{\ce{D}})^d}{(P_{\ce{A}})^a (P_{\ce{B}})^b} \] \[ K_p = K_c (RT)^{\Delta n} \]

For \( \ce{N2 + 3H2 <=> 2NH3} \), \( \Delta n = -2 \), at 298 K:

\[ RT = 0.0821 \cdot 298 \approx 24.46 \, \text{L·atm/mol} \]

Reaction Quotient (\( Q \))

For \( \ce{CO + H2O <=> CO2 + H2} \), initial \( 0.1 \, \text{M} \), \( K_c = 1.0 \):

\[ Q_c = \frac{(0)^2}{(0.1)^2} = 0 \]

Shifts right as \( Q < K \).

Le Chatelier’s Principle

Predicts equilibrium shifts under stress.

Concentration

For \( \ce{CO + 2H2 <=> CH3OH} \), increasing \( [\ce{CO}] \):

\[ Q_c = \frac{[\ce{CH3OH}]}{[\ce{CO}][\ce{H2}]^2} \]

Shifts right to form more \( \ce{CH3OH} \).

Pressure

For \( \ce{N2 + 3H2 <=> 2NH3} \), increasing pressure favors \( \ce{NH3} \).

Temperature

Exothermic \( \ce{2NO2 <=> N2O4} \), heating shifts left.

\[ \ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \]

Applications

Ammonia Synthesis

Haber process (\( \ce{N2 + 3H2 <=> 2NH3} \)):

\[ K_p = \frac{(P_{\ce{NH3}})^2}{(P_{\ce{N2}})(P_{\ce{H2}})^3} \]

Oxygen Transport

Hemoglobin: \( \ce{Hb + 4O2 <=> Hb(O2)4} \).

Ocean Chemistry

\( \ce{CO2 + H2O <=> H2CO3} \), rising \( \ce{CO2} \) shifts right, lowering pH.

Drug Solubility

Ibuprofen: \( \ce{HA <=> H+ + A-} \), \( K_a = 1.2 \times 10^{-5} \).