Questions: Chemical Exchange Kinetics from NMR Line Shapes

Question 1

You record ¹H NMR spectra of N,N-dimethylformamide (DMF) at temperatures from −30°C to 150°C. At −30°C you observe two sharp methyl peaks separated by 40 Hz. At 150°C you observe one sharp peak. Where does this single peak appear in the spectrum?

• A. At the frequency of the more upfield (shielded) methyl peak
• B. At the population-weighted average frequency of the two original peaks
• C. At the frequency of the more downfield (deshielded) methyl peak
• D. Anywhere between the two original peaks, depending on instrumental conditions

Answer: (B)

In the fast-exchange limit, the nucleus switches between both environments so rapidly that it reports only a time-averaged frequency. Since both methyl groups in DMF are present in equal populations (50% each), the single high-temperature peak appears exactly midway between the two slow-exchange peaks. If the two sites had unequal populations (e.g., 70%/30%), the averaged peak would appear 70% of the way toward the major-site frequency. This population-weighted averaging is a hallmark of fast exchange and is distinct from the two-peak slow-exchange pattern.

Question 2

A researcher claims that the exchange rate constant at coalescence equals the frequency separation between the two peaks (k = Δν). What is wrong with this statement?

• A. Nothing is wrong — this is the correct coalescence condition
• B. The correct condition is k = πΔν/√2, which differs numerically from k = Δν
• C. The exchange rate cannot be determined from the coalescence temperature alone
• D. Coalescence occurs when k equals π/Δν, the inverse of the frequency separation

Answer: (B)

The correct coalescence condition is k = πΔν/√2 ≈ 2.22Δν, not k = Δν. This is one of the most common quantitative errors in NMR exchange analysis. The coalescence condition is derived from the Bloch equations modified for exchange, and the factor of π/√2 arises from the mathematical criterion for the two peaks to just merge into a single broad hump. Using k = Δν underestimates the rate constant at coalescence by roughly a factor of 2.2, leading to significant errors in activation parameters extracted from Eyring analysis.

Question 3

In the fast-exchange limit, NMR reports a single peak at the population-weighted average of the two exchanging sites' resonance frequencies.

• T. True
• F. False

Answer: True

True. When the exchange rate greatly exceeds the frequency separation (k >> πΔν), a nucleus switches environments so rapidly that it effectively samples both during the measurement. The NMR spectrometer records the time-averaged precession frequency: ν_obs = p_A × ν_A + p_B × ν_B, where p_A and p_B are the fractional populations of the two sites. Equal populations (50/50) give a peak exactly midway; unequal populations skew it toward the more abundant site.

Question 4

Lowering the temperature of a sample showing fast-exchange NMR behavior will cause the single averaged peak to split immediately into two sharp peaks.

• T. True
• F. False

Answer: False

False. The transition from fast exchange to slow exchange does not produce an immediate clean splitting. Passing through intermediate exchange as temperature decreases, the single peak first *broadens*, then flattens and merges around the coalescence temperature, and only then resolves into two separate (initially broadened, then sharpening) peaks as the temperature continues to fall. The intermediate exchange regime always produces broadened lines — never an abrupt doubling of a sharp peak.

Question 5

Explain why intermediate exchange — when the exchange rate is comparable to the frequency separation — causes NMR peaks to broaden and eventually coalesce, rather than simply showing two sharp peaks or one sharp averaged peak.

Model answer: Broadening arises from the uncertainty principle applied to frequency. A nucleus must reside in a given environment for time τ to define its resonance frequency with precision ~1/τ. In intermediate exchange, τ is comparable to 1/Δν — the nucleus doesn't stay put long enough to define a precise frequency, introducing frequency uncertainty that manifests as line broadening. As the rate increases further, both environments are sampled so rapidly that neither gives a distinct frequency; a time-averaged frequency dominates and the line resharpens into one peak.

This is the energy-time uncertainty principle applied to NMR line shapes. Exchange contributes an additional dephasing mechanism: each exchange event interrupts the coherent precession of a spin, shortening its transverse coherence time and broadening its peak. When k ≈ Δν, exchange-induced dephasing is maximized (maximum line broadening and coalescence). When k >> Δν, rapid averaging 'motionally narrows' the line to give a single sharp peak. Variable-temperature NMR exploits this: the temperature-dependent line shape encodes the exchange rate at each temperature.