⚖️ Chemical & Ionic Equilibrium

Equilibrium chemistry represents the delicate balancing point where forward and backward reaction rates match. In competitive examinations, it is arguably the single highest-yield topic in physical chemistry, spanning coupled equilibria, buffer capacity bounds, simultaneous precipitation, and indicator selection.

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1. 🔄 Chemical Equilibrium & Le Chatelier's Principle

1.1 Equilibrium Constant Formulations

For the general reaction $aA+bB\rightleftharpoons cC+dD$:

$$K_c=\frac{[C{]}^c[D{]}^d}{[A{]}^a[B{]}^b},K_p=\frac{(P_C{)}^c(P_D{)}^d}{(P_A{)}^a(P_B{)}^b}$$$$K_p=K_c(RT{)}^{\mathrm{\Delta }{n}_g},K_x=K_p{P}^{-\mathrm{\Delta }{n}_g}$$

where $\mathrm{\Delta }{n}_g=(c+d)-(a+b)$ (gaseous species only).

1.2 Le Chatelier's Principle: Inert Gas Addition Matrix

Perturbation	Condition $\mathrm{\Delta }{n}_g>0$ ($PC{l}_5\rightleftharpoons PC{l}_3+C{l}_2$)	Condition $\mathrm{\Delta }{n}_g<0$ ($N_2+3H_2\rightleftharpoons 2N{H}_3$)	Condition $\mathrm{\Delta }{n}_g=0$ ($H_2+I_2\rightleftharpoons 2HI$)
Inert Gas added at Constant Volume ($V=\text{const}$)	NO EFFECT (partial pressures unchanged)	NO EFFECT	NO EFFECT
Inert Gas added at Constant Pressure ($P=\text{const}$)	Shifts FORWARD (towards more moles)	Shifts BACKWARD (towards fewer moles)	NO EFFECT
Increasing Total Pressure (Decreasing Volume)	Shifts BACKWARD	Shifts FORWARD	NO EFFECT
Increasing Temperature	If $\mathrm{\Delta }H>0⟹$ shifts FORWARD; If $\mathrm{\Delta }H<0⟹$ shifts BACKWARD

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2. 💧 Ionic Equilibrium & pH Formulations

2.1 The Autoionization of Water

$$2H_{2}O\rightleftharpoons H_3{O}^{+}+O{H}^{-}⟹K_w=[H^{+}][O{H}^{-}]=1.0\times {10}^{-14}\text{ at }{25}^{\circ }\text{C}$$

Note: As temperature increases ($T={90}^{\circ }\text{C}$), $K_w\approx {10}^{-12}⟹\text{pH of neutral pure water }=6.0$. Pure water at ${90}^{\circ }\text{C}$ is neutral, NOT acidic!

2.2 Ostwald's Dilution Law (Weak Monoprotic Acid $HA$)

$$HA\rightleftharpoons H^{+}+A^{-}(C(1-\alpha ),C\alpha ,C\alpha )$$$$K_a=\frac{C{\alpha }^2}{1-\alpha }⟹\alpha =\sqrt{\frac{K_a}{C}}(\text{valid when }\alpha \le 0.05)$$$$[H^{+}]=C\alpha =\sqrt{K_{a}C}⟹\mathbf{pH}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{[}{\text{pK}}_{\mathbf{a}}\mathbf{-}\log \mathbf{C}\mathbf{]}$$

When $C\le {10}^{-6}\text{ M}$

For very dilute acid solutions (e.g., ${10}^{-8}\text{ M }HCl$), water's autoionization $[H^{+}{]}_w$ cannot be neglected:

$$[H^{+}{]}_{\text{total}}={10}^{-8}+[H^{+}{]}_w⟹[H^{+}{]}^2-{10}^{-8}[H^{+}]-{10}^{-14}=0⟹[H^{+}]=1.05\times {10}^{-7}\text{ M}⟹\mathbf{pH}\mathbf{=}\mathbf{6.98}$$

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3. 🛡️ Buffer Solutions & Henderson-Hasselbalch Equation

A buffer solution resists changes in pH upon addition of small amounts of strong acid or base.

3.1 Acidic Buffer (Weak Acid + Its Conjugate Base / Salt)

$$HA+H_{2}O\rightleftharpoons H_3{O}^{+}+A^{-}$$$$\mathbf{pH}\mathbf{=}{\text{pK}}_{\mathbf{a}}\mathbf{+}\log \left(\frac{\mathbf{[}\text{Conjugate Base }{\mathbf{A}}^{\mathbf{-}}\mathbf{]}}{\mathbf{[}\text{Weak Acid }\mathbf{HA}\mathbf{]}}\right)\mathbf{=}{\text{pK}}_{\mathbf{a}}\mathbf{+}\log \left(\frac{\mathbf{[}\text{Salt}\mathbf{]}}{\mathbf{[}\text{Acid}\mathbf{]}}\right)$$

• Maximum Buffer Capacity: Occurs when $[\text{Salt}]=[\text{Acid}]⟹\mathbf{pH}\mathbf{=}{\text{pK}}_{\mathbf{a}}$.
• Effective Buffer Operating Range: $\text{pH}\mathbf{=}{\text{pK}}_{\mathbf{a}}\pm \mathbf{1}$.

3.2 Basic Buffer (Weak Base + Its Conjugate Acid / Salt)

$$\mathbf{pOH}\mathbf{=}{\text{pK}}_{\mathbf{b}}\mathbf{+}\log \left(\frac{\mathbf{[}\text{Salt}\mathbf{]}}{\mathbf{[}\text{Base}\mathbf{]}}\right)⟹\text{pH}=14-\text{pOH}$$

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4. 🧪 Salt Hydrolysis Master Matrix

Salt Category	Example	Hydrolyzing Ion	Hydrolysis Constant $K_h$	Degree of Hydrolysis $h$	Final pH Formula at ${25}^{\circ }\text{C}$
Strong Acid + Strong Base	$NaCl,KN{O}_3$	None (Neutral)	No hydrolysis	$0$	$\mathbf{pH}\mathbf{=}\mathbf{7.00}$
Weak Acid + Strong Base	$C{H}_{3}COONa$	Anion ($A^{-}$)	$K_h=\frac{K_w}{K_a}$	$h=\sqrt{\frac{K_w}{K_{a}C}}$	$\mathbf{pH}\mathbf{=}\mathbf{7}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{[}{\text{pK}}_{\mathbf{a}}\mathbf{+}\log \mathbf{C}\mathbf{]}$
Strong Acid + Weak Base	$N{H}_{4}Cl$	Cation ($B^{+}$)	$K_h=\frac{K_w}{K_b}$	$h=\sqrt{\frac{K_w}{K_{b}C}}$	$\mathbf{pH}\mathbf{=}\mathbf{7}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{[}{\text{pK}}_{\mathbf{b}}\mathbf{+}\log \mathbf{C}\mathbf{]}$
Weak Acid + Weak Base	$C{H}_{3}COON{H}_4$	Both Cation & Anion	$K_h=\frac{K_w}{K_a{K}_b}$	$h=\sqrt{\frac{K_w}{K_a{K}_b}}$	$\mathbf{pH}\mathbf{=}\mathbf{7}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{[}{\text{pK}}_{\mathbf{a}}\mathbf{-}{\text{pK}}_{\mathbf{b}}\mathbf{]}$ (Independent of $C$!)

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5. 🧊 Solubility Product ($K_{sp}$) & Simultaneous Equilibria

For a sparingly soluble salt $A_x{B}_y\rightleftharpoons x{A}^{y+}+y{B}^{x-}$:

$$K_{sp}=[A^{y+}{]}^x[B^{x-}{]}^y=(xs{)}^x(ys{)}^y=x^x{y}^y{s}^{x+y}$$

• For $1:1$ salt ($AgCl$): $K_{sp}=s^2⟹s=\sqrt{K_{sp}}$.
• For $1:2$ or $2:1$ salt ($A{g}_{2}Cr{O}_4,PbC{l}_2$): $K_{sp}=4s^3⟹s=(K_{sp}/4{)}^{1/3}$.
• For $1:3$ salt ($Al(OH{)}_3,Fe(OH{)}_3$): $K_{sp}=27s^4⟹s=(K_{sp}/27{)}^{1/4}$.
• For $2:3$ salt ($C{a}_3(P{O}_4{)}_2$): $K_{sp}=108s^5⟹s=(K_{sp}/108{)}^{1/5}$.

5.1 Common Ion Effect & Precipitation Criterion

• Condition for Precipitation: Ionic Product ($Q_{sp}>K_{sp}$).
• Unsaturated Solution: $Q_{sp}<K_{sp}$ (more salt dissolves).
• Saturated Dynamic Equilibrium: $Q_{sp}=K_{sp}$.