🪞 Isomerism & Stereochemistry

Stereochemistry is the three-dimensional geometry of organic molecules and reaction transitions. In IIT-JEE Advanced and Olympiads, questions test R/S priority rules on Fischer projections, enantiomeric excess ($ee$), number of stereoisomers ($2^n$ vs symmetric meso formulas), and cyclohexane chair conformations.

---

1. 📐 Geometrical Isomerism ($GI$)

Geometrical isomerism arises due to restricted rotation around a double bond ($C=C,C=N,N=N$) or in cyclic rings, with two different groups attached to each ring/double bond carbon ($abC=Cab$ or $abC=Ccd$).

1.1 The Cahn-Ingold-Prelog ($CIP$) Priority Rules for $(E/Z)$ System:

1. Rule 1 (Atomic Number): Higher atomic number atom directly bonded to stereocenter gets higher priority ($-I>-Br>-Cl>-S{O}_{3}H>-F>-OH>-N{H}_2>-C{H}_3>-H$).
2. Rule 2 (Isotopes): For isotopes, higher mass number gets priority ($-T>-D>-H$).
3. Rule 3 (Successive Exploration): If first atoms are identical, compare the atoms attached to them in decreasing priority order until a point of difference is reached ($-C{H}_{2}C{H}_3>-C{H}_3$; $-CH(C{H}_3{)}_2>-C{H}_{2}C{H}_{2}C{H}_3$).
4. Rule 4 (Multiple Bonds): Double/triple bonds are treated by duplicating/triplicating phantom atoms ($-CH=O\equiv -CH(O{)}_2$; $-C\equiv N\equiv -C(N{)}_3$; $-COOH\equiv -C(O{)}_3$).

• $Z$ (Zusammen = Together): Highest priority groups on the same side of double bond.
• $E$ (Entgegen = Opposite): Highest priority groups on opposite sides of double bond.

---

2. 🌀 Optical Isomerism & Chirality

A molecule is optically active (chiral) if and only if it is non-superimposable on its mirror image.

$$\text{Condition for Chirality: Absence of Plane of Symmetry }\mathbf{(}\sigma \mathbf{/}\mathbf{POS}\mathbf{)}\text{ and Center of Symmetry }\mathbf{(}\mathbf{i}\mathbf{/}\mathbf{COS}\mathbf{)}$$

2.1 Enantiomers vs. Diastereomers vs. Meso Compounds

Stereoisomer Class	Relationship	Physical Properties (BP, MP, Density, Refractive Index)	Optical Activity
Enantiomers	Non-superimposable mirror images	Identical (except direction of optical rotation $[\alpha ]$)	Optically Active (Equal & opposite $[\alpha ]$)
Diastereomers	Stereoisomers that are NOT mirror images	Completely Different (easily separated by fractional distillation)	Can be chiral or achiral
Meso Compounds	Contains $\ge 2$ chiral centers but possesses internal symmetry ($POS$ or $COS$)	Distinct single compound	Optically Inactive due to internal compensation
Racemic Mixture ($dl/\pm$)	$1:1$ equimolar mixture of $d$ and $l$ enantiomers	Different from pure enantiomers	Optically Inactive due to external compensation

$$\text{Enantiomeric Excess (\% ee)}=\frac{|[\alpha {]}_{\text{observed}}|}{[\alpha {]}_{\text{pure enantiomer}}}\times 100\mathrm{\%}=\frac{|d-l|}{d+l}\times 100\mathrm{\%}$$

---

3. 🧮 Number of Stereoisomers Calculation Formula

Let $n$ be the number of stereogenic units (chiral carbons + double bonds exhibiting $GI$):

1. Unsymmetrical Molecules (No symmetry in any conformation):$$\text{Total Stereoisomers}\mathbf{=}{\mathbf{2}}^{\mathbf{n}}(\text{All }{2}^n\text{ are optically active, Meso }=0)$$
2. Symmetrical Molecules with EVEN number of chiral centers ($n$ is even):$$\text{Optically Active Isomers }(a)=2^{n-1}$$$$\text{Meso Isomers }(m)=2^{(n/2)-1}$$$$\text{Total Stereoisomers }\mathbf{=}\mathbf{a}\mathbf{+}\mathbf{m}\mathbf{=}{\mathbf{2}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{+}{\mathbf{2}}^{\mathbf{(}\mathbf{n}\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{-}\mathbf{1}}$$Example: Tartaric Acid ($HOOC-CH(OH)-CH(OH)-COOH$, $n=2$):$$a=2^{2-1}=2\text{ (d and l)},m=2^{1-1}=1⟹\text{Total}=3.$$
3. Symmetrical Molecules with ODD number of chiral centers ($n$ is odd):$$\text{Optically Active Isomers }(a)=2^{n-1}-2^{(n-1)/2}$$$$\text{Meso Isomers }(m)=2^{(n-1)/2}$$$$\text{Total Stereoisomers }\mathbf{=}\mathbf{a}\mathbf{+}\mathbf{m}\mathbf{=}{\mathbf{2}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$$

---

4. 🪑 Conformational Analysis: Ethane, Butane & Cyclohexane

4.1 Conformational Stability Order

1. $n$-Butane (around $C_2-C_3$ bond):$$\text{Anti-Staggered }\mathbf{(}\theta \mathbf{=}{\mathbf{180}}^{\circ }\mathbf{)}\mathbf{>}\text{Gauche }\mathbf{(}\theta \mathbf{=}{\mathbf{60}}^{\circ }\mathbf{)}\mathbf{>}\text{Partially Eclipsed }\mathbf{(}\theta \mathbf{=}{\mathbf{120}}^{\circ }\mathbf{)}\mathbf{>}\text{Fully Eclipsed }\mathbf{(}\theta \mathbf{=}{\mathbf{0}}^{\circ }\mathbf{)}$$
   • Exception: Ethylene Glycol ($HO-C{H}_2-C{H}_2-OH$) and 2-Fluoroethanol: Gauche conformation is MORE stable than Anti due to intramolecular Hydrogen bonding!
2. Cyclohexane Conformations:$$\text{Chair}\mathbf{>}\text{Twist Boat}\mathbf{>}\text{Boat}\mathbf{>}\text{Half Chair}$$
   • In the chair conformation, all $C-H$ bonds are perfectly staggered with zero torsional strain.
   • Bulky substituents strictly prefer the equatorial position to eliminate $1,3$-diaxial steric repulsions.