§2 The Idea, Step by Step
Here is a strange fact: the nucleus of a hydrogen atom behaves like a tiny spinning bar magnet. On its own it points every which way, but slide it into a powerful magnet and it lines up — and now you can make it "sing." Tickle it with a radio wave of exactly the right pitch and it absorbs energy and flips. That special pitch is the whole trick behind NMR. The clever part is that not every hydrogen in a molecule sings the same note: each one is wrapped in a slightly different cloud of electrons, and that cloud shields it from the big magnet. So a molecule full of hydrogens hums a little chord, and reading that chord tells you how the molecule is built.
Build it up — the note each proton sings
The pitch is the Larmor frequency, $\nu = \dfrac{\gamma B_0}{2\pi}$, where $B_0$ is the magnet's field strength and $\gamma$ is a constant for hydrogen. The electron cloud knocks the field down a touch to $B_0(1-\sigma)$ — a richer cloud means more shielding ($\sigma$ larger) and a lower note. Rather than quote raw megahertz, chemists report a proton's position as the chemical shift $\delta$, in parts per million (ppm), measured from a reference (TMS) set to $\delta = 0$. One worked number: in a $B_0 = 7.05$ T magnet a bare proton sings at 300 MHz, and benzene's ring protons sit way down the scale at $\delta \approx 7.27$ ppm.
Make it precise — three numbers per peak
The full statement is $\nu = \dfrac{\gamma B_0}{2\pi}(1-\sigma)$ and $\delta = \dfrac{\nu_{sample}-\nu_{TMS}}{\nu_{spectrometer}}\times 10^6$. Every peak then carries three clues. Its position ($\delta$) reveals the electron environment — electron-poor protons are pushed downfield to high $\delta$. Its splitting reports the neighbours: $n$ equivalent neighbouring protons split a signal into $n+1$ lines with Pascal's-triangle intensities, spaced by the coupling constant $J$ (in Hz, and — unlike $\delta$ — independent of $B_0$). Its area counts how many protons sit there. In the sim the sliders map straight onto this: $B_0$ sets the spectrometer frequency, $\delta$ slides the peak along the axis, $J$ stretches the line spacing, and $n$ chooses the multiplet.
Try this in the sim above
(1) In Splitting Patterns, set $n=0$ for a lone singlet, then walk $n$ up to 3 and watch a quartet appear with the 1:3:3:1 pattern. (2) In ¹H NMR Spectrum, drag $B_0$ from 7 T toward 14 T: the spectrometer frequency roughly doubles toward 600 MHz, yet every peak stays put in ppm — proof that $\delta$ is field-independent. (3) Load the Ethanol preset and read the integration heights: CH₃ : CH₂ : OH come out 3 : 2 : 1, exactly the proton count of CH₃CH₂OH.
§3 Equation Derivation
NMR Resonance Condition & Chemical Shift
\[\nu = \frac{\gamma B_0}{2\pi}(1-\sigma)\qquad \delta = \frac{\nu_{sample}-\nu_{TMS}}{\nu_{spectrometer}}\times10^6\text{ ppm}\]
| Symbol | Meaning | Value / Unit |
|---|
| \(\nu\) | Larmor (resonance) frequency | Hz or MHz |
| \(\gamma\) | Gyromagnetic ratio (¹H: 2.675×10⁸ rad T⁻¹s⁻¹) | rad T⁻¹ s⁻¹ |
| \(B_0\) | External magnetic field strength | T (Tesla) |
| \(\sigma\) | Shielding constant (from electron density) | dimensionless |
| \(\delta\) | Chemical shift | ppm |
| \(J\) | Scalar coupling constant (spin-spin) | Hz |
| \(n\) | Number of equivalent neighbouring protons | — |
Step-by-Step Derivation
Step 1 — Nuclear spin in a magnetic field
A proton (spin I=½) in field B₀ has two energy states: \(E_{\pm} = \mp\frac{1}{2}\gamma\hbar B_0\)
Energy gap: \(\Delta E = \gamma\hbar B_0 = h\nu\)
Step 2 — Resonance condition (Larmor frequency)
Absorption occurs when RF frequency matches: \(\nu_0 = \dfrac{\gamma B_0}{2\pi}\)
For ¹H at B₀ = 7.05 T: ν₀ = (2.675×10⁸ × 7.05)/(2π) = 300 MHz
Step 3 — Chemical shielding (local electron current)
Electrons surrounding the nucleus create an opposing field: B_local = B₀(1−σ). The effective field at the nucleus is reduced → resonance at lower frequency. More electron-dense environments → higher σ → more shielded → smaller δ (upfield).
Step 4 — Chemical shift definition (ppm)
\(\delta(\text{ppm}) = \dfrac{\nu_{sample}-\nu_{TMS}}{\nu_{spectrometer}}\times10^6\)
TMS (tetramethylsilane) is reference at δ = 0. Scale is field-independent in ppm.
Step 5 — Spin-spin coupling (n+1 rule)
Neighbouring proton spins can be α (↑) or β (↓). For n equivalent neighbours, the signal splits into n+1 lines with binomial intensities (Pascal's triangle).
Doublet (n=1): 1:1 · Triplet (n=2): 1:2:1 · Quartet (n=3): 1:3:3:1
Step 6 — Coupling constant J (independent of B₀)
J is the frequency separation between lines in a multiplet, in Hz. J is mediated through bonds (not through space for scalar coupling) and is independent of B₀ — distinguishing it from chemical shift differences (which scale with B₀ in Hz but are constant in ppm).
Worked Example — Ethanol ¹H NMR
CH₃–CH₂–OH at 300 MHz (B₀ = 7.05 T)
| Group | δ (ppm) | Multiplicity | Integration |
| CH₃ | 1.2 | Triplet (n=2 from CH₂) | 3H |
| CH₂ | 3.7 | Quartet (n=3 from CH₃) | 2H |
| OH | 2.6 | Broad singlet (fast exchange) | 1H |
Reference: Clayden, Greeves & Warren — Organic Chemistry, 2nd Ed., Chapter 13: "¹H NMR" | Skoog et al. — Fundamentals of Analytical Chemistry, 9th Ed., Chapter 19
§4 Frequently Asked Questions
🔬 SimulationWhat does each simulation mode show?▼
The Spin Precession mode shows a ¹H nucleus precessing (rotating) around B₀ at the Larmor frequency — when RF matches this frequency, the spin flips from α to β. ¹H NMR Spectrum shows a realistic simulated spectrum with peaks at characteristic δ values for the selected molecule. Splitting Patterns shows how the n+1 rule generates doublets, triplets, quartets with correct Pascal's triangle intensities. Chemical Shift shows a δ reference chart with typical ranges. Ring Current shows the anisotropic shielding of aromatic rings that causes aromatic H to appear at δ 7–8.
Key: NMR peaks encode three pieces of information: position (δ = chemical environment), splitting (number of neighbours), and area (number of protons).
🌍 Real LifeWhere is NMR used in real life beyond the laboratory?▼
MRI (Magnetic Resonance Imaging) in hospitals is ¹H NMR of water in soft tissues — the same physics as laboratory NMR. Pharmaceutical companies use NMR daily to confirm the structure of newly synthesized drugs (identity testing, purity, stereochemistry). Food authentication: honey adulteration, olive oil purity, and wine analysis are all done by NMR. Petroleum geology uses low-field NMR to measure porosity in rock cores. In Bangladesh, universities and pharmaceutical companies (Square, Beximco) use NMR for drug structure confirmation and research.
Key: MRI is NMR of your body's water — the same Larmor frequency equation applies, just with water protons in tissue instead of organic molecules.
🧪 ConceptualWhy do aromatic protons appear at δ 7–8, further downfield than expected?▼
The 6 π electrons in benzene circulate in a ring current when placed in B₀. This circulation creates a secondary magnetic field that runs counter to B₀ inside the ring (shielding) but aligns with B₀ outside the ring (deshielding). Protons on the outside of the ring (all aromatic H in benzene) experience this deshielding effect, so they resonate at higher δ (7.27 ppm for benzene) than typical C–H (δ 0–4). Hypothetically, protons positioned above/below the ring would appear at very low δ (upfield) due to the shielding cone.
Key: Ring current deshielding is unique to aromatic systems — the δ 7–8 region is a reliable indicator of aromatic protons.
🧮 MathematicalHow does changing field strength B₀ affect the spectrum?▼
The Larmor frequency scales linearly with B₀: ν₀ = γB₀/2π. At B₀ = 7.05 T, ¹H resonates at 300 MHz; at 14.1 T → 600 MHz. Chemical shifts in ppm remain constant (by definition), so peaks appear at the same δ values regardless of field. However, coupling constants J (in Hz) are field-independent while chemical shift differences (in Hz) scale with B₀. Higher field separates overlapping peaks: two peaks 0.05 ppm apart differ by 15 Hz at 300 MHz but 30 Hz at 600 MHz — much easier to resolve.
Key: Higher field → better resolution (because Δν_Hz increases but J stays constant) → cleaner multiplets and separation of overlapping peaks.
💡 Non-ObviousWhy does the OH proton of ethanol sometimes appear as a singlet and sometimes as a triplet?▼
In dry, pure ethanol, the OH proton couples with the adjacent CH₂ (n=2 neighbours) and should appear as a triplet by the n+1 rule. However, in practice with trace acid, water, or at room temperature, the OH proton undergoes rapid intermolecular exchange (proton transfer between molecules). If exchange is faster than the coupling constant (J ≈ 5–7 Hz), the coupling is averaged to zero — appearing as a broad singlet. D₂O addition (D₂O shake) removes the OH peak entirely, confirming which peak is OH. This exchange-broadening is also why carboxylic acid COOH appears as a broad singlet.
Key: Exchangeable protons (OH, NH, COOH) often appear as broad singlets due to rapid proton exchange — not because they have no neighbours.
🌌 DeepHow does NMR determine 3D molecular structure?▼
2D NMR experiments like COSY (Correlation Spectroscopy) reveal which protons are coupled (3–4 bonds apart) — effectively tracing the carbon skeleton. NOESY (Nuclear Overhauser Effect Spectroscopy) detects protons that are close in space (< 5 Å) regardless of bonding — this reveals 3D folding. For proteins, these NOE distances are fed into computational algorithms (distance geometry, molecular dynamics) to calculate the 3D structure. This is how protein solution structures are determined without crystals. The complete structures of antibiotics, natural products, and therapeutic proteins are routinely solved by NMR.
Key: 2D NMR + NOE distances → 3D molecular structure in solution — this is how proteins are structurally characterized without X-ray crystallography.
Reference: Clayden, Greeves & Warren — Organic Chemistry, 2nd Ed., Chapters 13–14 | Skoog et al. — Fundamentals of Analytical Chemistry, 9th Ed., Chapter 19 | LibreTexts Chemistry — NMR Spectroscopy https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Organic_Chemistry/13%3A_Structure_Determination/13.04%3A_NMR_Spectroscopy
§5 Common Misconceptions
❌ Misconception: "The n+1 rule always works — every proton with n neighbours gives n+1 peaks."
✅ Correction: The n+1 rule applies only to protons with equivalent neighbours. If the neighbouring protons have different coupling constants (like diastereotopic CH₂ protons), the splitting is more complex (dd, ddd patterns). Also, aromatic protons show complex second-order multiplets when chemical shift differences are comparable to J. Magnetically equivalent protons do not split each other. The n+1 rule is a first-order approximation valid only when Δν (Hz) >> J (Hz).
📖 Reference: Clayden, Greeves & Warren — Organic Chemistry, 2nd Ed., Chapter 13: "Spin–spin coupling" — first-order approximation limits
❌ Misconception: "Downfield means lower field, so downfield protons have less electron density and are more shielded."
✅ Correction: This is exactly backwards. Downfield (higher δ, e.g. δ 7–12) means deshielded — the proton has LESS electron density around it, experiences a stronger effective B-field, and resonates at higher frequency. Upfield (lower δ, e.g. δ 0–2) means shielded — more electrons surround the proton, reducing the effective field. Electronegativity withdraws electrons from the proton → deshielded → downfield. Alkyl groups donate electrons → shielded → upfield.
📖 Reference: Clayden, Greeves & Warren — Organic Chemistry, 2nd Ed., Chapter 13: "Chemical shift and electron density"
❌ Misconception: "The area under an NMR peak tells you the total number of protons in the molecule."
✅ Correction: Integration gives the relative number of protons — the ratio between peaks, not the absolute number. If a molecule shows peaks with areas 3:2:1, the proton ratio is 3:2:1. To know the absolute number, you need additional information (molecular formula from mass spectrometry, or an internal standard of known concentration). A triplet and a quartet with integration ratio 3:2 tells you there are 3 protons in one group and 2 in the other, consistent with an ethyl group (CH₃:CH₂ = 3:2).
📖 Reference: Skoog et al. — Fundamentals of Analytical Chemistry, 9th Ed., §19B-4: "Integration of NMR signals"
❌ Misconception: "Chemically equivalent protons are the same as magnetically equivalent protons."
✅ Correction: Chemically equivalent protons have the same chemical shift (same δ). Magnetically equivalent protons are chemically equivalent AND have identical coupling to every other proton in the molecule. In para-disubstituted benzene (e.g., 4-chlorobromobenzene), H₂ and H₆ are chemically equivalent (same δ) but magnetically non-equivalent (different coupling to H₃). Magnetically non-equivalent protons cause complex second-order spectra (AA'BB' systems), not the simple n+1 patterns assumed for first-order spectra.
📖 Reference: Clayden, Greeves & Warren — Organic Chemistry, 2nd Ed., Chapter 13: "Magnetic and chemical equivalence"
❌ Misconception: "¹H NMR directly shows the structure of a molecule — you can read off the structure from the spectrum alone."
✅ Correction: NMR provides strong structural evidence but rarely unambiguous proof alone. The same spectrum can sometimes fit multiple structures, especially for complex molecules. Full structure determination requires combining ¹H NMR with ¹³C NMR (carbon skeleton), 2D NMR (COSY for connectivity, HSQC for C-H pairs, HMBC for long-range C-H), mass spectrometry (molecular formula), and IR (functional groups). Even then, absolute configuration (R/S) often requires additional methods like X-ray crystallography or optical rotation.
📖 Reference: Clayden, Greeves & Warren — Organic Chemistry, 2nd Ed., Chapter 13 + 14: "NMR is one piece of the structure puzzle"
Section 4 reference: Taber, K.S. — Chemical Misconceptions (RSC, 2002) | Clayden, Greeves & Warren — Organic Chemistry, 2nd Ed. | Hornak, J.P. — The Basics of NMR (online textbook, Rochester Institute of Technology) https://www.cis.rit.edu/htbooks/nmr/