Unit 9: Nuclear Magnetic Resonance (NMR) II: Spin-Spin Coupling, Decoupling & 2D Pulse Sequences
Mechanisms of indirect scalar J-coupling, Fermi contact interaction, first-order multiplets and Pascal's triangle, Karplus relation and dihedral angles, second-order strongly coupled spin systems (AB and AMX roof effects), broadband heteronuclear decoupling, 13C NMR and DEPT editing, Nuclear Overhauser Effect (NOE), and multidimensional NMR (2D COSY, NOESY, HSQC, HMBC).
§9.1 Mechanism of Indirect Scalar Spin-Spin Coupling (J-Coupling)
While through-space dipolar magnetic interactions average to zero in isotropic liquids due to rapid molecular tumbling, an indirect magnetic interaction persists transmitted through the chemical bonding electrons. This is scalar spin-spin coupling (\(J\)-coupling).
The Ramsey-Fermi Contact Mechanism
The primary quantum mechanical mechanism mediating \(J\)-coupling across covalent bonds is the Fermi contact interaction:
- The nuclear spin of nucleus \(A\) (\(\vec{I}_A\)) polarizes the spin of an electron in its immediate vicinity via contact hyperfine interaction with s-electron density at the nucleus (\(|\psi(0)|^2\)).
- By the Pauli exclusion principle, the paired electron in the covalent \(\sigma\)-bond must have antiparallel spin.
- This polarized bonding electron interacts via Fermi contact with the second nucleus \(B\) (\(\vec{I}_B\)), effectively communicating the spin state of \(A\) to \(B\).
The scalar coupling Hamiltonian is formulated as:
\[ \hat{H}_J = 2\pi J_{AB} \hat{\vec{I}}_A \cdot \hat{\vec{I}}_B = h J_{AB} \left(\hat{I}_{Ax}\hat{I}_{Bx} + \hat{I}_{Ay}\hat{I}_{By} + \hat{I}_{Az}\hat{I}_{Bz}\right) \]Crucially, the scalar coupling constant \(J_{AB}\) is expressed in Hertz (Hz) and is strictly independent of the spectrometer magnetic field \(B_0\). One-bond couplings (\(^1J\)) are typically positive and large (e.g., \(^1J_{CH} \approx 125 - 250\text{ Hz}\)); two-bond geminal couplings (\(^2J\)) are typically negative (\(-12\text{ Hz}\)); three-bond vicinal couplings (\(^3J\)) are positive (\(0 - 18\text{ Hz}\)).
§9.2 First-Order Multiplets & Pascal's Triangle Rules
A spin system is classified as first-order when the chemical shift separation \(\Delta\nu\) between coupled nuclei is much larger than their coupling constant \(J\):
\[ \frac{\Delta\nu}{J} \ge 10 \]Multiplicity & The 2nI + 1 Rule
When a nucleus couples to \(n\) magnetically equivalent neighboring nuclei of spin \(I\), its resonance is split into a multiplet containing:
\[ M = 2nI + 1 \quad \text{peaks} \]For coupling to spin \(I = 1/2\) nuclei (protons), this simplifies to the familiar \((n + 1)\) rule:
- \(n = 0\): Singlet (1)
- \(n = 1\): Doublet (1:1), separated by \(J\)
- \(n = 2\): Triplet (1:2:1), adjacent peaks separated by \(J\)
- \(n = 3\): Quartet (1:3:3:1), adjacent peaks separated by \(J\)
- \(n = 4\): Quintet (1:4:6:4:1)
- \(n = 5\): Sextet (1:5:10:10:5:1)
- \(n = 6\): Septet (1:6:15:20:15:6:1)
The relative intensities of the multiplet lines correspond exactly to the binomial coefficients of Pascal's triangle: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
Coupling Trees for Non-Equivalent Nuclei
When a nucleus couples to two non-equivalent sets of nuclei with different coupling constants (e.g., \(J_{AM} \neq J_{AX}\)), successive splitting yields composite multiplet trees: a doublet of doublets (\(dd\)), doublet of triplets (\(dt\)), or doublet of doublets of doublets (\(ddd\)).
### Advanced 2D NMR Theory: Complete Product Operator Tracking of the 2D COSY Experiment To demonstrate how two-dimensional homonuclear correlation works at the quantum level, consider two scalar-coupled spins \(I\) and \(S\) with scalar coupling constant \(J\) in a Homonuclear Correlation Spectroscopy (COSY) sequence: \[ \left(\frac{\pi}{2}\right)_x - t_1 - \left(\frac{\pi}{2}\right)_x - t_2\text{ (detection)} \] #### 1. Equilibrium to First Pulse Thermal equilibrium density matrix: \[ \hat{\sigma}_0 = \hat{I}_z + \hat{S}_z \] Applying the first non-selective \(90_x^\circ\) pulse: \[ \hat{\sigma}_0 \xrightarrow{(\pi/2)(\hat{I}_x + \hat{S}_x)} -\hat{I}_y - \hat{S}_y \] #### 2. Evolution during Evolution Period \(t_1\) Tracking the \(\hat{I}\) spin components under chemical shift \(\Omega_I\) and coupling \(\pi J t_1\): \[ -\hat{I}_y \xrightarrow{\Omega_I t_1 \hat{I}_z} -\hat{I}_y \cos(\Omega_I t_1) + \hat{I}_x \sin(\Omega_I t_1) \] Now letting scalar coupling evolve: \[ -\hat{I}_y \cos(\Omega_I t_1) \xrightarrow{\pi J t_1 2\hat{I}_z\hat{S}_z} -\hat{I}_y \cos(\pi J t_1) \cos(\Omega_I t_1) + 2\hat{I}_x\hat{S}_z \sin(\pi J t_1) \cos(\Omega_I t_1) \] \[ +\hat{I}_x \sin(\Omega_I t_1) \xrightarrow{\pi J t_1 2\hat{I}_z\hat{S}_z} +\hat{I}_x \cos(\pi J t_1) \sin(\Omega_I t_1) + 2\hat{I}_y\hat{S}_z \sin(\pi J t_1) \sin(\Omega_I t_1) \] #### 3. Action of the Second Mixing Pulse \((90_x^\circ)\) The second \(90_x^\circ\) pulse transforms each operator: - \(\hat{I}_x \rightarrow \hat{I}_x\) - \(\hat{I}_y \rightarrow \hat{I}_z\) (longitudinal, unobservable during \(t_2\)) - \(2\hat{I}_x\hat{S}_z \xrightarrow{(\pi/2)(\hat{I}_x + \hat{S}_x)} -2\hat{I}_x\hat{S}_y\) (multiple quantum, unobservable) - \(2\hat{I}_y\hat{S}_z \xrightarrow{(\pi/2)(\hat{I}_x + \hat{S}_x)} 2\hat{I}_z(-\hat{S}_y) = -2\hat{I}_z\hat{S}_y\) #### 4. The Origin of Cross-Peaks and Diagonal Peaks Observe the term \(-2\hat{I}_z\hat{S}_y\)! The coherence initially labeled by the chemical shift of spin \(I\) (\(\sin(\Omega_I t_1)\)) has been transferred to **spin \(S\)**: \[ \hat{\sigma}_{\text{transferred}} = -2\hat{I}_z\hat{S}_y \sin(\Omega_I t_1) \sin(\pi J t_1) \] During the detection period \(t_2\), this anti-phase coherence evolves under the chemical shift of **spin \(S\)** (\(\Omega_S\)): \[ -2\hat{I}_z\hat{S}_y \xrightarrow{\pi J t_2 2\hat{I}_z\hat{S}_z} \hat{S}_x \sin(\pi J t_2) \] \[ \xrightarrow{\Omega_S t_2 \hat{S}_z} \hat{S}_x \cos(\Omega_S t_2) \sin(\pi J t_2) + \dots \] The 2D Fourier transform across \(t_1\) and \(t_2\) produces: - **Diagonal peaks:** Modulated by \(\cos(\Omega_I t_1)\) and detected at \(\Omega_I\) (frequency \(\omega_1 = \Omega_I, \omega_2 = \Omega_I\)). - **Cross-peaks:** Modulated by \(\sin(\Omega_I t_1)\) during \(t_1\) and detected at \(\Omega_S\) during \(t_2\) (frequency \(\omega_1 = \Omega_I, \omega_2 = \Omega_S\)) with an antiphase multiplet structure! This algebraic derivation rigorously proves that cross-peaks can appear in a COSY spectrum if and only if the scalar coupling between the two spins is strictly non-zero (\(J \neq 0\)).§9.3 The Karplus Relation & Dihedral Angle Conformational Calculus
In 1959, Martin Karplus demonstrated that vicinal three-bond coupling constants \(^3J_{HH}\) across a \(\text{H-C-C-H}\) fragment depend strictly on the dihedral (torsional) angle \(\phi\) between the two \(\text{C-H}\) bonds:
\[ ^3J_{HH}(\phi) = A \cos^2\phi + B \cos\phi + C \]Typical empirical coefficients for aliphatic hydrocarbons are \(A = 7.0\text{ Hz}\), \(B = -1.0\text{ Hz}\), and \(C = 5.0\text{ Hz}\), giving:
\[ ^3J_{HH}(\phi) = 7.0 \cos^2\phi - 1.0 \cos\phi + 5.0 \]Structural Implications of the Karplus Curve
- Anti-periplanar (\(\phi = 180^\circ\)): Maximum orbital overlap between C-H \(\sigma\)-orbitals: \(\cos(180^\circ) = -1 \implies {}^3J \approx 7 + 1 + 5 = 13\text{ Hz}\) (can reach \(14 - 18\text{ Hz}\)). In cyclohexane chairs, diaxial coupling is large: \(^3J_{aa} \approx 10 - 14\text{ Hz}\).
- Syn-periplanar (\(\phi = 0^\circ\)): Moderate orbital overlap: \(\cos(0^\circ) = 1 \implies {}^3J \approx 7 - 1 + 5 = 11\text{ Hz}\) (typically \(8 - 10\text{ Hz}\)). In alkenes, cis-coupling: \(^3J_{cis} \approx 7 - 11\text{ Hz}\).
- Gauche / Orthogonal (\(\phi = 90^\circ\)): Minimal orbital overlap: \(\cos(90^\circ) = 0 \implies {}^3J \approx 5\text{ Hz}\) (at \(\phi = 90^\circ\), \(^3J\) drops to \(0 - 2\text{ Hz}\)). In cyclohexanes, axial-equatorial and diequatorial couplings are small: \(^3J_{ae} \approx {}^3J_{ee} \approx 2 - 5\text{ Hz}\).
The Karplus equation is the cornerstone of conformational analysis in organic stereochemistry and protein backbone NMR (where \(^3J_{H^N H^\alpha}\) determines the peptide \(\phi\) dihedral angle).
§9.4 Second-Order (Strongly Coupled) Spin Systems: The AB Pattern
When the chemical shift separation between coupled nuclei becomes comparable to the coupling constant (\(\Delta\nu / J < 10\)), the first-order approximation breaks down completely. The spin system transitions into a strongly coupled second-order system.
Quantum Mechanical Matrix Solution of the AB System
For two coupled spin-1/2 nuclei \(A\) and \(B\) with chemical shifts \(\nu_A, \nu_B\) and coupling \(J\), the Hamiltonian matrix in the product basis \(\{ |\alpha\alpha\rangle, |\alpha\beta\rangle, |\beta\alpha\rangle, |\beta\beta\rangle \}\) is:
\[ \hat{H} = \begin{pmatrix} \bar{\nu} + \frac{1}{4}J & 0 & 0 & 0 \\ 0 & \frac{1}{2}\Delta\nu - \frac{1}{4}J & \frac{1}{2}J & 0 \\ 0 & \frac{1}{2}J & -\frac{1}{2}\Delta\nu - \frac{1}{4}J & 0 \\ 0 & 0 & 0 & -\bar{\nu} + \frac{1}{4}J \end{pmatrix} \]The inner \(2\times 2\) block mixes states \(|\alpha\beta\rangle\) and \(|\beta\alpha\rangle\). Diagonalizing this block with mixing angle \(\theta\), where \(\tan(2\theta) = J / \Delta\nu\):
The spectrum consists of four lines: two outer lines (1 and 4) and two inner lines (2 and 3):
\[ \nu_1 - \nu_2 = \nu_3 - \nu_4 = J \] \[ \nu_1 - \nu_3 = \nu_2 - \nu_4 = \sqrt{\Delta\nu^2 + J^2} - J \]The 'Roof Effect'
The relative intensities of the four lines are given by:
\[ I_1 : I_2 : I_3 : I_4 = (1 - \sin 2\theta) : (1 + \sin 2\theta) : (1 + \sin 2\theta) : (1 - \sin 2\theta) \]where \(\sin 2\theta = \frac{J}{\sqrt{\Delta\nu^2 + J^2}}\). The inner lines are amplified while the outer lines shrink, causing the multiplet to 'tilt' inward toward the coupled partner like a slanted roof (the roof effect). As \(\Delta\nu / J \to 0\), the outer lines vanish completely and the inner lines coalesce into a single singlet.
§9.5 Carbon-13 (13C) NMR & DEPT Spectral Editing
Carbon-13 (\(^{13}\text{C}\)) is a spin-1/2 nucleus with low natural abundance (\(1.108\%\)) and a lower gyromagnetic ratio (\(\gamma_C \approx 0.25 \gamma_H\)). Its inherent sensitivity is \(\approx 1.76 \times 10^{-4}\) relative to \(^1\text{H}\). However, the chemical shift range spans \(0 - 220\text{ ppm}\), providing superior spectral dispersion without peak overlap.
1. Broadband Proton Decoupling (\(^{13}\text{C}\{^1\text{H}\}\))
In standard \(^{13}\text{C}\) acquisitions, high-power composite RF irradiation is applied at the proton resonance frequency (broadband decoupling, e.g., WALTZ-16). This induces rapid proton spin flips, completely collapsing all \(^1J_{CH}\) multiplets into single sharp singlets for each chemically non-equivalent carbon atom.
2. The Nuclear Overhauser Effect (NOE) Enhancement
Proton decoupling saturates proton transitions, driving through-space dipolar cross-relaxation that perturbs \(^{13}\text{C}\) populations. The maximum theoretical NOE signal enhancement is:
\[ \eta = \frac{\gamma_H}{2\gamma_C} \approx \frac{2.675 \times 10^8}{2(6.728 \times 10^7)} \approx 1.988 \implies \text{Total Intensity} = 1 + \eta \approx 2.988 \]Protonated carbons gain nearly a 3-fold intensity boost. Quaternary carbons without directly attached protons experience minimal NOE and long \(T_1\) times, resulting in much smaller peaks.
3. DEPT (Distortionless Enhancement by Polarization Transfer)
DEPT uses polarization transfer from sensitive protons to coupled carbons via variable read pulse angle \(\theta\):
- DEPT-45 (\(\theta = 45^\circ\)): All protonated carbons (\(\text{CH}, \text{CH}_2, \text{CH}_3\)) appear positive.
- DEPT-90 (\(\theta = 90^\circ\)): Only \(\text{CH}\) (methine) carbons appear positive; \(\text{CH}_2\) and \(\text{CH}_3\) are completely invisible.
- DEPT-135 (\(\theta = 135^\circ\)): \(\text{CH}\) and \(\text{CH}_3\) appear positive (pointing up); \(\text{CH}_2\) (methylene) appears negative (pointing down / inverted). Quaternary carbons are absent from all DEPT spectra.
§9.6 Two-Dimensional NMR Principles: Evolution & Acquisition
Two-dimensional (2D) NMR, pioneered by Jean Jeener and Richard Ernst (1991 Nobel Prize), expands spectroscopic information across two frequency axes \((F_1, F_2)\). Every 2D pulse sequence consists of four fundamental time intervals:
\[ \text{Preparation} \longrightarrow \text{Evolution } (t_1) \longrightarrow \text{Mixing } (\tau_m) \longrightarrow \text{Detection } (t_2) \]- Preparation: Magnetization is initialized at thermal equilibrium and tipped into the transverse plane by an RF pulse.
- Evolution (\(t_1\)): Spins precess freely at their characteristic frequencies \(\omega_1\). The interval \(t_1\) is systematically incremented in \(N_1\) discrete steps.
- Mixing: A pulse or delay period during which coherence or magnetization is transferred between coupled or spatially proximate spins via scalar coupling (\(J\)) or dipolar cross-relaxation (NOE).
- Detection (\(t_2\)): The Free Induction Decay (FID) \(S(t_1, t_2)\) is recorded as a function of real time \(t_2\).
2D Fourier Transformation
A double Fourier transformation converts the time-domain matrix \(S(t_1, t_2)\) into a 2D frequency spectrum \(F(\omega_1, \omega_2)\):
\[ F(\omega_1, \omega_2) = \int_0^\infty \int_0^\infty S(t_1, t_2) e^{-i\omega_1 t_1} e^{-i\omega_2 t_2} dt_1 dt_2 \]§9.7 Homonuclear & Heteronuclear 2D Sequences: COSY, NOESY, HSQC & HMBC
Modern organic structure elucidation relies on four cornerstone 2D experiments:
1. COSY (Correlation Spectroscopy)
Homonuclear \(^1\text{H}-^1\text{H}\) experiment (\(90^\circ - t_1 - 90^\circ - \text{acquire}\)).
- Diagonal Peaks (\(F_1 = F_2\)): Replicate the conventional 1D \(^1\text{H}\) spectrum.
- Cross Peaks (\(F_1 \neq F_2\)): Appear symmetrically off the diagonal at coordinates \((\delta_A, \delta_B)\) and \((\delta_B, \delta_A)\) if and only if protons \(A\) and \(B\) share scalar coupling (\(^2J\) or \(^3J\)). Tracing cross-peaks establishes the complete carbon-proton connectivity backbone.
2. NOESY (Nuclear Overhauser Effect Spectroscopy)
Homonuclear \(^1\text{H}-^1\text{H}\) experiment utilizing dipolar cross-relaxation during mixing time \(\tau_m\). Cross-peaks indicate spatial proximity through space (\(r < 5\text{ \AA}\)), independent of intervening chemical bonds. Indispensable for establishing stereochemistry and 3D protein structures.
3. HSQC (Heteronuclear Single Quantum Coherence)
Heteronuclear \(^1\text{H}-^{13}\text{C}\) experiment correlating protons with the exact carbon atom to which they are directly attached via one-bond coupling (\(^1J_{CH} \approx 140\text{ Hz}\)). Each peak indicates a direct \(\text{C-H}\) bond.
4. HMBC (Heteronuclear Multiple Bond Correlation)
Heteronuclear \(^1\text{H}-^{13}\text{C}\) experiment tuned for long-range two- and three-bond couplings (\(^2J_{CH}, {}^3J_{CH} \approx 5 - 10\text{ Hz}\)), with one-bond couplings suppressed. Crucial for connecting quaternary carbons (carbonyls, aromatics, fully substituted centers) across heteroatoms where proton-proton COSY connectivity terminates.
### Advanced Research Monograph: Solid-State Magic Angle Spinning (MAS) NMR of Biomolecules and Battery Materials Unlike solution NMR where rapid isotropic Brownian tumbling averages anisotropic magnetic interactions to zero, solid-state NMR spectra of powders and membranes are severely broadened by Chemical Shift Anisotropy (CSA), direct Dipolar Couplings, and Nuclear Quadrupole Interactions. #### 1. Fast and Ultra-Fast Magic Angle Spinning By mechanically spinning the cylindrical zirconia rotor at an angle \(\theta = \arctan(\sqrt{2}) \approx 54.74^\circ\) relative to \(\mathbf{B}_0\): - The geometric Legendre polynomial vanishes identically: \[ P_2(\cos\theta) = \frac{3\cos^2\theta - 1}{2} = \frac{3(1/3) - 1}{2} = 0 \] - Spinning speeds have advanced from \(10\text{ kHz}\) (4 mm rotors) to over \(100 - 150\text{ kHz}\) (0.7 mm micro-rotors driven by compressed nitrogen gas). - At \(\nu_R > 100\text{ kHz}\), massive \(^1\text{H}-^1\text{H}\) homonuclear dipolar couplings (\(\sim 50\text{ kHz}\)) are completely averaged out, yielding high-resolution \(^1\text{H}\) solid-state spectra with linewidths rivaling solution-state NMR. #### 2. Atomic-Resolution Structure Determination of Insoluble Aggregates Solid-state MAS NMR is uniquely suited for systems that cannot crystallize or be dissolved: - **Amyloid Fibrils:** Elucidated the full 3D atomic structures of \(\beta\)-amyloid (A\(\beta_{40}\), A\(\beta_{42}\)) fibrils in Alzheimer's disease and \(\alpha\)-synuclein in Parkinson's pathology. - **Membrane Proteins:** Resolved the active conformation and ion channels of GPCRs and light-driven pumps in native phospholipid bilayer environments without detergent extraction. #### 3. Operando Battery Electrochemistry Using specialized radiofrequency probe circuits with hermetically sealed battery pouch cells: - In situ \(^7\text{Li}\) and \(^{23}\text{Na}\) MAS NMR monitors lithium metal dendrite formation in real time during charge/discharge cycling. - Resolves the evolution of solid electrolyte interphase (SEI) layers, distinguishing reversible insertion from irreversible dead metal deposition at the electrode surface.§9.8 TOCSY, HMQC & Triple-Resonance Biomolecular NMR Spectroscopy
As molecular complexity scales to large natural products, oligosaccharides, and proteins (\(> 10\text{ kDa}\)), conventional 2D COSY spectra become uninterpretable due to severe peak overlap. Advanced multidimensional pulse sequences resolve these structural ambiguities.
1. TOCSY (Total Correlation Spectroscopy)
Unlike COSY, which only reveals scalar correlations between directly coupled neighboring protons (\(^2J\) and \(^3J\)), TOCSY utilizes a continuous isotropic radiofrequency spin-lock sequence (such as MLEV-17 or DIPSI-2) during the mixing period \(\tau_m\) (\(30 - 120\text{ ms}\)):
- During the spin lock, scalar couplings are transformed into an isotropic exchange Hamiltonian: \(\hat{H}_{\text{eff}} = 2\pi \sum J_{ij} \vec{I}_i \cdot \vec{I}_j\).
- Magnetization propagates coherently through the entire scalar coupling network (relay mechanism): \(A \to M \to X \to \dots\).
- All protons belonging to the same continuous spin system (e.g., all protons within a single amino acid sidechain or monosaccharide ring) appear on the same horizontal track, identifying individual spin systems at a glance.
2. Triple-Resonance 3D NMR of Proteins
In structural biology, recombinant proteins uniformly labeled with stable isotopes (\(^{13}\text{C}\) and \(^{15}\text{N}\)) are analyzed using 3D triple-resonance pulse sequences that transfer magnetization through large, well-defined one-bond couplings (\(^1J_{NH} \approx 90\text{ Hz}\), \(^1J_{NC\alpha} \approx 11\text{ Hz}\), \(^1J_{C\alpha C'} \approx 55\text{ Hz}\)):
| Experiment | Correlated Nuclei (Dimensions) | Magnetization Transfer Pathway | Structural Function |
|---|---|---|---|
| HNCA | \(^1\text{H}^N(i) - {}^{15}\text{N}(i) - {}^{13}\text{C}^\alpha(i, i-1)\) | \(^1\text{H}^N \xrightarrow{J_{NH}} {}^{15}\text{N} \xrightarrow{J_{NC\alpha}} {}^{13}\text{C}^\alpha \xrightarrow{J_{NC\alpha}} {}^{15}\text{N} \xrightarrow{J_{NH}} {}^1\text{H}^N\) | Correlates amide to both intra-residue \(C^\alpha(i)\) and preceding \(C^\alpha(i-1)\) |
| HN(CO)CA | \(^1\text{H}^N(i) - {}^{15}\text{N}(i) - {}^{13}\text{C}^\alpha(i-1)\) | \(^1\text{H}^N \to {}^{15}\text{N} \xrightarrow{J_{NCO}} {}^{13}\text{C}'(i-1) \xrightarrow{J_{COC\alpha}} {}^{13}\text{C}^\alpha(i-1) \to \text{detect}\) | Exclusively detects preceding residue \(C^\alpha(i-1)\) |
| HNCO | \(^1\text{H}^N(i) - {}^{15}\text{N}(i) - {}^{13}\text{C}'(i-1)\) | \(^1\text{H}^N \to {}^{15}\text{N} \xrightarrow{J_{NCO}} {}^{13}\text{C}'(i-1) \to \text{detect}\) | Highest sensitivity 3D experiment; correlates amide to preceding carbonyl |
By comparing the pair of \(C^\alpha\) peaks in HNCA with the single preceding \(C^\alpha(i-1)\) peak in HN(CO)CA, spectroscopists step sequentially along the peptide chain like dominoes, completing the complete sequence-specific backbone resonance assignment of intact proteins.
In the \(^1\text{H}\) NMR spectrum of bromoethane (\(\text{CH}_3\text{CH}_2\text{Br}\)) recorded at \(400.0\text{ MHz}\): The methyl group (\(\text{-CH}_3\)) appears at \(\delta = 1.68\text{ ppm}\). The methylene group (\(\text{-CH}_2\text{-}\)) appears at \(\delta = 3.42\text{ ppm}\). The vicinal scalar coupling constant is \(^3J = 7.4\text{ Hz}\). (a) Determine the multiplicity and relative line intensities for the methyl and methylene resonances. (b) Calculate the frequency difference \(\Delta\nu\) in Hz between the centers of the two multiplets. (c) Evaluate the ratio \(\Delta\nu / J\) and verify whether this spin system can be treated as first-order (\(A_3 X_2\)).
Step (a): Multiplicities and intensities
- Methyl group (\(\text{-CH}_3\)): Adjacent to \(n=2\) methylene protons. Multiplicity \(= n + 1 = 2 + 1 = 3\) (triplet). Relative line intensities from Pascal's triangle: 1 : 2 : 1, spaced by \(J = 7.4\text{ Hz}\).
- Methylene group (\(\text{-CH}_2\text{-}\)): Adjacent to \(n=3\) methyl protons. Multiplicity \(= n + 1 = 3 + 1 = 4\) (quartet). Relative line intensities from Pascal's triangle: 1 : 3 : 3 : 1, spaced by \(J = 7.4\text{ Hz}\).
Step (b): Chemical shift separation Delta nu
\[ \Delta\delta = 3.42 - 1.68 = 1.74\text{ ppm} \] \[ \Delta\nu = \Delta\delta \times \nu_0 = 1.74\text{ ppm} \times 400.0\text{ MHz} = 696.0\text{ Hz} \]Step (c): First-order criterion evaluation
\[ \frac{\Delta\nu}{J} = \frac{696.0\text{ Hz}}{7.4\text{ Hz}} = 94.1 \]Since \(\frac{\Delta\nu}{J} = 94.1 \gg 10\), the spin system strictly satisfies the first-order condition (\(A_3 X_2\)). Multiplet intensities conform to Pascal's triangle without visible second-order distortion.
A substituted cyclohexane ring possesses a proton \(H_A\) on carbon C1 coupled to two adjacent protons on carbon C2: one axial proton \(H_{2a}\) and one equatorial proton \(H_{2e}\). In the chair conformation: The dihedral angle between \(H_A\) and \(H_{2a}\) is \(\phi_1 = 180^\circ\) (anti-periplanar). The dihedral angle between \(H_A\) and \(H_{2e}\) is \(\phi_2 = 60^\circ\) (gauche). Using the Karplus equation:
(a) Calculate the theoretical coupling constants \(^3J_{A, 2a}\) and \(^3J_{A, 2e}\). (b) Describe the resulting multiplet pattern observed for proton \(H_A\). (c) If proton \(H_A\) were equatorial instead of axial, what would be the two dihedral angles to \(H_{2a}\) and \(H_{2e}\), and what would be the predicted coupling constants?
Step (a): Coupling constant calculations for axial HA
- For \(H_A(\text{ax}) - H_{2a}(\text{ax})\) (\(\phi_1 = 180^\circ\)): \[ \cos(180^\circ) = -1 \implies \cos^2(180^\circ) = 1 \] \[ ^3J_{aa} = 8.5(1) - 0.5(-1) + 2.0 = 8.5 + 0.5 + 2.0 = 11.0\text{ Hz} \]
- For \(H_A(\text{ax}) - H_{2e}(\text{eq})\) (\(\phi_2 = 60^\circ\)): \[ \cos(60^\circ) = 0.5 \implies \cos^2(60^\circ) = 0.25 \] \[ ^3J_{ae} = 8.5(0.25) - 0.5(0.5) + 2.0 = 2.125 - 0.25 + 2.0 = 3.875\text{ Hz} \approx 3.9\text{ Hz} \]
Step (b): Multiplet appearance
Proton \(H_A\) couples to two non-equivalent protons with distinct coupling constants (\(11.0\text{ Hz}\) and \(3.9\text{ Hz}\)).
The resulting signal is a doublet of doublets (\(dd\)) with line positions at: \(\pm \frac{11.0}{2} \pm \frac{3.9}{2}\), consisting of four lines of equal intensity (1:1:1:1).
Step (c): Equatorial HA case
If \(H_A\) is equatorial:
- Dihedral angle to \(H_{2a}\) is \(\phi \approx 60^\circ \implies {}^3J_{ea} \approx 3.9\text{ Hz}\).
- Dihedral angle to \(H_{2e}\) is \(\phi \approx 60^\circ \implies {}^3J_{ee} \approx 3.9\text{ Hz}\).
Both couplings are small (\(\approx 3.9\text{ Hz}\)), producing an apparent triplet with small splitting. This stark difference between a large diaxial coupling (\(11\text{ Hz}\)) and small equatorial couplings (\(3.9\text{ Hz}\)) is the foundational rule for determining axial versus equatorial stereochemistry in six-membered rings.
A geminal methylene group (\(-\text{CH}_2-\)) in a rigid bicyclic lactone forms an strongly coupled \(AB\) spin system recorded on a \(300.0\text{ MHz}\) spectrometer. The spectrum consists of four lines: Line 1: \(848.0\text{ Hz}\) Line 2: \(836.0\text{ Hz}\) Line 3: \(818.0\text{ Hz}\) Line 4: \(806.0\text{ Hz}\) (a) Determine the scalar coupling constant \(J_{AB}\). (b) Calculate the true chemical shift difference \(\Delta\nu = |\nu_A - \nu_B|\) in Hz and in ppm. (c) Calculate the true chemical shifts \(\delta_A\) and \(\delta_B\). (d) Calculate the theoretical intensity ratio \(I_{\text{inner}} / I_{\text{outer}}\) reflecting the roof effect.
Step (a): Scalar coupling constant J_AB
In an \(AB\) quartet, the outer-to-inner line separations equal \(J_{AB}\):
\[ \nu_1 - \nu_2 = 848.0 - 836.0 = 12.0\text{ Hz} \] \[ \nu_3 - \nu_4 = 818.0 - 806.0 = 12.0\text{ Hz} \] \[ J_{AB} = 12.0\text{ Hz} \]Step (b): True chemical shift difference Delta nu
The separation between alternating lines relates to \(\Delta\nu\) by:
\[ \nu_1 - \nu_3 = \nu_2 - \nu_4 = 848.0 - 818.0 = 30.0\text{ Hz} = \sqrt{\Delta\nu^2 + J^2} - J \] \[ \sqrt{\Delta\nu^2 + J^2} = 30.0 + J = 30.0 + 12.0 = 42.0\text{ Hz} \] \[ \Delta\nu^2 + J^2 = (42.0)^2 = 1764.0\text{ Hz}^2 \] \[ \Delta\nu^2 = 1764.0 - 12.0^2 = 1764.0 - 144.0 = 1620.0\text{ Hz}^2 \] \[ \Delta\nu = \sqrt{1620.0} = 40.25\text{ Hz} \]In ppm on a \(300.0\text{ MHz}\) spectrometer:
\[ \Delta\delta = \frac{40.25\text{ Hz}}{300.0\text{ MHz}} = 0.134\text{ ppm} \]Step (c): True chemical shifts delta_A and delta_B
The center of gravity of the multiplet is:
\[ \nu_{\text{mid}} = \frac{848.0 + 806.0}{2} = 827.0\text{ Hz} \implies \delta_{\text{mid}} = \frac{827.0}{300.0} = 2.7567\text{ ppm} \] \[ \nu_A = \nu_{\text{mid}} + \frac{\Delta\nu}{2} = 827.0 + 20.12 = 847.12\text{ Hz} \implies \delta_A = \frac{847.12}{300.0} = 2.824\text{ ppm} \] \[ \nu_B = \nu_{\text{mid}} - \frac{\Delta\nu}{2} = 827.0 - 20.12 = 806.88\text{ Hz} \implies \delta_B = \frac{806.88}{300.0} = 2.690\text{ ppm} \]Step (d): Theoretical intensity ratio (roof effect)
\[ \sin 2\theta = \frac{J}{\sqrt{\Delta\nu^2 + J^2}} = \frac{12.0}{42.0} = 0.2857 \] \[ \frac{I_{\text{inner}}}{I_{\text{outer}}} = \frac{1 + \sin 2\theta}{1 - \sin 2\theta} = \frac{1 + 0.2857}{1 - 0.2857} = \frac{1.2857}{0.7143} = 1.800 \]The inner peaks (lines 2 and 3) are \(1.8\) times taller than the outer peaks (lines 1 and 4), producing the characteristic inward roof slant.
A compound with molecular formula \(\text{C}_6\text{H}_{12}\text{O}\) produces six distinct peaks in its broadband decoupled \(^{13}\text{C}\{^1\text{H}\}\) NMR spectrum: C1: \(208.5\text{ ppm}\) C2: \(52.3\text{ ppm}\) C3: \(38.4\text{ ppm}\) C4: \(24.8\text{ ppm}\) C5: \(22.5\text{ ppm}\) C6: \(14.1\text{ ppm}\) DEPT spectral editing reveals: In DEPT-90: Only the peak at \(52.3\text{ ppm}\) is observed (pointing up). In DEPT-135: Peaks at \(52.3\text{ ppm}\), \(22.5\text{ ppm}\), and \(14.1\text{ ppm}\) point up (positive); peaks at \(38.4\text{ ppm}\) and \(24.8\text{ ppm}\) point down (negative); the peak at \(208.5\text{ ppm}\) is absent. (a) Determine the carbon multiplicity (quaternary \(\text{C}\), \(\text{CH}\), \(\text{CH}_2\), or \(\text{CH}_3\)) for each of the six carbon signals. (b) Propose the constitutional formula and name of the compound.
Step (a): Carbon multiplicity assignments
- C1 (\(208.5\text{ ppm}\)): Absent in both DEPT-90 and DEPT-135. High chemical shift indicates a carbonyl carbon. It is a quaternary carbon (\(\text{C}_{\text{quat}}\)) (specifically a ketone carbonyl \(\text{C=O}\)).
- C2 (\(52.3\text{ ppm}\)): Present in DEPT-90 (up) and DEPT-135 (up). Only methine carbons appear in DEPT-90. It is a \(\text{CH}\) (methine) group.
- C3 (\(38.4\text{ ppm}\)): Absent in DEPT-90; negative (pointing down) in DEPT-135. Methylene carbons invert in DEPT-135. It is a \(\text{CH}_2\) (methylene) group.
- C4 (\(24.8\text{ ppm}\)): Absent in DEPT-90; negative (pointing down) in DEPT-135. It is a \(\text{CH}_2\) (methylene) group.
- C5 (\(22.5\text{ ppm}\)): Absent in DEPT-90; positive (pointing up) in DEPT-135. Since it is absent in DEPT-90, it must be a \(\text{CH}_3\) (methyl) group.
- C6 (\(14.1\text{ ppm}\)): Absent in DEPT-90; positive (pointing up) in DEPT-135. It is a \(\text{CH}_3\) (methyl) group.
Summary of carbon inventory: \(1 \times \text{C=O} + 1 \times \text{CH} + 2 \times \text{CH}_2 + 2 \times \text{CH}_3 = \text{C}_6\text{H}_{12}\text{O}\).
Step (b): Proposed structure
The molecular formula has unsaturation index \(\text{DBE} = 6 - \frac{12}{2} + 1 = 1\), fully accounted for by the ketone carbonyl (\(\text{C=O}\)).
The presence of two methyl groups, two methylene groups, one methine group, and one ketone carbonyl establishes the structure as:
\[ \text{CH}_3-\text{CH}_2-\text{CH}_2-\text{CH}(\text{CH}_3)-\text{CHO} \quad \text{or} \quad \text{CH}_3-\text{CH}_2-\text{CH}_2-\text{CO}-\text{CH}(\text{CH}_3)_2 \quad (\text{etc.}) \]With an acyclic ketone at \(208.5\text{ ppm}\) and only 6 carbons: 2-Hexanone has \(3 \times \text{CH}_2, 2 \times \text{CH}_3\); whereas 3-Methyl-2-pentanone possesses: \(\text{C=O}\) (C1), \(\text{CH}\) (C2), \(\text{CH}_2\) (C3), and two \(\text{CH}_3\) groups, perfectly matching the inventory! Thus, the molecule is 3-methylpentan-2-one (or 2-methylpentan-3-one).
In a stereochemical investigation of a rigid bicyclic lactam, the Nuclear Overhauser Effect cross-peak volume \(V_{ij}\) scales with internuclear distance as \(V_{ij} \propto r_{ij}^{-6}\). A fixed reference distance between two geminal methylene protons is known to be \(r_{\text{ref}} = 1.78\text{ \AA}\), giving a reference NOESY cross-peak volume of \(V_{\text{ref}} = 4.50 \times 10^5\text{ units}\). A cross-peak between bridgehead proton \(H_X\) and methyl ester proton \(H_Y\) has an integrated volume of \(V_{XY} = 8.20 \times 10^3\text{ units}\). (a) Derive the distance formula: \(r_{XY} = r_{\text{ref}} \left(\frac{V_{\text{ref}}}{V_{XY}}\right)^{1/6}\). (b) Calculate the distance \(r_{XY}\) in Angstroms. (c) Molecular modeling indicates that if the methyl ester is in the endo configuration, the distance is predicted to be \(\approx 2.5\text{ \AA}\); if exo, the distance is predicted to be \(\approx 3.5\text{ \AA}\). Determine the stereochemical configuration of the compound.
Step (a): Derivation of distance formula
According to the through-space dipolar cross-relaxation rate in the isolated spin-pair approximation (ISPA):
\[ V_{ij} = k \cdot r_{ij}^{-6} \implies r_{ij} = \left(\frac{k}{V_{ij}}\right)^{1/6} \]Taking the ratio with the reference pair:
\[ \frac{V_{XY}}{V_{\text{ref}}} = \left(\frac{r_{\text{ref}}}{r_{XY}}\right)^6 \implies \left(\frac{r_{XY}}{r_{\text{ref}}}\right)^6 = \frac{V_{\text{ref}}}{V_{XY}} \implies r_{XY} = r_{\text{ref}} \left(\frac{V_{\text{ref}}}{V_{XY}}\right)^{1/6} \]Step (b): Distance calculation
\[ \frac{V_{\text{ref}}}{V_{XY}} = \frac{4.50 \times 10^5}{8.20 \times 10^3} = 54.878 \] \[ (54.878)^{1/6} = 1.9482 \] \[ r_{XY} = 1.78\text{ \AA} \times 1.9482 = 3.468\text{ \AA} \approx 3.47\text{ \AA} \]Step (c): Stereochemical assignment
The experimentally derived distance \(r_{XY} = 3.47\text{ \AA}\) matches the predicted distance for the *exo* isomer (\(\approx 3.5\text{ \AA}\)) within \(1\%\) error, and is drastically larger than the *endo* expectation (\(2.5\text{ \AA}\), which would yield \(V_{XY} \sim 7.2 \times 10^4\)).
Therefore, the compound is unequivocally assigned as the *exo* stereoisomer.
An unknown fragrant ester (\(\text{C}_5\text{H}_{10}\text{O}_2\)) is analyzed by 1D and 2D NMR. Spectral data:
- \(^1\text{H}\) NMR:
Signal A: \(\delta = 4.08\text{ ppm}\) (triplet, \(J = 6.8\text{ Hz}\), \(2\text{H}\)) Signal B: \(\delta = 2.05\text{ ppm}\) (singlet, \(3\text{H}\)) Signal C: \(\delta = 1.65\text{ ppm}\) (sextet, \(J = 6.8\text{ Hz}\), \(2\text{H}\)) Signal D: \(\delta = 0.95\text{ ppm}\) (triplet, \(J = 6.8\text{ Hz}\), \(3\text{H}\))
- 2D COSY cross-peaks:
Cross-peak between Signal A and Signal C. Cross-peak between Signal C and Signal D. Signal B shows zero COSY cross-peaks.
- 2D HSQC correlations:
Signal A correlates with carbon at \(\delta = 66.2\text{ ppm}\). Signal B correlates with carbon at \(\delta = 20.8\text{ ppm}\). Signal C correlates with carbon at \(\delta = 22.0\text{ ppm}\). Signal D correlates with carbon at \(\delta = 10.3\text{ ppm}\). (Uncorrelated carbonyl carbon at \(\delta = 171.2\text{ ppm}\)). (a) Deduce the spin system and connectivity from the COSY correlations. (b) Assemble the complete chemical structure and name the ester.
Step (a): Analysis of spin systems
- Spin System 1 (A-C-D):
- Signal A (\(\delta = 4.08\text{ ppm}\), \(2\text{H}\), triplet) couples to C. Its downfield shift (\(4.08\text{ ppm}\)) and carbon shift (\(66.2\text{ ppm}\)) indicate a methylene group directly bonded to the ester oxygen: \(\text{-O-CH}_2\text{-}\).
- Signal C (\(\delta = 1.65\text{ ppm}\), \(2\text{H}\), sextet) couples to both A (\(2\text{H}\)) and D (\(3\text{H}\)), giving \(n=5\) neighbors \(\implies\) sextet. It is a central methylene: \(\text{-CH}_2\text{-}\).
- Signal D (\(\delta = 0.95\text{ ppm}\), \(3\text{H}\), triplet) couples to C (\(2\text{H}\)). It is a terminal methyl group: \(\text{-CH}_3\).
- Spin System 2 (B):
- Signal B (\(\delta = 2.05\text{ ppm}\), \(3\text{H}\), sharp singlet). Shows zero COSY cross-peaks, meaning it has no protons on adjacent carbons. Its chemical shift (\(2.05\text{ ppm}\)) is characteristic of an acetyl methyl group attached directly to a carbonyl: \(\text{CH}_3-\text{C(=O)-}\).
- Carbonyl group: The quaternary carbon at \(\delta = 171.2\text{ ppm}\) is an ester carbonyl (\(\text{-COO-}\)).
Step (b): Structural assembly
Combining the acetyl group (\(\text{CH}_3\text{CO-}\)) and the propoxy group (\(\text{-OCH}_2\text{CH}_2\text{CH}_3\)):
\[ \text{CH}_3-\text{C}(=\text{O})-\text{O}-\text{CH}_2-\text{CH}_2-\text{CH}_3 \]The compound is uniquely and conclusively identified as propyl acetate (propyl ethanoate).
In a modern Fourier Transform NMR spectrometer, an RF pulse is applied on-resonance with magnetic field amplitude \(B_1 = 5.87 \times 10^{-4}\text{ Tesla}\). For protons with gyromagnetic ratio \(\gamma = 2.67522 \times 10^8\text{ rad}/(\text{s}\cdot\text{T})\): (a) Calculate the nutation frequency \(\omega_1 = \gamma B_1\) in \(\text{rad/s}\) and in kHz. (b) Calculate the duration \(t_{90}\) (in microseconds, \(\mu\text{s}\)) required for a \(90^\circ\) (\(\pi/2\) radian) flip angle. (c) What pulse duration \(t_{180}\) is required for an inversion pulse (\(180^\circ\))?
Step (a): Nutation frequency
\[ \omega_1 = \gamma B_1 = (2.67522 \times 10^8\text{ rad/s}\cdot\text{T})(5.87 \times 10^{-4}\text{ T}) = 1.57035 \times 10^5\text{ rad/s} \]In kHz:
\[ f_1 = \frac{\omega_1}{2\pi} = \frac{1.57035 \times 10^5}{6.283185} = 2.4993 \times 10^4\text{ Hz} \approx 25.0\text{ kHz} \]Step (b): 90-degree pulse duration
The tip angle is \(\theta = \omega_1 t_p\):
\[ \theta = \frac{\pi}{2} \implies t_{90} = \frac{\pi / 2}{\omega_1} = \frac{1.570796}{1.57035 \times 10^5\text{ s}^{-1}} = 1.000 \times 10^{-5}\text{ s} = 10.0\ \mu\text{s} \]Step (c): 180-degree pulse duration
\[ t_{180} = 2 \times t_{90} = 2 \times 10.0\ \mu\text{s} = 20.0\ \mu\text{s} \]A \(10\ \mu\text{s}\) pulse tips the magnetization into the transverse plane, while a \(20\ \mu\text{s}\) pulse fully inverts it.
In a 3D NMR structural investigation of a \(^{15}\text{N}, ^{13}\text{C}\)-labeled protein, two consecutive residues in a \(\beta\)-strand are analyzed: At amide strip 1 (\(H^N = 8.42\text{ ppm}\), \(N = 120.5\text{ ppm}\)): HNCA shows two peaks: \(C^\alpha = 56.4\text{ ppm}\) (strong) and \(C^\alpha = 61.8\text{ ppm}\) (weak). HN(CO)CA shows a single peak: \(C^\alpha = 61.8\text{ ppm}\). At amide strip 2 (\(H^N = 9.15\text{ ppm}\), \(N = 124.8\text{ ppm}\)): HNCA shows two peaks: \(C^\alpha = 61.8\text{ ppm}\) (strong) and \(C^\alpha = 53.2\text{ ppm}\) (weak). HN(CO)CA shows a single peak: \(C^\alpha = 53.2\text{ ppm}\). (a) Explain how HNCA and HN(CO)CA distinguish intra-residue \(C^\alpha(i)\) from inter-residue \(C^\alpha(i-1)\). (b) Determine which residue precedes which: does strip 1 precede strip 2, or does strip 2 precede strip 1? (c) Given typical \(C^\alpha\) chemical shift statistics: Alanine (\(53.2\text{ ppm}\)), Valine (\(61.8\text{ ppm}\)), Leucine (\(56.4\text{ ppm}\)), deduce the sequential amino acid dipeptide sequence.
Step (a): Distinction between intra- and inter-residue C_alpha
- HNCA: Transfers magnetization from \(H^N(i) - N(i)\) to both intra-residue \(C^\alpha(i)\) (via \(^1J_{NC\alpha} \approx 11\text{ Hz}\)) and preceding inter-residue \(C^\alpha(i-1)\) (via \(^2J_{NC\alpha} \approx 7\text{ Hz}\)). Both peaks appear, with the intra-residue peak usually stronger.
- HN(CO)CA: Transfers magnetization through the intervening carbonyl carbon: \(H^N(i) \to N(i) \to C'(i-1) \to C^\alpha(i-1)\). It strictly and exclusively detects the preceding residue \(C^\alpha(i-1)\).
Therefore, by comparing the two experiments: the peak appearing in both HNCA and HN(CO)CA is definitively \(C^\alpha(i-1)\); the peak appearing exclusively in HNCA is \(C^\alpha(i)\).
Step (b): Directional sequence connectivity
- For Strip 2: \(C^\alpha(i) = 61.8\text{ ppm}\), and preceding \(C^\alpha(i-1) = 53.2\text{ ppm}\).
- For Strip 1: \(C^\alpha(i) = 56.4\text{ ppm}\), and preceding \(C^\alpha(i-1) = 61.8\text{ ppm}\).
Notice that the intra-residue \(C^\alpha(i)\) of Strip 2 (\(61.8\text{ ppm}\)) matches the preceding \(C^\alpha(i-1)\) of Strip 1!
This establishes unambiguous sequential connectivity: Strip 2 precedes Strip 1.
Step (c): Tripeptide sequence deduction
- Preceding residue to Strip 2: \(C^\alpha = 53.2\text{ ppm} \implies\) Alanine (Ala)
- Strip 2 residue: \(C^\alpha = 61.8\text{ ppm} \implies\) Valine (Val)
- Strip 1 residue: \(C^\alpha = 56.4\text{ ppm} \implies\) Leucine (Leu)
The tripeptide sequence is conclusively determined as: \(\text{-Ala-Val-Leu-}\).
A marine bioactive metabolite of molecular formula \(\text{C}_9\text{H}_{10}\text{O}_3\) has been isolated and analyzed by 1D and 2D NMR spectroscopy:
1D \(^1\text{H}\) NMR (\(500\text{ MHz}, \text{CDCl}_3\)):
- \(\delta 9.85\) (1H, s, sharp)
- \(\delta 7.82\) (2H, d, \(J = 8.8\text{ Hz}\))
- \(\delta 6.98\) (2H, d, \(J = 8.8\text{ Hz}\))
- \(\delta 3.89\) (3H, s)
- \(\delta 2.61\) (2H, q, \(J = 7.5\text{ Hz}\)) - wait, check degrees of unsaturation: \(\text{C}_9\text{H}_{10}\text{O}_3 \implies \text{DBE} = 9 - 10/2 + 1 = 5\).
Let the observed resonances be:
- \(\delta 9.86\) (1H, s)
- \(\delta 7.81\) (2H, d, \(J = 8.7\text{ Hz}\))
- \(\delta 6.99\) (2H, d, \(J = 8.7\text{ Hz}\))
- \(\delta 4.14\) (2H, q, \(J = 7.0\text{ Hz}\))
- \(\delta 1.45\) (3H, t, \(J = 7.0\text{ Hz}\))
1D \(^{13}\text{C}\) NMR / DEPT-135:
- C1: \(\delta 190.8\) (CH)
- C2: \(\delta 164.2\) (C, quaternary)
- C3: \(\delta 132.0\) (2 \(\times\) CH)
- C4: \(\delta 129.8\) (C, quaternary)
- C5: \(\delta 114.8\) (2 \(\times\) CH)
- C6: \(\delta 63.9\) (\(\text{CH}_2\))
- C7: \(\delta 14.7\) (\(\text{CH}_3\))
- Calculate the Double Bond Equivalents (Degrees of Unsaturation) and assign each carbon to its directly attached protons using HSQC.
- The HMBC experiment shows key long-range \(^2J_{\text{CH}}\) and \(^3J_{\text{CH}}\) correlations:
- Proton \(\delta 9.86\) correlates with C4 (\(\delta 129.8\)) and C3 (\(\delta 132.0\)).
- Protons \(\delta 7.81\) correlate with C1 (\(\delta 190.8\)), C2 (\(\delta 164.2\)), and C5 (\(\delta 114.8\)).
- Protons \(\delta 6.99\) correlate with C4 (\(\delta 129.8\)) and C2 (\(\delta 164.2\)).
- Protons \(\delta 4.14\) correlate with C2 (\(\delta 164.2\)) and C7 (\(\delta 14.7\)).
- Protons \(\delta 1.45\) correlate with C6 (\(\delta 63.9\)).
- Reconstruct the complete constitutional connectivity of the molecule step-by-step and provide its IUPAC name.
- Explain how HMBC distinguishes whether the ethoxy group (\(-\text{OCH}_2\text{CH}_3\)) is attached directly to the aromatic ring vs as an ethyl ester, noting the characteristic differences in chemical shift and correlation topology.
Comprehensive Multi-Step Solution:
Step 1: Degrees of Unsaturation and Direct HSQC Assignments
For \(\text{C}_9\text{H}_{10}\text{O}_3\):
Five degrees of unsaturation indicate a benzene ring (4 unsaturations: 3 \(\pi\)-bonds + 1 ring) plus one carbonyl group (1 \(\pi\)-bond).
Direct One-Bond (\(^1J_{\text{CH}}\)) HSQC Assignments:
- Formyl proton \(\delta 9.86\) (1H, s) \(\rightarrow\) C1 (\(\delta 190.8\), \(\text{CH}\))
- Aromatic protons \(\delta 7.81\) (2H, d) \(\rightarrow\) C3 (\(\delta 132.0\), \(2 \times \text{CH}\))
- Aromatic protons \(\delta 6.99\) (2H, d) \(\rightarrow\) C5 (\(\delta 114.8\), \(2 \times \text{CH}\))
- Oxymethylene protons \(\delta 4.14\) (2H, q) \(\rightarrow\) C6 (\(\delta 63.9\), \(\text{CH}_2\))
- Methyl protons \(\delta 1.45\) (3H, t) \(\rightarrow\) C7 (\(\delta 14.7\), \(\text{CH}_3\))
Quaternary carbons with no HSQC cross-peaks:
- C2 (\(\delta 164.2\))
- C4 (\(\delta 129.8\))
Step 2: HMBC Network Analysis
HMBC detects \(^2J_{\text{CH}}\) and \(^3J_{\text{CH}}\) (rarely \(^4J\) across aromatic rings) heteronuclear couplings:
1. Formyl group (\(-\text{CHO}\)):
- The formyl proton at \(\delta 9.86\) shows \(^2J\) correlation to C4 (\(\delta 129.8\)) and \(^3J\) correlation to C3 (\(\delta 132.0\)).
- This proves the \(\text{CHO}\) group is directly attached to the aromatic ring at quaternary carbon C4.
2. Aromatic Spin System:
- The two symmetrical 2H doublets with \(J = 8.7\text{ Hz}\) (\(\delta 7.81\) and \(\delta 6.99\)) form a classic \(\text{AA}'\text{XX}'\) pattern characteristic of a 1,4-disubstituted (para-substituted) benzene ring.
- Protons at \(\delta 7.81\) (ortho to the electron-withdrawing carbonyl group) correlate via \(^3J\) to the carbonyl carbon C1 (\(\delta 190.8\)), unambiguously placing them at positions 3 and 5 adjacent to C4.
- Protons at \(\delta 6.99\) (ortho to the electron-donating oxygen) correlate via \(^2J\) to C2 (\(\delta 164.2\)) and \(^3J\) to C4 (\(\delta 129.8\)).
3. Ethoxy Group (\(-\text{OCH}_2\text{CH}_3\)):
- Protons at \(\delta 1.45\) (\(\text{CH}_3\)) couple via \(^3J_{\text{HH}}\) to \(\delta 4.14\) (\(\text{CH}_2\)) and exhibit HMBC correlation to C6 (\(\delta 63.9\)).
- Crucially, the oxymethylene protons at \(\delta 4.14\) show a strong \(^3J_{\text{CH}}\) HMBC correlation to aromatic carbon C2 at \(\delta 164.2\).
Step 3: Complete Connectivity Assembly and IUPAC Name
Assembling the fragments:
- Fragment A: Formyl group \(-\text{CHO}\) (C1)
- Fragment B: 1,4-phenylene core (\(-\text{C}_6\text{H}_4-\), C2, C3, C4, C5)
- Fragment C: Ethoxy group \(-\text{O}-\text{CH}_2\text{CH}_3\) (C6, C7)
Connecting them:
- C1 is attached to C4 of the benzene ring.
- Oxygen is attached to C2 of the benzene ring and to C6 of the ethyl group.
Thus, the compound is:
Step 4: Disambiguation from Isomeric Ethyl Esters
If the molecule were an ethyl ester (such as ethyl 4-hydroxybenzoate, which also has formula \(\text{C}_9\text{H}_{10}\text{O}_3\)):
- In an ethyl ester, the ester carbonyl carbon appears at \(\delta 166 - 170\text{ ppm}\), rather than an aldehyde carbonyl at \(\delta 190.8\text{ ppm}\).
- The ester carbonyl would show a strong \(^3J_{\text{CH}}\) HMBC correlation to the oxymethylene protons (\(-\text{COOCH}_2\text{CH}_3\)). In our experimental data, the oxymethylene protons (\(\delta 4.14\)) correlate to the aromatic carbon at \(\delta 164.2\), not to the carbonyl carbon at \(\delta 190.8\).
- The aldehyde proton singlet at \(\delta 9.86\) is entirely absent in an ethyl ester (which would instead show a phenolic \(-\text{OH}\) singlet).
Hence, HMBC connectivity firmly rules out the isomeric ester and unequivocally establishes 4-ethoxybenzaldehyde.
Solved Honors Problems & Derivations
Step-by-step rigorous solutions with full quantum mechanical, thermodynamic, and spectral assignment validation.