= Mixture ambiguity in aromatic-proton counts
\b[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.<Nomenclature of Tocopherols and Related Compounds by IUPAC-IUB Joint Commission on Biochemical Nomenclature>[Recommendations 1.2 and 2–5]{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 gives
$$
x_\alpha+x_\beta+x_\gamma+x_\delta=1,\qquad
r=x_\beta+x_\gamma+2x_\delta.
$$
The 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 $d=x_\delta$ gives
$$
x_\alpha=1-r+d,\qquad x_\beta+x_\gamma=r-2d.
$$
Nonnegative fractions imply
$$
\max(0,r-1)\le d\le\frac r2,\qquad 0\le r\le2.
$$
For $r=1.30$, 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.
\Image[/tocopherol-count-feasible-mixtures.png]
{title=Different tocopherol mixtures with the same aromatic-proton count}
{description=Original feasible-region diagram for a synthetic normalized count $r=1.30$. Every blue point has the same count; the remaining mole fraction can still be divided between beta- and gamma-tocopherol without changing that count. The gray triangle imposes nonnegative fractions summing to one.}
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