1What a qNMR purity is

A qNMR assay reports a mass fraction: how many grams of the analyte are present per gram of the solid that was weighed into the tube. Water, residual crystallisation solvent, inorganic salts and silica fines give no usable signal, yet every milligram of them is part of the weighed mass — so they lower the purity. A sample that is 95 % analyte and 5 % water assays at 95 %, even though the water is nearly invisible in the spectrum.

This is the opposite behaviour to a chromatographic area %. An HPLC purity is the analyte's share of the detector response, and anything the detector does not see — water, salts, sorbents, anything without a chromophore at the working wavelength — simply drops out of the calculation, pushing the figure up. The two numbers answer different questions, which is why a material can carry an HPLC purity of 99.5 % and a qNMR assay of 97 % with neither being wrong.

NMR signal area is proportional to the number of nuclei, with the same proportionality for every compound in the tube — there are no response factors and no calibration curve against the analyte itself. The comparison runs against any well-chosen certified standard, which is why the international metrology bodies class qNMR as a potential primary ratio method, alongside gravimetry, titrimetry and coulometry, and why it works for compounds that have no reference standard of their own.

2The measurement equation

With subscript a for the analyte and s for the internal standard, the assay computes:

Read it as two mole ratios, as in Part 1: the first two factors are the mole ratio the spectrum measures, the next two are the mole ratio the balance fixed, and the assay is the first over the second, scaled by the standard's own purity. Each term has its own error sources:

The molar mass must be that of the solid actually weighed. A hydrochloride weighed as the salt but entered as the free base biases the assay by the full mass ratio of the two forms, and no spectroscopy afterwards can correct it.

3The internal standard

The standard appears in every ratio of the equation, and its selection criteria are settled: certified purity with a stated uncertainty, traceable to a primary standard; chemical inertness towards the analyte and the solvent; adequate solubility; no hygroscopicity or volatility that would falsify the weighing; and few signals, placed where the analyte has none. Certified reference materials meeting these criteria are produced against the primary standards of the national metrology institutes — NIST's benzoic acid primary standard is certified at a mass fraction of 0.99992 g/g, and the corresponding Japanese (NMIJ) materials at 0.9993–0.9998 — with commercial CRM lines certified against them at relative expanded uncertainties of roughly 0.1–0.3 %.

Common certified internal standards for ¹H qNMR

The T₁ column is indicative only: T₁ is a property of a compound in a particular solution, and it varies with solvent, concentration, temperature and field. It is listed because the slowest-relaxing signal in the tube, standard included, sets the length of the whole experiment, as §4 shows; the single aromatic proton of tetrachloronitrobenzene relaxes in roughly ten seconds and more, and regularly requires a longer delay than the analyte would. For the measurement itself, T₁ is measured, not assumed.

Beyond the table, the deciding factor is spectral space: the standard's signals must fall where the analyte, its impurities, the residual solvent and water are all absent. In practice the analyte's ordinary ¹H spectrum is recorded first, and the standard is chosen so that its singlet falls into a clear region.

4Acquisition that makes an integral true

Part 1 listed what a trustworthy integral needs. For an absolute assay those requirements acquire numbers.

The relaxation delay. After each pulse the magnetisation recovers exponentially with the time constant T₁, and any signal read before it has recovered integrates low. The recovery is 95.0 % after 3 × T₁, 99.3 % after 5 × T₁, 99.9 % after 7 × T₁ and better than 99.99 % after 10 × T₁ — counted for the slowest-relaxing signal in the tube, which is frequently the standard's isolated singlet rather than the analyte. Routine quantitative work runs at five to seven times T₁ with a 90° pulse; metrology laboratories run at ten to fifteen. A quantitative acquisition is therefore long: at seven times a 10 s T₁, each scan includes over a minute of waiting.

Signal-to-noise. The signal-to-noise ratio bounds the precision of any integral drawn through noise. The validation literature puts the requirement at 150:1 and above for a target uncertainty of 1 %, with 250:1 quoted as the working figure for routine ¹H assays; laboratories certifying reference materials work near 1000:1. Signal-to-noise grows with the number of scans, and each scan includes the relaxation delay above — the two requirements together set the length of the experiment.

The ¹³C satellites are part of the signal. Every C–H resonance carries two satellites, one either side at half the one-bond C–H coupling — typically some 60–100 Hz out — from the 1.1 % of molecules whose carbon is ¹³C. Each satellite holds 0.55 % of the parent's area. At the 1 % level they cannot be ignored: either the integration range includes them for analyte and standard alike, or they are collapsed into the parent by ¹³C decoupling. They also have a use: each satellite is an intensity reference of 0.55 % of one proton, against which a small impurity peak can be judged directly.

The remaining requirements are brief: a tuned and matched probe; temperature regulated through the run, held to a drift of 0.1 K; the standard and analyte at roughly similar molar amounts; no sample spinning, so that spinning sidebands never appear; and no solvent presaturation, which suppresses more than the solvent. Each requirement protects the same proportionality of area to number of nuclei.

5Processing the spectrum

The spectrum on the screen is produced from the free-induction decay by a chain of processing choices, and each choice affects the integrals. Quantitative processing is conservative: two- to three-fold zero filling for digital resolution; exponential line broadening no stronger than 0.3 Hz; phasing done manually, because an automatic phase that is almost right produces the dispersive tails that ruin adjacent integrals; and a baseline corrected deliberately, since any residual offset is integrated along with every peak that sits on it.

Then the limits are drawn. An NMR line is Lorentzian, and a large share of a Lorentzian's area lies far from the peak. Part 1 stated this qualitatively; the exact fractions are:

Share of a pure Lorentzian line's area inside the integration range

A range that looks generous on screen — ten line widths either side — still clips three per cent of the area. The classical working rule, an integration range of 64 line widths in total, recovers 99.0 %; the metrology laboratories extend to 76 line widths each side and further. Two consequences follow. First, both signals must be cut the same way: identical relative ranges for analyte and standard cancel most of the clipped tails in the ratio. Second, ranges this wide include whatever else lies in the window — satellites, as above, and any impurity — which is why line fitting (deconvolution) is used where signals are too close for a clean integration range.

6What the assay looks like in practice

A published validation shows the parts assembled. Tadalafil was assayed in DMSO-d₆ against 2,4-dinitrotoluene as the internal standard: the two compounds were chosen so that three aromatic tadalafil signals stand clear of the standard's, the methyl signals were rejected for quantification precisely because of their long T₁, and the method's limit of quantification was set from the signal-to-noise requirement of the validation literature. The assay agreed with HPLC on the same materials — an orthogonal confirmation, since the two methods share no physics.

7The uncertainty of the number

A purity figure is complete only with its uncertainty. The budget behind a qNMR assay has few entries: the certified uncertainty of the standard, entering every result unchanged; the two weighings — the reason metrology work uses a microbalance and sets minimum portions: at a ten-milligram portion, a weighing uncertainty of a few micrograms contributes a few hundredths of a per cent; and the integration itself, carrying the signal-to-noise, baseline and range effects of §4 and §5, which with reasonable care is near a tenth of a per cent, and dominates when it is not.

Combined, carefully executed routine assays report relative expanded uncertainties around 0.5–1 %. Specialised high-precision work does better: a validated procedure on 2.5 mg of material has been reported at a combined uncertainty of 0.15 %, and the laboratories that certify reference materials by qNMR state relative expanded uncertainties down to 0.1 %. Those figures describe specialised practice, not every certificate. A purity of 99.5 % from an assay with a 1 % uncertainty is a different statement from the same number with 0.2 %.

8Where qNMR fails

The limits follow from the assumptions. Overlap is the most common: the quantified signals of analyte and standard must each stand on clear baseline, and in a crowded aromatic region, or a natural-product extract, no standard may have a singlet to place. Exchangeable protons fail differently — a signal in exchange with water quantifies the exchange pool, not the molecule, and signals adjoining the water resonance are compromised with it. Broad lines invalidate the integration arithmetic of §5, which assumed a narrow Lorentzian; a line broadened by dynamics or by aggregation retains its area but spreads it beyond any practical range. Molecules with few protons for their mass provide little signal per milligram, and the achievable precision falls. And below roughly the satellite level, one part in a few hundred per proton, ¹H integration runs out of dynamic range for impurity work — there chromatography, or heteronuclear qNMR on a clearer axis, is the better tool.

None of these is rare. A competent report states which of them applied and what was done about it — a change of solvent, a different standard, a fitted rather than integrated signal.

9Recognition

The internal-standard assay is codified. The European Pharmacopoeia's NMR chapter (2.2.33) provides for quantitation against an intensity standard under conditions of essentially complete relaxation; the United States Pharmacopeia's chapter <761> defines the same procedure as its internal standard method; the Japanese Pharmacopoeia assigns the content of some two dozen of its crude-drug assay reagents by qNMR against SI-traceable standards; and ISO 24583:2022 specifies the ¹H internal-standard method for purity determination in the food field. The BIPM classes qNMR as a potential primary reference measurement procedure, and the primary standards it is anchored to — NIST's benzoic acid at 0.99992 g/g among them — are certified by methods independent of NMR. A purity measured this way is traceable, through the certificate of one weighed standard, to the SI.