6.4 Optical Activity, Diastereomers, Meso Compounds & Fischer Projections

6.4 Optical Activity, Diastereomers, Meso Compounds & Fischer Projections

Optical Activity, Polarimetry & Racemic Mixtures

Enantiomers possess identical thermodynamic properties (melting point, boiling point, density, heat of combustion, $pK_a$, chromatographic retention times on achiral stationary phases) because their internal atomic distances and bond dipoles are identical. They differ in only two physical phenomena: their interaction with other chiral molecules (such as enzyme active sites) and their rotation of plane-polarized light.

Polarimetry and Specific Rotation

Normal light vibrates in all planes perpendicular to its direction of propagation. Passing monochromatic light (typically the sodium D-line, $\lambda = 589.3\text{ nm}$) through a polarizing filter isolates light oscillating in a single plane. When plane-polarized light passes through a solution containing a chiral substance, the plane of polarization rotates by an observed angle $\alpha_{\text{obs}}$:

• Dextrorotatory ($d$ or $(+)$): The compound rotates plane-polarized light clockwise (to the right).
• Levorotatory ($l$ or $(-)$): The compound rotates plane-polarized light counterclockwise (to the left).

Fundamental MCAT Rule: Absolute configuration $(R)$ or $(S)$ has ZERO direct correlation to the sign of optical rotation $(+)$ or $(-)$. An $(R)$ stereocenter can be dextrorotatory or levorotatory; the optical rotation sign must be determined empirically using a polarimeter.

The Biot Equation for Specific Rotation

To standardize observed optical rotation against variable concentration and path length, optical activity is reported as specific rotation $[\alpha]_\lambda^T$:

$$[\alpha]_\lambda^T = \frac{\alpha_{\text{obs}}}{c \cdot l}$$

where:

• $\alpha_{\text{obs}}$ is the experimentally measured rotation angle in degrees ($^\circ$),
• $c$ is the concentration of the chiral solute in grams per milliliter ($\text{g/mL}$),
• $l$ is the path length of the polarimeter sample tube in decimeters ($\text{dm}$) ($1\text{ dm} = 10\text{ cm} = 0.1\text{ m}$),
• $T$ is temperature (typically $20^\circ\text{C}$ or $25^\circ\text{C}$) and $\lambda$ is the wavelength (sodium D-line).

Racemic Mixtures (Racemates) and Enantiomeric Excess

• Racemic Mixture ($\pm$ or $dl$): An equimolar ($50:50$) mixture of two enantiomers. Because enantiomers possess equal and opposite specific rotations ($[\alpha]_{(+)} = -[\alpha]_{(-)}$), optical rotation cancels completely, yielding zero net optical activity ($[\alpha]_{\text{racemic}} = 0^\circ$).
• Enantiomeric Excess ($ee$): Quantifies optical purity when one enantiomer is in excess: $$\% ee = \frac{|[\alpha]_{\text{sample}}|}{|[\alpha]_{\text{pure enantiomer}}|} \times 100\% = \% \text{ major enantiomer} - \% \text{ minor enantiomer}$$
• Chiral Resolution: Separation of enantiomers cannot be achieved by simple distillation or achiral chromatography because enantiomers have identical physical constants. Resolution requires reacting the racemate with an enantiopure chiral resolving agent (e.g., $(2R,3R)$-tartaric acid) to generate a pair of diastereomeric salts, which possess distinct solubilities and can be separated by fractional crystallization before cleaving the resolving agent.

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Diastereomers, Cis/Trans, (E)/(Z) Systems & Meso Symmetry

Diastereomers are stereoisomers that are not mirror images of one another. They arise when a molecule possesses two or more stereocenters and some, but not all, stereocenters are inverted.

The $2^n$ Rule for Maximum Stereoisomers

For a molecule possessing $n$ distinct chiral stereocenters, the theoretical maximum number of stereoisomers is:

$$\text{Maximum Number of Stereoisomers} = 2^n$$

For example, a molecule with 3 distinct chiral centers possesses $2^3 = 8$ stereoisomers, organized into 4 pairs of enantiomers. Any two stereoisomers that are not enantiomers are diastereomers.

Meso Compounds: Symmetry Cancelling Optical Activity

When a molecule contains $n \ge 2$ chiral stereocenters with symmetrical substitution patterns, the actual number of stereoisomers is less than $2^n$ due to the existence of meso compounds.

• A meso compound contains chiral stereocenters (typically an $(R,S)$ pair) and an internal plane of symmetry ($\sigma$) or center of inversion ($i$).
• Because the top half of the molecule is the exact mirror image of the bottom half, the molecule is superimposable on its mirror image.
• Meso compounds are achiral overall and completely optically inactive ($[\alpha] = 0^\circ$) via internal compensation.

Stereochemical Properties of Tartaric Acid Stereoisomers

Alkene Stereoisomerism: $(E)$ vs. $(Z)$ Nomenclature

Rotation about carbon-carbon double bonds is restricted by the $260\text{ kJ/mol}$ barrier of the $\pi$ bond. When both $sp^2$ carbons carry two different substituents, diastereomeric alkenes exist:

• Cis / Trans: Applicable only when each double-bonded carbon possesses an identical reference group (or a hydrogen atom). Cis indicates identical groups on the same side; trans indicates opposite sides.
• $(E)$ / $(Z)$ CIP Algorithm: Applicable to all polysubstituted alkenes:
  1. Assign CIP priorities (1 or 2) to the two substituents on carbon A.
  2. Assign CIP priorities (1 or 2) to the two substituents on carbon B independently.
  3. If both priority 1 groups are on the same side of the double bond axis, the alkene is $(Z)$ (zusammen, German for together).
  4. If the priority 1 groups are on opposite sides, the alkene is $(E)$ (entgegen, German for opposite).

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Fischer Projections & D/L Stereochemical Convention

Fischer projections provide a standardized two-dimensional method for depicting three-dimensional carbohydrates, amino acids, and polyols.

The Spatial Rules of Fischer Projections

• Vertical Bonds Recede: Bonds drawn vertically project away from the viewer (equivalent to dashes in 3D wedge-dash notation).
• Horizontal Bonds Emerge: Bonds drawn horizontally project toward the viewer (equivalent to wedges in 3D wedge-dash notation).
• Orientation Convention: The most oxidized carbon of the parent chain (e.g., $-\text{CHO}$ or $-\text{COOH}$) is placed at or near the top of the vertical backbone.

Permitted Manipulations of Fischer Projections

1. $180^\circ$ In-Plane Rotation: Preserves the true stereochemical configuration (rotates vertical dashes to vertical dashes).
2. $90^\circ$ Rotation Forbidden: Inverts the stereochemical configuration (swaps vertical dashes for horizontal wedges).
3. Pairwise Group Swapping: Swapping any two substituents on a stereocenter inverts the configuration from $(R)$ to $(S)$ or vice versa. Performing a second swap restores the original configuration.

The Rosanoff D/L Stereochemical Convention

The D/L naming system is an older convention based on relative configuration relative to glyceraldehyde and remains universally utilized in biochemistry:

• Monosaccharides (Aldoses & Ketoses): Focus on the highest-numbered chiral carbon (the stereocenter farthest from the carbonyl group, such as C5 in glucose). In a standard vertical Fischer projection with C1 at the top:
  • If the $-\text{OH}$ group is on the right, the sugar belongs to the D-family (natural mammalian sugars are almost exclusively D-sugars, e.g., D-glucose, D-ribose).
  • If the $-\text{OH}$ group is on the left, the sugar belongs to the L-family.
• Amino Acids: Focus on the $\alpha$-carbon in a standard Fischer projection with the carboxylate ($-\text{COO}^-$) at the top and side chain ($-\text{R}$) at the bottom:
  • If the amino group ($-\text{NH}_3^+$) is on the left, it is an L-amino acid.
  • If the amino group is on the right, it is a D-amino acid.

MCAT High-Yield Fact: All 19 chiral proteinogenic amino acids in human proteins are L-amino acids. Under CIP rules, all natural L-amino acids possess the $(S)$-absolute configuration, with one single exception: L-cysteine, which is $(R)$ because the sulfur-containing thiol methyl sidechain ($-\text{CH}_2\text{SH}$, $Z=16$) has higher CIP priority than the carboxyl group ($-\text{COOH}$, bonded to oxygen $Z=8$). Glycine is achiral.