Chemical Equilibrium, Acid–Base & Solubility Equilibria – Comprehensive Note

Chemical Equilibrium: Core Ideas

A chemical reaction rarely goes to completion; most reactions are reversible and can reach a condition in which no macroscopic change is observed. This state—chemical equilibrium—has four defining features:

1. A closed system (no matter in or out) is required.

2. Forward and reverse reactions continue simultaneously (dynamic character).

3. The rates of the forward and reverse processes are equal.

4. Once equilibrium is reached, the measurable amounts (concentrations or partial-pressures) of all species become constant—though not necessarily equal.

Texts frequently pose three leading questions to motivate the concept: (i) Do reactants always completely disappear? (ii) Are reactions always one-way? (iii) Can unconsumed reactant coexist with no further observable change? The answer to all three is embodied in the equilibrium concept.

Classes of Equilibria

1. Homogeneous equilibrium – all species share the same phase (e.g.
   $\mathrm{H<em>2(g) + I</em>2(g) \leftrightarrow 2HI(g)}$). Two sub-cases arise:
   • equal mole numbers on each side; • unequal.

2. Heterogeneous equilibrium – reactants/products exist in different phases, e.g.
   $\mathrm{CaCO<em>3(s) \leftrightarrow CaO(s)+CO</em>2(g)}$.

3. Phase equilibrium – single substance distributed among phases (liquid–vapour water in a stoppered flask).

4. Solution equilibrium – dissolution ↔ recrystallisation of a sparingly-soluble solid in its saturated solution.

Attaining Equilibrium & Rate Behaviour

• At the start the forward rate is maximal; as product accumulates it slows, while the reverse rate accelerates until equality is achieved.
• Graphs of rate vs. time for forward and reverse lines cross at the equilibrium point.

Equilibrium Constant Expressions (Law of Mass Action)

For a general balanced equation
$aA+bB\leftrightarrow cC+dD$
rate laws at equilibrium give
$K<em>{\text{eq}} = \frac{[C]^c[D]^d}{[A]^a[B]^b}$ (activities may be used for strict thermodynamic work). Pure solids and liquids appear with activity 1 and therefore drop out of $K$. Two special forms: • $K</em>c$ – written with molar concentrations (mol·L$^{-1}$).
• $K<em>p$ – written with gas partial pressures; related to $K</em>c$ by
$K<em>p = K</em>c(RT)^{\Delta n},\qquad \Delta n = \sum n<em>{\text{g,prod}}-\sum n</em>{\text{g,react}}$.

Meaning of $K_{eq}$

• $K\gg1$ → products dominate; reaction is said to be product-favoured.
• $K\ll1$ → reactants dominate.
• $K\approx1$ → appreciable amounts of both sides.

$K$ is temperature-dependent but independent of catalyst and of initial concentrations.

Reaction Quotient $Q_c$

Identical algebraic form to $K_c$ but evaluated with any set of instantaneous concentrations. Comparison with $K$ predicts direction:
• $Q=K$ : system is at equilibrium.
• $QK$ : reverse reaction proceeds (shift left).

Manipulating Equilibrium Equations & $K$

Rule set:

1. Reverse the equation → $K' = 1/K$.

2. Multiply coefficients by $n$ → $K' = K^n$.

3. Add equations → multiply corresponding $K$ values.

Full worked problems (e.g.
combining $K$ for dissolution of $\mathrm{Ag<em>2CO</em>3}$ with an ammonia complexation equilibrium) illustrate these rules.

Gas-Phase Equilibria: $K_p$ Examples

Ideal-gas substitution $P=[A]RT$ leads directly to the $K<em>p–K</em>c$ link above. Worked sample: $\mathrm{N<em>2(g)+3H</em>2(g)\leftrightarrow2NH<em>3(g)}$ at $500\,^{\circ}!\mathrm{C}$ with $K</em>c=6.0\times10^{-2}$ gives $K_p=1.5\times10^{-5}$ because $\Delta n=-2$.

Le Châtelier’s Principle & Factors Shifting Equilibrium

A stress (concentration, pressure/volume, temperature change) induces a shift that partially counteracts that stress.

1. Concentration: Adding reactant → drive right; removing product → drive right; adding product or removing reactant → drive left.

2. Volume/Pressure (gases): Decreasing volume (raising pressure) favours the side with fewer moles of gas; opposite for volume increase. No effect when gaseous mole numbers are equal.

3. Temperature: Treat heat as reactant (endothermic) or product (exothermic). Raising $T$ favours endothermic direction; lowering $T$ favours exothermic. Only temperature changes $K$.

4. Catalysts: lower $E_a$ equally for forward & reverse; no change in $K$—equilibrium is attained faster, not at a different position.

5. Inert gas at constant volume: no effect on $Q$, therefore no shift.

Equilibrium Calculations (ICE/ICEP tables)

Systematic steps: list Initial concentrations or pressures, define Change (often $x$), write Equilibrium expressions, substitute into the $K$ expression, solve (quadratic or approximations) and verify that any small-$x$ approximation (usually valid if <5 % change) is justified.

Acid–Base Theory Overview

• Arrhenius: acids generate $\mathrm{H^+}$; bases provide $\mathrm{OH^-}$ in aqueous medium.
• Brønsted–Lowry: acid = proton donor, base = proton acceptor; defines conjugate acid–base pairs (e.g.
$\mathrm{NH<em>3/NH</em>4^+}$, $\mathrm{H<em>2O/OH^-}$). • Lewis: acid = electron-pair acceptor, base = electron-pair donor (e.g. $\mathrm{BF</em>3+NH_3}$ forms a dative bond).

Strength of Acids & Bases

Strong acids: $\mathrm{HCl,\,HBr,\,HI,\,HNO<em>3,\,HClO</em>4,\,H<em>2SO</em>4}$ (first step). Strong bases: group 1 hydroxides plus $\mathrm{Ca(OH)<em>2, Sr(OH)</em>2, Ba(OH)<em>2}$. Complete dissociation → $[\text{ion}]=C</em>{\text{formal}}$.

Weak species establish equilibrium constants:
$\mathrm{HA+H<em>2O\leftrightarrow H</em>3O^+ + A^-},\qquad K<em>a$ $\mathrm{B+H</em>2O\leftrightarrow BH^+ + OH^-},\qquad K<em>b$ Their strengths are conveyed by $pK</em>a=-\log K_a$ etc.; smaller $pK$ → stronger.

Relationship: $K<em>aK</em>b = K_w = 1.0\times10^{-14}\;(25\,^{\circ}!\mathrm{C}).$

Trends in Acid Strength

1. Binary acids HX – increase down a group (bond length) and left-to-right with electronegativity. Order: \mathrm{HI>HBr>HCl>HF}.

2. Oxoacids $\mathrm{H<em>mXO</em>n}$ – greater number of terminal O atoms (and higher oxidation state on X) increases acidity; e.g.
   \mathrm{HClO4>HClO3>HClO_2>HClO}. More electronegative central atoms also stabilise $A^-$, enhancing strength.

3. Organic acids – electron-withdrawing substituents (e.g.
   $\mathrm{ClCH_2COOH}$) and resonance increase acidity.

4. Amines as bases – electron-donating alkyl groups increase basicity; electronegative substituents decrease.

pH, pOH and the Water Autoionisation Constant

$K<em>w=[H</em>3O^+][OH^-]=1.0\times10^{-14}$ at $25\,^{\circ}!\mathrm{C}.$
Definitions: $\text{pH}=-\log[H<em>3O^+],\; \text{pOH}=-\log[OH^-],$ with $\text{pH+pOH}=14.$ Examples: 0.0025 M HCl → $[H</em>3O^+]$ = 0.0025 M, pH = 2.60.

Hydrolysis of Salts & Solution pH

Nature of the conjugate ions determines pH:
• Strong-acid + strong-base salt → neutral ($\text{pH}=7$).
• Strong-base + weak-acid salt (e.g.
$\mathrm{NaC<em>2H</em>3O<em>2}$) → basic (anion hydrolyses). • Weak-base + strong-acid salt (e.g. $\mathrm{NH</em>4Cl}$) → acidic (cation hydrolyses).
• Weak-weak salts: compare $K<em>a$ of cation to $K</em>b$ of anion.

Buffer Solutions & Henderson–Hasselbalch Equation

A buffer (weak acid + conjugate base or weak base + conjugate acid) resists pH changes. Derivation from $K<em>a$: $pH = pK</em>a + \log\frac{[\text{base}]}{[\text{acid}]}.$
Operating limits: the ratio inside the logarithm should lie between 0.1 and 10, and each component’s concentration should exceed $K_a$ by ≳100×.

Example design: obtain pH 5.10 with 0.300 L of 0.25 M acetic acid. Calculated acetate concentration $0.56\,\text{M}$ ⇒ dissolve 14 g NaC$2$H$3$O$_2$.

pH change on adding strong acid/base is computed by stoichiometric adjustment of buffer components followed by H–H evaluation.

Acid–Base Titrations & Indicators

Key points:
• Equivalence point where moles $H^+$ = moles $OH^-$ (not necessarily pH 7).
• End point signalled by colour change of an indicator, itself a weak acid/base: $\mathrm{HIn \leftrightarrow H^+ + In^-}$, colour depends on $\frac{[In^-]}{[HIn]}$.
Selection rules: choose an indicator whose pH transition falls on the steep part of the titration curve – examples:
strong acid vs strong base: phenolphthalein or bromothymol blue; weak acid vs strong base: phenolphthalein; strong acid vs weak base: methyl orange.

Polyprotic titrations display multiple equivalence points, each corresponding to a stepwise deprotonation (e.g.
carbonic acid).

Solubility Equilibria & $K_{sp}$

For a sparingly-soluble salt $\mathrm{M<em>pX</em>q(s)}$:
$\mathrm{M<em>pX</em>q(s) \leftrightarrow pM^{n+} + qX^{m-}}$
$K_{sp} = [M^{n+}]^p [X^{m-}]^q.$

Relation to molar solubility $s$ (example $\mathrm{PbI<em>2}$): $K</em>{sp}=s(2s)^2=4s^3 \Rightarrow s = \sqrt[3]{K_{sp}/4}.$

Common-Ion Effect in Solubility

Adding a common ion decreases solubility. PbI$_2$ in 0.10 M KI gives $s\approx7.1\times10^{-7}\;\text{M}$ instead of $1.2\times10^{-3}\;\text{M}.$

Precipitation Criteria

Compute the ionic reaction quotient $Q<em>{sp}$ with initial (mixed) ion concentrations: • Q{sp}>K{sp} → precipitation occurs. • Q{sp}<K_{sp} → no precipitate.

Fractional precipitation is exploitable for selective ion separation.

Other Factors Affecting Solubility

pH (for salts containing basic/acidic ions) and complex-ion formation (e.g.
$\mathrm{Ag^+ + 2NH<em>3 \leftrightarrow [Ag(NH</em>3)_2]^+}$) can dramatically increase apparent solubility.

Summary of Frequently-Used Equations

• $K<em>{eq}=\dfrac{\prod [\text{products}]^{\nu}}{\prod [\text{reactants}]^{\nu}}\quad(\text{omit pure }s,\,l).$ • $K</em>p=K<em>c(RT)^{\Delta n}.$ • $Q</em>c$ has same form as $K<em>c$; compare to predict direction. • $pH=-\log[H</em>3O^+],\, pOH=-\log[OH^-],\, pH+pOH=14.$
• $K<em>w=[H</em>3O^+][OH^-]=1.0\times10^{-14}\;(25^{\circ}!\text{C}).$
• $K<em>aK</em>b=K<em>w$ (conjugate pairs). • Henderson–Hasselbalch: $pH=pK</em>a+\log\dfrac{[\text{base}]}{[\text{acid}]}.$
• $K_{sp}=[M^{n+}]^p[X^{m-}]^q.$

These relations, together with ICE-table logic, provide the computational framework for equilibrium problems encountered in General & Physical Chemistry.