Equilibrium revision notes

Physical Equilibrium

Physical equilibrium exists when there is a phase transformation and the rates of opposing physical processes become equal.

• Dynamic Nature: Physical equilibrium is not static; intense activity occurs at the boundary between phases. Proven experimentally by using radioactive sugar isotopes in a saturated solution; radioactivity soon appears in the solid phase due to continuous exchange.
• Solid-Liquid Equilibrium: Occurs at the normal melting/freezing point (at 1 atm/1.013 bar pressure). The mass of the two phases remains constant in an insulated system (no heat exchange).
• Liquid-Vapour Equilibrium: Attained when evaporation rate equals condensation rate. The pressure exerted by the vapour in equilibrium with its liquid is the equilibrium vapour pressure.
  • More volatile liquids have higher vapour pressures and lower boiling points.
  • JEE TIPBoiling point relies on atmospheric pressure. At high altitudes, atmospheric pressure is lower, causing water to boil at temperatures below 100°C.
• Solid-Vapour Equilibrium: Observed in substances that sublime, such as solid iodine (purple vapour), camphor, and $NH_4Cl$.
• Solids in Liquids: Characterized by a saturated solution where the rate of dissolution equals the rate of crystallization. The concentration of solute is constant at a given temperature.
• Gases in Liquids (Henry’s Law): The mass of a gas dissolved in a given mass of a solvent is proportional to the pressure of the gas above the solvent.
  • JEE TIPSolubility of gases decreases with an increase in temperature. This is why soda becomes "flat" when left open in a warm room.

Chemical Equilibrium & Reaction Extent

Chemical equilibrium is attained when forward and backward reaction rates become equal, keeping reactant and product concentrations constant.

• Dynamic Nature Proof: Demonstrated by Haber using isotopic scrambling. If equilibrium is reached with $H_2, N_2,$ and $NH_3$, and then $D_2$ (deuterium) is introduced, the system eventually contains scrambled species: $H_2, HD, D_2$ and $NH_3, NH_2D, NHD_2, ND_3$.JEE TIPThis mixing strictly proves that forward and backward reactions continue indefinitely; they do not statically halt.
• Extent of Reaction based on $K_c$:
  • $K_c > 10^3$: Products heavily predominate; reaction goes nearly to completion (e.g., $H_2 + O_2 \rightleftharpoons H_2O$ has $K_c = 2.4 \times 10^{47}$).
  • $K_c < 10^{-3}$: Reactants heavily predominate; reaction rarely proceeds (e.g., $H_2O \rightleftharpoons H_2 + \frac{1}{2}O_2$ has $K_c = 4.1 \times 10^{-48}$).
  • $10^{-3} < K_c < 10^3$: Appreciable concentrations of both reactants and products exist (e.g., $H_2 + I_2 \rightleftharpoons 2HI$ has $K_c = 57.0$).

Homogeneous and Heterogeneous Equilibria

• Homogeneous Equilibria: All reactants and products are in the same phase.
  • Gas phase: $N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$
  • Aqueous phase: $CH_3COOC_2H_5(aq) + H_2O(l) \rightleftharpoons CH_3COOH(aq) + C_2H_5OH(aq)$
• Heterogeneous Equilibria: Involves substances in multiple phases.
  • Example: $CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$.
  • Example: $Ni(s) + 4CO(g) \rightleftharpoons Ni(CO)_4(g)$.

Equilibrium Constants ($K_c$, $K_p$) & Reaction Quotient ($Q$)

For a reaction $aA + bB \rightleftharpoons cC + dD$:

• $K_c$ and $K_p$ Expressions: $K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$ and $K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b}$.
• Relationship: $K_p = K_c(RT)^{\Delta n}$.
  • $\Delta n = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)}$.
  • JEE TIPPressure must strictly be in bar (1 bar = $10^5$ Pa) to align with standard state definitions, and $R = 0.0831 \text{ bar L mol}^{-1} \text{K}^{-1}$.
• Modifying Constants:
  • Reverse reaction: $K_{rev} = 1/K_{fwd}$.
  • Multiply by factor $n$: $K_{new} = (K_{fwd})^n$.
  • Adding reactions: $K_{net} = K_1 \times K_2$.
• Reaction Quotient ($Q_c$ / $Q_p$): Calculated using concentrations/pressures at any arbitrary time $t$.
  • $Q < K$: Reaction proceeds forward.
  • $Q > K$: Reaction proceeds backward.
  • $Q = K$: Equilibrium state.

Thermodynamics of Equilibrium

• Gibbs Free Energy ($\Delta G$): The driving force of a reaction. $\Delta G = \Delta G^\circ + RT \ln Q$.
• At equilibrium, $\Delta G = 0$ and $Q = K_c$. This yields: $\Delta G^\circ = -RT \ln K$ or $K = e^{-\Delta G^\circ / RT}$.
  • If $\Delta G^\circ < 0$, $K > 1$ (spontaneous forward, products favored).
  • If $\Delta G^\circ > 0$, $K < 1$ (non-spontaneous forward, reactants favored).

Le Chatelier’s Principle & Industrial Factors

If a system at equilibrium is subjected to a change, it shifts to counteract the effect of the change.

• Concentration: Adding reactant/removing product pushes the reaction forward.
• Pressure/Volume: Increasing pressure (decreasing volume) shifts equilibrium toward fewer moles of gas. If $\Delta n = 0$, pressure has no effect.
• Temperature: The only factor that alters the numerical value of $K$.
  • Exothermic ($\Delta H < 0$): $K$ decreases as $T$ increases. (Cold favored).
  • Endothermic ($\Delta H > 0$): $K$ increases as $T$ increases. (Hot favored).
• Effect of Inert Gas: Adding an inert gas at constant volume does not change partial pressures or shift the equilibrium.
• Catalyst: Lowers activation energy for forward and backward reactions equally. It does NOT alter $K$ or the equilibrium composition, just the speed of attaining it.
• Industrial Applications:
  • Haber Process ($NH_3$ synthesis): Exothermic ($\Delta H = -92.38 \text{ kJ/mol}$). Favored by high pressure (200 atm), optimal moderate temperature (500°C), and an iron catalyst.
  • Contact Process ($SO_3$ synthesis): Exothermic. Uses Platinum ($Pt$) or Divanadium pentoxide ($V_2O_5$) as a catalyst to speed up the sluggish oxidation of $SO_2$.
• Color Change Reactions (Equilibrium Shift Proofs):
  • $Fe^{3+} \text{(yellow)} + SCN^- \text{(colourless)} \rightleftharpoons [Fe(SCN)]^{2+} \text{(deep red)}$.JEE TIPAdding oxalic acid removes $Fe^{3+}$ forming $[Fe(C_2O_4)_3]^{3-}$, fading the red color. Adding $HgCl_2$ removes $SCN^-$ forming $[Hg(SCN)_4]^{2-}$, also fading the red color.
  • $2NO_2 \text{(brown)} \rightleftharpoons N_2O_4 \text{(colourless)}$. Exothermic. Immersing in ice turns it colourless; hot water turns it deep brown.
  • $[Co(H_2O)_6]^{3+} \text{(pink)} + 4Cl^- \rightleftharpoons [CoCl_4]^{2-} \text{(blue)} + 6H_2O$. Endothermic. Freezing mixture turns it pink; room temp makes it blue.

Ionic Equilibrium, Electrolytes & Acid-Base Theories

Michael Faraday classified substances as electrolytes (conduct electricity in aqueous solutions) and non-electrolytes.

• Arrhenius Concept: Acids yield $H^+_{(aq)}$ in water; bases yield $OH^-_{(aq)}$. (Limited to aqueous media).
• Brönsted-Lowry Concept: Acids are proton ($H^+$) donors; bases are proton acceptors.
  • Conjugate Pairs: Differ by one proton. A strong acid yields a very weak conjugate base ($HCl \rightarrow Cl^-$), and a strong base yields a weak conjugate acid.
• Lewis Concept: Acids are electron-pair acceptors (e.g., $BF_3, AlCl_3, Co^{3+}, Mg^{2+}, H^+$); bases are electron-pair donors (e.g., $H_2O, NH_3, OH^-, F^-$).

pH, Water Auto-ionization ($K_w$) & Acid Strength

• Auto-ionization of Water: $H_2O + H_2O \rightleftharpoons H_3O^+ + OH^-$.
  • $K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14} M^2$ at 298 K. $pK_w = pH + pOH = 14$.
  • JEE TIP$K_w$ increases with temperature. However, pH variations with temperature are small and often ignored.
  • Molarity of pure water $\approx 55.55 M$.
• Weak Acids ($K_a$) and Bases ($K_b$):
  • $K_a = \frac{c\alpha^2}{1-\alpha}$. For very weak acids, $\alpha = \sqrt{K_a/c}$.
• Conjugate Pair Relation: $K_a \times K_b = K_w \implies pK_a + pK_b = pK_w = 14$ (at 298K).
• Factors Affecting Acid Strength:
  • Down a group: H-A bond strength decreases (size increases), making cleavage easier. Acidity: $HF < HCl < HBr < HI$.
  • Across a period: H-A electronegativity difference increases, marking easier charge separation. Acidity: $CH_4 < NH_3 < H_2O < HF$.
• Polyprotic Acids: Have successive constants. $K_{a1} \gg K_{a2} \gg K_{a3}$.
• Indicators: Compounds like phenolphthalein and bromothymol blue are weak acids ($HIn \rightleftharpoons H^+ + In^-$) where the unionized form and conjugate base form display different colors.

Salt Hydrolysis & Buffer Solutions

• Salt Hydrolysis: Salts from strong acids + strong bases ($NaCl$) do not hydrolyse (pH = 7).
  • Weak Acid + Strong Base ($CH_3COONa$): Hydrolyses to give $OH^-$. Alkaline, pH > 7.
  • Strong Acid + Weak Base ($NH_4Cl$): Hydrolyses to give $H^+$. Acidic, pH < 7.
  • Weak Acid + Weak Base ($CH_3COONH_4$): Both ions hydrolyse. $pH = 7 + \frac{1}{2}(pK_a - pK_b)$.
• Buffer Solutions: Solutions resisting pH change upon addition of small amounts of acid/alkali or upon dilution.
  • Acidic Buffer (Weak Acid + Conjugate Base Salt): e.g., $CH_3COOH + CH_3COONa$. Henderson-Hasselbalch: $pH = pK_a + \log \frac{[\text{Salt}]}{[\text{Acid}]}$.
  • Basic Buffer (Weak Base + Conjugate Acid Salt): e.g., $NH_4OH + NH_4Cl$. $pOH = pK_b + \log \frac{[\text{Conjugate Acid}]}{[\text{Base}]}$.

Solubility Product ($K_{sp}$) & Common Ion Effect

• Solubility Categories: Soluble ($>0.1M$), Slightly Soluble ($0.01M - 0.1M$), Sparingly Soluble ($<0.01M$).
• Solubility Product ($K_{sp}$): For a sparingly soluble salt $M_xX_y \rightleftharpoons xM^{p+} + yX^{q-}$ with molar solubility $S$: $K_{sp} = [M^{p+}]^x[X^{q-}]^y = (xS)^x(yS)^y = x^x y^y S^{(x+y)}$.
  • JEE TIPExample: Zirconium phosphate $(Zr^{4+})_3(PO_4^{3-})_4 \implies K_{sp} = 3^3 \cdot 4^4 \cdot S^7 = 6912 S^7$.
• Ionic Product ($Q_{sp}$): If $Q_{sp} > K_{sp}$, precipitation occurs.
• Common Ion Effect: Decreases solubility.
• Effect of pH on Solubility: Salts of weak acids (like phosphates, cyanides) show increased solubility at lower pH (more acidic). The $H^+$ ions protonate the weak acid anion, actively removing it from the solubility equilibrium.
  • $S = \left[ K_{sp} \frac{([H^+] + K_a)}{K_a} \right]^{1/2}$.

Key Concepts & Definitions

Dynamic Equilibrium:

Macroscopic properties are constant, but microscopic forward and reverse processes occur exactly at the same rate.

Isotopic Scrambling:

Experimental use of isotopes (D2D_2D2​ instead of H2H_2H2​, or radioactive sugar) to prove reactions continue dynamically at equilibrium.

Electrolytes (Strong vs. Weak):

Strong electrolytes dissociate almost completely (∼100%\sim 100\%∼100%) in water; weak electrolytes dissociate only partially (<5%<5\%<5%).

Hydronium Ion (H3O+H_3O^+H3​O+):

A bare proton (H+H^+H+) is too intensely charged to exist freely; it bonds with water's lone pairs.

Conjugate Acid-Base Pair:

An acid and a base that differ by exactly one proton (H+H^+H+).

Buffer Solution:

A solution resisting pH change upon addition of small amounts of acid/alkali or upon dilution.

Important Rules, Laws & Principles

• Law of Mass Action: Equilibrium constant is the product of product concentrations divided by reactant concentrations, each raised to their stoichiometric coefficients.
• Henry’s Law: The mass of a gas dissolved in a solvent is directly proportional to the partial pressure of the gas above the solvent.
• Le Chatelier’s Principle: When an equilibrium system is subjected to a change in concentration, pressure, or temperature, it shifts in a direction that minimizes or counteracts the change.

Formulae & Equations

1. $K_p = K_c(RT)^{\Delta n}$
2. $\Delta G = \Delta G^\circ + RT \ln Q$
3. $\Delta G^\circ = -RT \ln K$ $\implies K = e^{-\Delta G^\circ/RT}$
4. $K_w = [H^+][OH^-] = 10^{-14} M^2$ (at 298K)
5. $pK_a + pK_b = pK_w = 14$ (for conjugate pairs)
6. $pH = 7 + \frac{1}{2}(pK_a - pK_b)$ (Salt of Weak Acid & Weak Base)
7. $pH = pK_a + \log \frac{[\text{Salt}]}{[\text{Acid}]}$ (Henderson-Hasselbalch, Acidic Buffer)
8. $K_{sp} = x^x y^y S^{(x+y)}$ (General formula for $M_xX_y$ salt solubility)

⚠️ EXCEPTIONS & ANOMALIES

• Pure Solids and Liquids in $K_{eq}$: Normally, concentration terms are included in $K$, BUT the molar concentrations of pure solids and pure liquids are absolutely constant regardless of their amount. They are mathematically folded into the $K$ value and conventionally assigned an active mass of exactly 1.
• Dimensionless Units of $K$: $K_c$ and $K_p$ are frequently written with units like $mol/L$ or $bar$, BUT strictly thermodynamically, if referenced against standard states (1 bar, 1 M), they are completely dimensionless quantities.
• Inert Gas Addition: Adding an inert gas typically changes total pressure, BUT if added at constant volume, it exerts zero shift on the equilibrium because the partial pressures/molarities of the reacting species are totally undisturbed.
• The Bare Proton Anomaly: $H^+$ is often written in equations, BUT its extremely small size radius ($\sim 10^{-15} m$) generates an intense electric field making it impossible to exist freely. It is uniquely heavily hydrated into $H_3O^+$, $H_5O_2^+$, $H_7O_3^+$, and $H_9O_4^+$.
• Solubility and pH: Normally, adding ions decreases solubility via the common ion effect, BUT adding $H^+$ (lowering pH) to a salt containing a weak acid anion actually increases its solubility by protonating the anion and dragging the equilibrium forward.
• Polyprotic Acid Dissociation: One might assume multiple protons leave a molecule with similar energy, BUT $K_{a1}$ is always vastly greater than $K_{a2}$. Once the first proton leaves, electrostatic forces fiercely hold onto the remaining protons in the now-negatively charged anion.
• Water's Unique Dual Role: Most substances are strictly acidic or basic, BUT water is anomalous as an amphoteric solvent, seamlessly acting as an acid in the presence of $NH_3$ and a base in the presence of $HCl$, and even simultaneously reacting with itself (auto-ionization).
• Temperature and pH: $K_w$ strictly depends on temperature (increases as T increases), BUT the variation in the pH scale with temperature is so mathematically slight that it is practically ignored in general calculations.

Previous Year JEE Topics

• Calculation of $K_p$ from $K_c$ and vice versa (especially ensuring proper R values and units, calculating $\Delta n$ for gaseous species only).
• Gibbs Free Energy & $K_{eq}$: Applying $\Delta G^\circ = -RT \ln K$ and identifying spontaneous vs non-spontaneous conditions.
• Le Chatelier's Principle: Predicting direction of shift based on Temperature, Pressure/Volume, and inert gas addition at constant $V$ vs constant $P$.
• Reaction Colors: Recalling deep red $[Fe(SCN)]^{2+}$, brown $NO_2$, and pink/blue cobalt complexes to predict shift direction based on observation.
• Conjugate Acid-Base Pairs: Identifying formulas differing by exactly one proton.
• Buffer Solutions: Calculating pH using Henderson-Hasselbalch equations, particularly identifying when a half-neutralization creates a buffer naturally.
• Solubility Product ($K_{sp}$): Calculating molar solubility for asymmetric salts (like $A_2B_3$, or $Zr_3(PO_4)_4$) and determining simultaneous solubility in presence of a common ion.