Chemical Equilibrium – Comprehensive Bullet-Point Notes

Types of Chemical Reactions

• Irreversible reaction
  • One-way; proceeds only from reactants (A) to products (B).
  • Representation: $A \rightarrow B$
  • Examples: $\text{CH}<em>4 + 2\text{O}</em>2 \rightarrow \text{CO}<em>2 + 2\text{H}</em>2\text{O}$, $\text{NaOH} + \text{HCl} \rightarrow \text{NaCl} + \text{H}_2\text{O}$
• Reversible reaction
  • Two-way; products can reconvert to reactants.
  • Representation: $A \rightleftharpoons B$
  • Examples: $\text{PCl}<em>5(g) \rightleftharpoons \text{PCl}</em>3(g)+\text{Cl}<em>2(g)$, $\text{SO}</em>3(g) \rightleftharpoons \text{SO}<em>2(g)+\tfrac12\text{O}</em>2(g)$

State of Equilibrium

• Achieved when rate of forward process equals rate of backward process: $r<em>f = r</em>b$.
• At equilibrium, measurable properties (concentration, pressure, density, colour) become time-invariant for given conditions.
• Requires a closed system.

Law of Mass Action & Rate Expressions

• For an elementary reaction $aA + bB \rightarrow cC + dD$:
  • Forward rate: $r<em>f = k</em>f [A]^a [B]^b$
  • Backward rate: $r<em>b = k</em>b [C]^c [D]^d$.
• At equilibrium: $k<em>f [A]^{a}</em>e [B]^{b}<em>e = k</em>b [C]^{c}<em>e [D]^{d}</em>e$.

Equilibrium Constants

• Concentration form (Kc): $K_c = \dfrac{[C]^c [D]^d}{[A]^a [B]^b}$.
• Pressure form (Kp) (only gases): $K<em>p = \dfrac{P</em>C^{c} P<em>D^{d}}{P</em>A^{a} P_B^{b}}$.
• Relation: $K<em>p = K</em>c (RT)^{\Delta n<em>g}$ with $\Delta n</em>g = (c+d)-(a+b)$ (difference in gaseous stoichiometric coefficients).
• Independence: $K_{eq}$ is unaffected by initial concentrations, catalysts, inert gases (at constant V), or whether the system starts from reactant or product side.

Worked Set-Up Examples

1. $\text{PCl}<em>5(g) \rightleftharpoons \text{PCl}</em>3(g)+\text{Cl}<em>2(g)$ • Initial: $n</em>0=PCl<em>5=n$, none of products. • Equilibrium: $n</em>{PCl<em>5}=n-x$, $n</em>{PCl<em>3}=n</em>{Cl<em>2}=x$. • $K</em>c = \dfrac{[PCl<em>3][Cl</em>2]}{[PCl_5]} = \dfrac{x^2}{(n-x)}$.
2. $\text{H}<em>2(g)+\text{I}</em>2(g) \rightleftharpoons 2\text{HI}(g)$
   • $K<em>c = \dfrac{[HI]^2}{[H</em>2][I_2]} = \dfrac{4x^2}{(a-x)(b-x)}$.
3. $\text{N}<em>2(g)+3\text{H}</em>2(g) \rightleftharpoons 2\text{NH}<em>3(g)$ • $K</em>c = \dfrac{[NH<em>3]^2}{[N</em>2][H_2]^3} = \dfrac{4x^2}{(a-x)(b-3x)^3}$.

Characteristics of Equilibrium State

• Dynamic; forward and reverse reactions continue at equal rates.
• Can be attained from any initial composition (reactants, products, or mixture).
• Catalyst lowers time to attain equilibrium but does not change $K_{eq}$.
• Activity (active mass) of pure solids/liquids is taken as constant (unity).
• Only temperature changes $K_{eq}$.

Factors Affecting $K_{eq}$

• Mode of writing reaction: reversing reaction inverts $K<em>{eq}$; multiplying stoichiometry by n raises $K</em>{eq}$ to $n^{th}$ power.
• Temperature dependence – van’t Hoff equation:
  $\dfrac{d\ln K<em>{eq}}{dT}=\dfrac{\Delta H^\circ}{RT^2}$ Integrated form: $\ln\dfrac{K</em>2}{K<em>1}=\dfrac{\Delta H^\circ}{R}\left(\dfrac{1}{T</em>1}-\dfrac{1}{T<em>2}\right)$ • Endothermic ( $\Delta H^\circ>0$ ): $K</em>{eq}$ increases with temperature.
  • Exothermic ( $\Delta H^\circ<0$ ): $K<em>{eq}$ decreases with temperature. • $\Delta H^\circ=0$ : $K</em>{eq}$ independent of T.

Reaction Quotient (Q)

• Same algebraic form as $K_c$, but concentrations at any instant.
• Comparison:
  • $Q<K<em>{eq}$ → reaction proceeds forward. • $Q>K</em>{eq}$ → reaction proceeds backward.
  • $Q=K_{eq}$ → system at equilibrium.

Gibbs Free Energy & Equilibrium

• $\Delta G = \Delta G^\circ + RT \ln Q$.
• At equilibrium $\Delta G = 0$, $Q=K<em>{eq}$, hence $\Delta G^\circ = -RT \ln K</em>{eq}$.
• Significance:
  • $K<em>{eq}>1$ → $\Delta G^\circ<0$ (product-favoured). • $K</em>{eq}<1$ → $\Delta G^\circ>0$ (reactant-favoured).

Activation Energy (Eₐ) & Energy Profile

• Minimum energy needed for reactants to form activated complex.
• Energy diagram parameters:
  • $E<em>a$ (forward), $E</em>a'$ (reverse).
  • Enthalpy change $\Delta H = E<em>p - E</em>r$.
  • Endothermic: $\Delta H>0$, peak closer to products; Exothermic: $\Delta H<0$.

Degree of Dissociation (α) & Molecular Mass

• Definition: $\alpha = \dfrac{\text{moles dissociated}}{\text{initial moles}}$.
• For $\text{PCl}<em>5 \rightleftharpoons \text{PCl}</em>3+\text{Cl}_2$, if one mole starts:
  • Equilibrium moles = $1-\alpha + 2\alpha = 1+\alpha$.
• Relation with molar mass or vapour density:
  $\alpha = \dfrac{M<em>t - M</em>o}{M<em>o (n-1)} = \dfrac{d</em>t - d<em>o}{d</em>o (n-1)}$
  where $n$ = total moles after complete dissociation per mole of reactant (here $n=2$), $M<em>t$ theoretical molar mass, $M</em>o$ observed.

Calculational Caveats

• In α problems, initial moles may be taken as 1, but use actual initial concentration or pressure values (do NOT set to 1) when writing $K_{eq}$ expressions.

Le Châtelier’s Principle

• If a system at equilibrium experiences a change in concentration, pressure/volume, or temperature, it shifts in the direction that counteracts the change.

Concentration Changes

• Adding reactant / removing product → shift forward.
• Removing reactant / adding product → shift backward.

Pressure / Volume Changes (gaseous systems)

• Decrease V (increase P): equilibrium shifts to side with fewer moles of gas.
• Increase V (decrease P): shifts to side with more moles of gas.
• Cases:
  • $\Delta n<em>g=0$ (e.g., $\text{H}</em>2+I<em>2 \rightleftharpoons 2\text{HI}$) → no effect. • $\Delta n</em>g<0$ (e.g., $2\text{NO}<em>2 \rightleftharpoons \text{N}</em>2\text{O}<em>4$) → increasing P shifts right. • $\Delta n</em>g>0$ (e.g., $\text{PCl}<em>5 \rightleftharpoons \text{PCl}</em>3+\text{Cl}_2$) → increasing P shifts left.
• Addition of inert gas:
  • At constant V: no effect on $K_{eq}$ or position.
  • At constant P: behaves like volume increase, apply mole-based rule.

Temperature Changes

• Endothermic: raise T → shift forward; lower T → shift backward.
• Exothermic: opposite trend.
• $\Delta H = 0$: no shift.

Applications to Physical Equilibria

• Liquid ⇌ Vapour (boiling): endothermic; raising P raises boiling point.
• Solid ⇌ Liquid (melting):
  • For water/ice (volume decreases on melting), pressure lowers melting point.
  • For metals (volume increases on melting), pressure raises melting point.
• Gas ⇌ Solution (Henry’s law): increased external P increases gas solubility.
• Endothermic dissolution (e.g., $\text{NH}_4\text{Cl}$ in water): solubility rises with T.
• Exothermic dissolution (e.g., $\text{NaOH}$): solubility decreases with T.

Simultaneous & Sequential Equilibria

• Simultaneous: two reactions sharing common species occur together; each has its own $K<em>{eq}$. Example: $A \rightleftharpoons B + C$ ( $K</em>1$ ) and $B \rightleftharpoons D + E$ ( $K_2$ ).
  Overall relationships can be derived via algebraic combination of individual equilibria.
• Sequential: product of first becomes reactant for second; treat stepwise, using intermediate concentrations.

Summary of Key Formulae

• $r = k [\text{active mass}]^{\text{coeff}}$ (Law of mass action).
• $K<em>c,\;K</em>p,\;K<em>p=K</em>c(RT)^{\Delta n_g}$.
• $Q$ vs $K_{eq}$ for predicting direction.
• $\Delta G^\circ = -RT \ln K_{eq}$.
• van’t Hoff: $\ln\dfrac{K<em>2}{K</em>1}=\dfrac{\Delta H^\circ}{R}\left(\dfrac{1}{T<em>1}-\dfrac{1}{T</em>2}\right)$.
• Degree of dissociation–molar mass relation: $\alpha = \dfrac{M<em>t-M</em>o}{M_o(n-1)}$.

Ethical & Practical Implications

• Industrial optimisation (Haber, Contact processes) relies on manipulating T, P, and catalysts per Le Châtelier to maximise yield while balancing economic cost.
• Understanding equilibrium is foundational for biochemical homeostasis, environmental chemistry (ozone balance, CO₂ solubility in oceans), and safety (pressure vessels, reactors).