= Stilbene isomer composition measurement
A \b[stilbene isomer composition measurement] estimates how much <cis-Stilbene> and <trans-Stilbene> a preparation contains. Total <stilbene> amount is insufficient: the two <geometric isomers> share a <molecular formula> and <molar mass>, while differing in arrangement around the central <double bond>. NIST records distinguish their structures and identifiers.<cis-Stilbene in the NIST Chemistry WebBook>[Identity, Z-isomer structure and stereoisomer cross-reference]{ref}<trans-Stilbene in the NIST Chemistry WebBook>[Identity, E-isomer structure and stereoisomer cross-reference]{ref}
An original two-wavelength model illustrates the inference. Measure a homogeneous solution in a fixed-path <spectrophotometer>, subtract the appropriate background and divide <absorbance> by path length. If only these two absorbing species contribute and the <Beer-Lambert law> applies,
$$
a_1=\epsilon_{E1}c_E+\epsilon_{Z1}c_Z,\qquad
a_2=\epsilon_{E2}c_E+\epsilon_{Z2}c_Z.
$$
The <molar absorption coefficients> must be calibrated in the same solvent and conditions. Their values are not supplied by the shared molecular formula.
Writing $D=\epsilon_{E1}\epsilon_{Z2}-\epsilon_{E2}\epsilon_{Z1}$, the solution for $D\ne0$ is
$$
c_E=\frac{\epsilon_{Z2}a_1-\epsilon_{Z1}a_2}{D},\qquad
c_Z=\frac{\epsilon_{E1}a_2-\epsilon_{E2}a_1}{D}.
$$
In chosen consistent scaled units, take the coefficient rows $(2,1)$ and $(1,3)$ and observations $(a_1,a_2)=(0.70,0.60)$. Then $D=5$, $c_E=0.30$ and $c_Z=0.10$. The modeled cis fraction is $c_Z/(c_E+c_Z)=0.25$. These coefficients and observations are illustrative, not experimental stilbene spectra.
The <determinant> condition is also a measurement-design warning. If both wavelengths respond to the two isomers in the same proportion, the equations contain only one independent composition constraint. A very small determinant relative to the coefficient scale can amplify reading or calibration errors. More repeated measurements cannot restore information missing from the response model.
A third wavelength provides a <stilbene composition consistency residual>: compare its observation with $\epsilon_{E3}c_E+\epsilon_{Z3}c_Z$ predicted from the first two. A discrepancy can reveal an unmodeled absorber, a changing preparation, an incorrect background or incompatible coefficients, without identifying which cause occurred. Timing and illumination should be controlled when testing a preparation capable of <photoisomerization>. Chemical identity, calibration, measurement geometry and model adequacy are separate requirements for a defensible composition estimate.
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