Mixture ambiguity in aromatic-proton counts

Mixture ambiguity in aromatic-proton counts is the inability of a total aromatic proton nuclear magnetic resonance integral to identify all components of a tocopherol mixture. The named substitution patterns leave zero, one, one and two aromatic hydrogens in alpha-tocopherol, beta-tocopherol, gamma-tocopherol and delta-tocopherol, respectively.^[ref]

Let their mole fractions be $x_{\alpha },x_{\beta },x_{\gamma },x_{\delta }$. Under an ideal quantitative response model, an aromatic integral normalized per tocopherol molecule givesThe normalization requires an independently established molecular amount or an equivalent valid calibration. A raw integral alone is not automatically a hydrogen count per molecule. This model also assumes that the selected response includes the relevant aromatic protons with compatible quantitative acquisition conditions.

There are four unknown fractions but only two displayed linear constraints. In particular, exchanging any allowed amount of beta-tocopherol for gamma-tocopherol preserves both constraints: their equal counts cannot distinguish these constitutional isomers.

An original feasible-range calculation makes the remaining ambiguity explicit. Setting $d=x_{\delta }$ givesNonnegative fractions implyFor $r=1.30$, any $d$ between $0.30$ and $0.65$ can fit. The first endpoint has no alpha-tocopherol and a combined beta/gamma fraction of $0.70$; the second has alpha fraction $0.35$ and no beta/gamma component. Intermediate values and beta/gamma exchanges produce further mixtures with the same count.

Resolving additional chemical shifts or using an independently calibrated chromatography measurement can provide new component-sensitive evidence. Repeating the same total integral more precisely only narrows the uncertainty around $r$; it does not add the missing independent composition constraints. The calculation expresses structural possibilities from the named molecules, not an assignment of measured peak positions.