= Stilbene composition consistency residual
A \b[stilbene composition consistency residual] is the difference between an additional optical observation and the value predicted from a fitted <stilbene> isomer mixture. It tests agreement with the measurement model rather than directly measuring an impurity or proving sample purity.
In a <stilbene isomer composition measurement>, two calibrated wavelengths can determine modeled concentrations $c_E,c_Z$ of <trans-Stilbene> and <cis-Stilbene>. Let $a_j$ be background-subtracted <absorbance> divided by path length, and let $E_{12}$ contain the first two wavelengths' <molar absorption coefficients>. For nonsingular $E_{12}$,
$$
\widehat c=E_{12}^{-1}\begin{pmatrix}a_1\\a_2\end{pmatrix},\qquad
r_3=a_3-e_3^{\mathsf T}\widehat c.
$$
The vector $e_3$ contains the third wavelength's coefficients. That third observation was not used to fit the two concentrations, so it provides an additional consistency condition.
An original example uses coefficient rows $(2,1)$, $(1,3)$ and $(1,1)$ in consistent scaled units. Observations $(a_1,a_2)=(0.70,0.60)$ give $(\widehat c_E,\widehat c_Z)=(0.30,0.10)$ and third-channel prediction $0.40$. An observed $a_3=0.43$ gives $r_3=0.03$. These are constructed coefficients, not measured stilbene spectra; the primary identity records establish that the two <geometric isomers> are chemically distinct despite their shared formula.<cis-Stilbene in the NIST Chemistry WebBook>[Identity and stereoisomer fields]{ref}<trans-Stilbene in the NIST Chemistry WebBook>[Identity and stereoisomer fields]{ref}
The fitted concentrations also carry reading error. In this example,
$$
r_3=a_3-0.4a_1-0.2a_2.
$$
With $q=(-0.4,-0.2,1)^{\mathsf T}$ and observation <covariance matrix> $\Sigma$, linear <uncertainty propagation> gives $u_r^2=q^{\mathsf T}\Sigma q$ when the coefficients are treated as fixed. Independent readings with equal <standard uncertainty> $0.01$ give $u_r=0.01\sqrt{1.2}\simeq0.01095$. The residual is then about $2.74$ such standard uncertainties; interpreting that as a probability statement needs an additional distribution model. Calibration-coefficient uncertainty must also be included when material.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Sections 5.1–5.2, correlated input quantities]{ref}
A common additive offset $b$ changes this residual by $0.4b$, so it does not cancel automatically. A nonzero residual can reflect background, changing illumination, coefficient errors or an unmodeled absorber. Conversely, an extra component whose responses mimic a combination of the two modeled spectra can remain hidden even when the residual is zero. Consistency is evidence about a stated model, not a unique identification of the sample's full composition.
Back to article page
