Chemical Equilibrium & Reactions

Chemical equilibrium describes the state in which forward and reverse reaction rates are equal, resulting in constant macroscopic concentrations despite continued molecular-level reactions. This note covers equilibrium thermodynamics, acid-base chemistry, solubility, kinetics, combustion, and electrochemistry.

1. Chemical Equilibrium

Dynamic Equilibrium Concept

At equilibrium, the system reaches a state where the rate of the forward reaction equals the rate of the reverse reaction. For a reversible reaction:

aA + bB \right leftharpoons cC + dD

The system approaches equilibrium from either direction and remains unchanged at the macroscopic level, though individual molecules continue to react. This is characterized by constant concentrations, constant pressure (if gaseous), and no observable property changes.

Equilibrium Constant K

The equilibrium constant is derived from thermodynamic considerations. At equilibrium, the Gibbs free energy change is zero. For the above reaction, the equilibrium expression is:

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

where $[X]$denotes concentration in mol/L at equilibrium. For gas-phase reactions, use partial pressures:

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

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

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

where $\Delta n = (c + d) - (a + b)$is the change in moles of gas. The magnitude of K indicates reaction extent: $K \gg 1$favors products, $K \ll 1$favors reactants.

2. Le Chatelier’s Principle

When a system at equilibrium is disturbed, it shifts to counteract the disturbance and re-establish equilibrium.

Concentration Changes: Increasing reactant concentration shifts equilibrium toward products; decreasing it shifts toward reactants. Products respond oppositely.

Pressure and Volume: For reactions with $\Delta n \neq 0$, increasing pressure shifts equilibrium toward the side with fewer gas moles (decreases $n$). For $\Delta n = 0$, pressure has no effect on K but affects absolute concentrations.

Temperature: Changes in temperature alter K itself (unlike concentration and pressure, which do not). For exothermic reactions ($\Delta H < 0$), increasing temperature shifts equilibrium toward reactants (decreases K). For endothermic reactions ($\Delta H > 0$), increasing temperature shifts toward products (increases K).

Catalysts: Catalysts accelerate both forward and reverse reactions equally; they do not change K or shift equilibrium, only the rate at which it is reached.

3. Gibbs Free Energy and Equilibrium

The relationship between reaction quotient Q and standard Gibbs free energy is:

$\Delta G = \Delta G^\circ + RT \ln Q$

where $\Delta G^\circ$is the standard free energy change, R = 8.314 J/(mol·K), T is absolute temperature (K), and Q is the reaction quotient:

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

At equilibrium, $\Delta G = 0$and $Q = K$, yielding:

$0 = \Delta G^\circ + RT \ln K$

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

This fundamental relationship connects equilibrium position to thermodynamic favorability. If $\Delta G^\circ < 0$, the reaction is spontaneous and $K > 1$. If $\Delta G^\circ > 0$, $K < 1$.

4. Acid-Base Chemistry

Brønsted-Lowry Theory

An acid is a proton (H⁺) donor and a base is a proton acceptor. Water autoionization illustrates this:

2H_2O \right leftharpoons H_3O^+ + OH^-

with equilibrium constant:

$K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14} \text{ at } 25°C$

pH is defined as:

$\text{pH} = -\log[H^+], \quad \text{pOH} = -\log[OH^-], \quad \text{pH} + \text{pOH} = 14$

Weak Acid and Base Equilibria

For a weak acid HA:

HA \right leftharpoons H^+ + A^-

The acid dissociation constant is:

$K_a = \frac{[H^+][A^-]}{[HA]}$

Similarly, for a weak base B:

B + H_2O \right leftharpoons BH^+ + OH^-

$K_b = \frac{[BH^+][OH^-]}{[B]}$

The relationship between Ka and Kb for a conjugate pair is:

$K_a \times K_b = K_w = 1.0 \times 10^{-14}$

Buffer Systems

A buffer resists pH change when small amounts of acid or base are added. The Henderson-Hasselbalch equation describes buffer pH:

$\text{pH} = \text{p}K_a + \log\frac{[A^-]}{[HA]}$

Effective buffering occurs when pH ≈ pKa (within ±1 unit). Buffers are most effective when $[A^-] \approx [HA]$.

5. Solubility Equilibria

Solubility Product Constant

For a sparingly soluble salt:

AB_n(s) \right leftharpoons A^{n+}(aq) + nB^-(aq)

the solubility product constant is:

$K_{sp} = [A^{n+}][B^-]^n$

Ksp is constant at a given temperature and independent of the concentration of other ions (pure water). The salt precipitates when the ion product Q > Ksp and dissolves when Q < Ksp.

Common Ion Effect

Addition of a common ion (one already present in the dissolution equilibrium) shifts equilibrium toward the solid, decreasing solubility. For example, adding NaCl to a NaF solution decreases NaF solubility.

Precipitation and Selective Precipitation

Two salts with different Ksp values can be separated by controlling [precipitating ion]. For instance, in qualitative analysis, Ag⁺ selectively precipitates Cl⁻ (AgCl, Ksp = 1.8×10⁻¹⁰) before Br⁻ (AgBr, Ksp = 5.0×10⁻¹³ requires lower [Ag⁺]).

6. Reaction Kinetics and Connection to Equilibrium

Rate Laws and Rate Constants

The rate of a reaction is proportional to reactant concentrations raised to experimental powers:

$\text{Rate} = k[A]^m[B]^n$

where k is the rate constant and m, n are reaction orders (determined experimentally, not from stoichiometry). Units of k depend on overall order.

Arrhenius Equation

Temperature dependence of the rate constant follows the Arrhenius equation:

$k = Ae^{-E_a/RT}$

or in logarithmic form:

$\ln k = \ln A - \frac{E_a}{RT}$

where Ea is activation energy (J/mol), A is the pre-exponential factor, and R is the gas constant. A 10°C rise typically increases k by a factor of 2-4.

Kinetics and Thermodynamics

Thermodynamics determines whether a reaction can occur (K > 1 means $\Delta G < 0$); kinetics determines how fast it occurs (Ea). A reaction may be thermodynamically favorable but kinetically slow. Catalysts lower Ea without changing K, accelerating both forward and reverse reactions equally to reach equilibrium faster.

7. Combustion Stoichiometry

Balanced Combustion Equations

Combustion of hydrocarbons produces CO₂ and H₂O:

$C_xH_y + \left(x + \frac{y}{4}\right)O_2 \rightarrow xCO_2 + \frac{y}{2}H_2O$

For complete combustion in excess O₂, no reactants remain. Incomplete combustion (limited O₂) produces CO and C instead of CO₂.

Enthalpy of Combustion

$\Delta H_{\text{comb}} = \Sigma(\text{bond energies broken}) - \Sigma(\text{bond energies formed})$

Enthalpy of combustion is typically negative (exothermic). Standard values are tabulated for common fuels. Bomb calorimetry determines experimental values at constant volume.

Adiabatic Flame Temperature

In an adiabatic combustion process (no heat loss), all energy released heats the products:

$\Delta H_{\text{comb}} = n_{\text{products}} C_p \Delta T$

Adiabatic flame temperature is the maximum theoretical temperature achievable and is reached when assuming complete combustion and no heat loss. Actual flame temperatures are lower due to heat loss and side reactions.

8. Electrochemistry Basics

Standard Reduction Potentials

Reduction potentials measure the tendency of a species to gain electrons. The standard cell potential for a redox reaction is:

$E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}$

If $E^\circ_{\text{cell}} > 0$, the reaction is spontaneous. All potentials are referenced to the standard hydrogen electrode (SHE), where $E^\circ = 0$V.

Nernst Equation

The cell potential under non-standard conditions is given by the Nernst equation:

$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q = E^\circ_{\text{cell}} - \frac{0.0592}{n} \log Q \text{ (at 25°C)}$

where n is the number of electrons transferred and F is Faraday’s constant (96,485 C/mol). At equilibrium, $E_{\text{cell}} = 0$and Q = K, relating electrochemical potentials to chemical equilibrium.

Relationship to Gibbs Free Energy

The maximum electrical work is:

$\Delta G^\circ = -nFE^\circ_{\text{cell}}$

This connects electrochemical potentials to thermodynamic favorability, with negative ΔG (positive E) indicating spontaneity.

Summary Table

Concept	Key Equation	Domain
Equilibrium constant	$K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$	General equilibrium
Gibbs relation	$\Delta G^\circ = -RT \ln K$	Thermodynamics
Reaction quotient	$\Delta G = \Delta G^\circ + RT \ln Q$	Thermodynamics
Henderson-Hasselbalch	$\text{pH} = \text{p}K_a + \log\frac{[A^-]}{[HA]}$	Acid-base
Solubility product	$K_{sp} = [A^{n+}][B^-]^n$	Solubility
Arrhenius equation	$k = Ae^{-E_a/RT}$	Kinetics
Nernst equation	$E = E^\circ - \frac{0.0592}{n} \log Q$	Electrochemistry

Cross-Links

• [[thermodynamics]] — Gibbs free energy, entropy, enthalpy foundations
• [[acid-base-titration]] — experimental pH titration curves
• [[redox-reactions]] — electron transfer and half-reactions
• [[reaction-kinetics]] — mechanism, collision theory, catalysis
• [[phase-diagrams]] — pressure-temperature equilibrium (liquid-vapor, solid-liquid)