Atomic and Molecular Orbitals and Organic Molecular Properties of Organic Molecules Study Guide

Atomic and Molecular Orbitals

• Definition of an Orbital: An orbital is the defined region in three-dimensional space around a nucleus where an electron is most likely to be found. Orbitals vary in size and shape depending on the energy of the electron they contain. These orbitals determine the chemical behavior of atoms.
• Ground-State Electronic Configuration of Carbon: The ground-state configuration is represented as $1s^2 2s^2 2p^2$.
  • Electrons of opposite spin are denoted by arrows pointing up and down.
  • The filling typically follows the Aufbau principle, the Pauli exclusion principle, and Hund’s rule.
• Characteristics of s-Orbitals:
  • 2s Orbital: Viewed as a cutaway, it shows the positive (peak-containing) region of the electron wave in blue and the negative (trough-containing) region in green. It can be described as two concentric spheres of electron density.
  • Size Comparison: A 2s orbital is significantly larger than a 1s orbital. Specifically, $90\%$ of the electron density of a 2s orbital resides within $3.9\,\text{\AA}$ of the nucleus.
• p-Orbitals: Along with s-orbitals, these determine the geometric shapes of atomic structures.

Covalent Bonding and Molecular Orbital Theory

• Bond Formation and Energy: As two hydrogen atoms approach each other, energy changes occur.
  • Bond Length: The internuclear distance at the point of minimum potential energy is defined as the length of the $H-H$ covalent bond.
  • Orbital Overlap: Electrons shared in a covalent bond result from the overlap of atomic orbitals (AOs) to form a new molecular orbital (MO).
• Molecular Orbitals (MOs):
  • When two atomic orbitals overlap, two molecular orbitals are formed: one lower in energy and one higher in energy than the original AOs.
  • Bonding MO: Formed when two $1s$ orbitals combine "in-phase." Electrons in this MO result in a stable covalent bond. In $H_2$, both electrons occupy the bonding MO.
  • Antibonding MO: Formed when two $1s$ orbitals combine "out-of-phase." In a ground-state $H_2$ molecule, the antibonding MO remains empty.

Sigma ($\sigma$) and Pi ($\pi$) Bonds

• Sigma ($\sigma$) Bonds:
  • These are strong bonds formed by the "head-on" overlap of two atomic orbitals.
  • Electrons in $1s$ and $2s$ orbitals combine to yield these bonds.
  • All single bonds in organic compounds are $\sigma$-bonds.
• Pi ($\pi$) Bonds:
  • These are weaker bonds formed by "side-on" overlap of two p-orbitals.
  • They are only found in multiple bonds (double or triple bonds).
• Bond Composition:
  • Single bond: One $\sigma$-bond.
  • Double bond: One $\sigma$-bond and one $\pi$-bond.
  • Triple bond: One $\sigma$-bond and two $\pi$-bonds.

Hybrid Orbitals and Hybridization

• Concept: Hybridization is a process proposed by Linus Pauling in 1931 where a carbon atom mixes its $2s$ and $2p$ atomic orbitals to form new hybrid orbitals.
• Carbon's Ground vs. Excited State:
  • Ground state: $1s^2 2s^2 2p_x^1 2p_y^1$.
  • Excited state (for bonding): $2s^1 2p_x^1 2p_y^1 2p_z^1$.
• Types of Hybridization:
  • $sp^3$ Hybridization: Mixed orbitals resulting in four equivalent orbitals (seen in methane, $CH_4$).
  • $sp^2$ Hybridization:
    • Occurs when carbon forms three single $\sigma$-bonds and one $\pi$-bond (e.g., ethene, $H_2C=CH_2$).
    • The three $sp^2$ orbitals point towards the corners of a triangle with bond angles of $120^\circ$.
    • The remaining p-orbital is perpendicular to the $sp^2$ plane.
    • Ethene contains $4 \times C-H\ \sigma$-bonds, $1 \times C-C\ \sigma$-bond, and $1 \times C-C\ \pi$-bond.
  • $sp$ Hybridization:
    • Occurs when carbon forms two single $\sigma$-bonds and two $\pi$-bonds (e.g., ethyne, $HC\equiv CH$).
    • The two $sp$ orbitals point in opposite directions with a bond angle of $180^\circ$.
    • The two p-orbitals are perpendicular to each other and to the $sp$ plane.
    • Ethyne contains $2 \times C-H\ \sigma$-bonds, $1 \times C-C\ \sigma$-bond, and $2 \times C-C\ \pi$-bonds.

Bond Dissociation Energy ($D$) and Bond Energy ($E$)

• Bond Dissociation Energy ($D$): The amount of energy consumed (when broken) or liberated (when formed) for a specific bond.
  • Bond breaking is endothermic ($D > 0$).
  • Bond formation is exothermic ($D < 0$).
  • Example: The $H-H$ bond requires $+435\,kJ/mol$ (or $104\,kcal/mol$) to cleave and releases $-435\,kJ/mol$ when formed.
• Comparison of Energies (kcal/mol):
  • Homolytic Cleavage ($A-B \rightarrow A \cdot + B \cdot$):
    • $H-H$: $104$
    • $CH_3-H$: $104$
    • $H-F$: $136$
    • $CH_3-Cl$: $84$
  • Heterolytic Cleavage ($A-B \rightarrow A^+ + B^-$):
    • $H-H$: $401$
    • $CH_3-H$: $313$
    • $H-F$: $370$
    • $CH_3-Cl$: $227$
• Bond Energy ($E$): A single average value of the different bond dissociation energies of a molecule. For methane ($CH_4$), the total energy to break all four bonds is $397\,kcal/mol$. The bond energy ($E$) is therefore $\frac{397}{4} = 99\,kcal/mol$.

Polarity of Bonds and Molecules

• Polar Covalent Bonds: Occur when two nuclei do not share electrons equally. The electron cloud is denser around the more electronegative atom.
  • Symbols: $\delta+$ (partial positive) and $\delta-$ (partial negative).
• Dipole Moment ($\mu$): Calculated as the product of the charge ($q$) and the distance between charges ($r$).
  • $\mu = q \times r$
  • Units: Reported in Debye ($D$).
  • $1.0\,D = 1.0 \times 10^{-18}\,esu\,cm$.
  • The charge on an electron is $4.80 \times 10^{-10}\,esu$.
• Electronegativity Difference (Continuum):
  • Nonpolar covalent: Electrons shared equally (e.g., $C-C$, $C-H$; difference $< 0.5$).
  • Polar covalent: Difference between $0.5$ and $1.7$ (e.g., $N-H$, $O-H$).
  • Ionic: Electrons are not shared; opposite charges attract (e.g., $Na^+Cl^-$, $K^+F^-$; difference $> 2.0$).
• Examples of Molecular Dipole Moments:
  • Ammonia ($NH_3$): $\mu = 1.46\,D$
  • Water ($H_2O$): $\mu = 1.84\,D$
  • Nitrogen trifluoride ($NF_3$): $\mu = 0.24\,D$
• Physical Effects: Polarity affects melting point (MP), boiling point (BP), and solubility. It also dictates the kind of chemical reactions that occur at specific bonds.

Energy of Activation ($E_{act}$) and the Transition State

• Energy of Activation ($E_{act}$): The minimum amount of energy that must be provided by a collision for a reaction to occur. Its source is the kinetic energy of moving particles.
  • Reaction Example: $CH_4 + Cl \cdot$
    • Cleaving $CH_3-H$ requires $104\,kcal/mol$.
    • Forming $H-Cl$ releases $103\,kcal/mol$.
    • Net energy required is $1\,kcal/mol$, but experiment shows $4\,kcal/mol$ must be supplied. This extra energy is necessary for the transition state.
• Collision Theory: Reactions require collisions of sufficient energy ($E_{act}$) and proper orientation.
• Transition State (T.S.): Found at the top of the energy hump in a potential energy diagram. It is the peak energy point in the transformation from reactants to products.
  • $E_{act} = \text{Energy of Transition State} - \text{Energy of Reactants}$.
  • $\Delta H = \text{Energy of Products} - \text{Energy of Reactants}$.
• Hammond Postulate (Reactivity vs. T.S.):
  • Easy (Exothermic) Reaction: Reagent of high reactivity; T.S. is reached early and resembles the reactants.
  • Difficult (Endothermic) Reaction: Reagent of low reactivity; T.S. is reached late and resembles the products.

Concept of Resonance and Delocalization

• Resonance: Occurs whenever a molecule can be represented by two or more structures differing only in electron arrangement (same nuclei positions).
  • Resonance Contributors: Individual structures that contribute to the actual molecule.
  • Resonance Hybrid: The actual molecule, which is more stable than any single contributor.
• Resonance Energy: The difference in energy between the hybrid and the most stable contributor.
  • Stability: The more stable the contributors, and the more equal they are in energy, the greater the resonance energy.
• Allylic Systems:
  • Allyl Radical: The p-orbital on the central carbon overlaps equally with p-orbitals on neighbors. Molecular orbitals include bonding (no nodes), nonbonding (node at $C2$), and antibonding (two nodes).
  • Stability: Allylic carbocations are roughly as stable as a nonallylic carbocation with one additional alkyl branch (e.g., a secondary allylic is as stable as a tertiary nonallylic).
• Delocalization: Resonance structures symbolize electron delocalization, which is a stabilizing effect because it results in lower-energy bonding molecular orbitals.

Aromaticity and Huckel's 4n+2 Rule

• Criteria for Aromaticity:
  1. The compound must be cyclic with a cyclic arrangement of p-orbitals.
  2. Every atom in the ring must possess a p-orbital (usually $sp^2$ hybridized).
  3. The ring must be planar.
  4. Huckel's 4n+2 Rule: The cyclic system must contain $4n+2\ \pi$-electrons (where $n = 0, 1, 2, \dots$). This means $2, 6, 10, 14, \dots\ \pi$-electrons.
• Benzene: Contains six $Cs$ and six delocalized $\pi$-electrons ($n=1$). It gains significant stabilization by occupying the three lowest energy MOs.
• Heterocycles:
  • Pyridine: The unshared electron pair on Nitrogen is in an $sp^2$ orbital and is not part of the $4n+2$ system.
  • Pyrrole: The Nitrogen is $sp^2$ hybridized, and its lone pair in the p-orbital is part of the $4n+2$ system (totaling 6 $\pi$-electrons).
  • Pyrrolium Ion: Not aromatic because protonation changes the N to $sp^3$, removing the lone pair from the cyclic interaction.

Electronic Effects in Organic Molecules

• Inductive Effect ($I$): Transmission of charge through $\sigma$-bonds due to electronegativity differences.
  • $-I$ effect: Electron-withdrawing group.
  • $+I$ effect: Electron-donating group.
• Mesomeric Effect ($M$) / Resonance Effect: Movement of electrons through the $\pi$-bond network.
  • $+M$ effect: A group or system that donates electrons (e.g., lone pairs or $\pi$-systems donating electrons).
  • $-M$ effect: A system that accepts electrons.
  • Comparison: Mesomeric effects are generally stronger than inductive effects and can operate over longer distances through conjugation (alternating single/double bonds).
• Hyperconjugation: The stabilization of a neighboring carbocation by the donation of electrons from nearby $C-H$ or $C-C\ \sigma$-bonds into a vacant p-orbital. This is also described as a form of resonance (delocalization).

Questions and Discussion

• Q: Are the following covalent bonds sharing electrons equally?
  • A: Bonds like $H-H$ and $C-C$ share electrons equally (nonpolar). Bonds like $H-F$, $C-H$, and $C-OH$ do not share equally (polar).
• Q: Which bond is more polar: $H-CH_3$ or $Cl-CH_3$?
  • A: $Cl-CH_3$ is more polar due to the higher electronegativity of chlorine compared to hydrogen.
• Q: Draw the energy profile for $CH_4 + Br \cdot \rightarrow CH_3 \cdot + HBr$ given $E_{act} = 18\,kcal$ and $\Delta H = +16\,kcal$.
  • Context: Since $\Delta H$ is positive and $E_{act}$ is high, the transition state is reached late and resembles the products.
• Q: Explain why pyrrole is aromatic but the pyrrolium ion is not.
  • A: In pyrrole, the Nitrogen lone pair is part of the 6 $\pi$-electron system. In the pyrrolium ion, the Nitrogen uses that lone pair to bond with a proton, making it $sp^3$ hybridized and breaking the cyclic p-orbital overlap.