Result

UV-visible spectroscopy

T-080Home CU-302Threads quantum · structure

Statement

Electronic transitions and chromophores.

Why it matters

rotational-spectroscopy and vibrational-ir-spectroscopy probe a molecule's lowest-energy motions, tumbling and then bond stretching/bending; ultraviolet-visible spectroscopy moves up the energy hierarchy once again to probe electronic structure directly — which molecular orbitals are occupied and how far apart in energy the frontier orbitals sit. beer-lambert-law supplies the quantitative tool, used throughout this technique and the others in the unit, for converting a measured absorbance into a chromophore's solution concentration.

Because the technique is directly sensitive to conjugation and electronic delocalisation, it is a standard, practical tool for characterising dyes, pigments, and any conjugated organic system, complementing rather than duplicating the structural information available from mass-spectrometry and nmr-chemical-shift.

Hypotheses

Absorption requires a photon whose energy exactly matches the gap between two electronic states.This is a resonance condition analogous to, but at a far higher energy scale than, the resonance conditions governing rotational-spectroscopy and vibrational-ir-spectroscopy; only a photon of the correct energy can drive the specific electronic transition in question. The molecule must possess a chromophore: a functional group or extended conjugated system with an accessible, sufficiently low-energy electronic transition.Fully saturated hydrocarbons, having only high-energy \(\sigma\to\sigma^*\) transitions, absorb only in the far, vacuum-ultraviolet region and appear essentially transparent to an ordinary UV-visible instrument, which typically covers only roughly \(200\)-\(800\,\text{nm}\). Interpreting the absorbance axis quantitatively, not merely qualitatively, additionally requires beer-lambert-law's own conditions — dilute solution and monochromatic light — to hold.

Proof

\[ \Delta E_{\text{electronic}} = h\nu = \frac{hc}{\lambda} \]

A photon is absorbed only if its energy matches the gap between an occupied and an unoccupied molecular orbital, the electronic analogue of the resonance conditions used throughout this spectroscopy unit. A

\[ \sigma\to\sigma^* > n\to\sigma^* > \pi\to\pi^* > n\to\pi^* \quad (\text{decreasing transition energy}) \]

Common organic chromophore transitions rank in this order of decreasing energy; \(\pi\to\pi^*\) transitions (typical of isolated C=C or C=O groups) and the generally lower-energy, often weaker \(n\to\pi^*\) transitions dominate the ordinary UV-visible region for most organic chromophores. A

\[ \text{Extending conjugation lowers the HOMO-LUMO gap, shifting absorption to longer wavelength (a "red shift").} \]

As more \(\pi\) orbitals delocalise and mix along a chain of alternating double bonds, the resulting molecular-orbital energy levels crowd closer together, narrowing the HOMO-LUMO gap and shifting the associated absorption progressively toward longer wavelength as conjugation length increases. A

\[ A=\varepsilon c l \]

beer-lambert-law relates the measured absorbance at a chromophore's characteristic wavelength to its solution concentration, given the molar absorptivity \(\varepsilon\) characteristic of that specific electronic transition. A

\[ \text{Relaxation: fluorescence, phosphorescence, or non-radiative decay return the excited state to the ground state.} \]

An excited electronic state can return to the ground state either by re-emitting a photon (fluorescence, same spin multiplicity; phosphorescence, different spin multiplicity, typically slower) or, more commonly, by dissipating the absorbed energy as heat without emitting a photon at all. B

Result

\[ \Delta E=h\nu=\frac{hc}{\lambda}\ (\text{electronic transition}); \qquad A=\varepsilon c l\ (\text{quantitation}) \]

Reading. Absorption wavelength is set by the chromophore's HOMO-LUMO-type gap, itself tuned systematically by conjugation; the measured absorbance at that wavelength quantifies the chromophore's concentration.

Scope. Requires a genuine accessible chromophore (Hypotheses); quantitation additionally requires the dilute, monochromatic conditions established in beer-lambert-law.

Corollaries & converses

Increasing conjugation length is the standard, reliable way to shift absorption from the UV into the visible range, which is why extensively conjugated molecules are visibly coloured while simple, unconjugated organics are not.
The same resonance-absorption logic used here, in vibrational-ir-spectroscopy, and in rotational-spectroscopy differs essentially only in the energy scale of the transition involved, which is why the techniques give complementary rather than redundant structural information.
A molecule's molar absorptivity \(\varepsilon\) is a property of the specific electronic transition, so different chromophores within the same molecule can be probed selectively at their own characteristic wavelengths.
An auxochrome — a substituent such as \(-\text{OH}\) or \(-\text{NH}_2\) that does not itself absorb strongly in the UV-visible range but shifts and intensifies a nearby chromophore's absorption when attached to it, typically by donating electron density into the conjugated system — is a further, systematic modifier of \(\lambda_{\max}\) layered on top of the basic conjugation-length trend of Step 3.

Fails without

No accessible chromophore (Hypotheses): a molecule with only \(\sigma\) bonds and no conjugation has no low-energy electronic transition within the ordinary UV-visible range, and shows essentially no absorption there at all, however carefully measured, regardless of concentration.
Drop beer-lambert-law's dilute, monochromatic conditions (\(t3\)): at high concentration or with polychromatic light, the absorbance-concentration relationship becomes non-linear, and the simple wavelength-to-transition-energy reading of Step 1 remains valid but the quantitative concentration determination of Step 4 becomes unreliable.

Common errors

Assuming every organic molecule absorbs somewhere in the ordinary UV-visible range; molecules with no chromophore beyond isolated \(\sigma\) bonds absorb only far outside this range.
Confusing a \(\pi\to\pi^*\) transition (generally more intense) with an \(n\to\pi^*\) transition (generally weaker) when interpreting relative peak intensities.
Assuming increasing conjugation shifts absorption toward shorter wavelength; the well-established trend (Step 3) is the opposite, a red shift with increasing conjugation.
Treating an observed absorption maximum as a single, sharply defined electronic state, when vibrational fine structure and solvent effects usually broaden a UV-vis band considerably compared with a much sharper rotational line.

Discussion

The systematic empirical rules relating chromophore structure, especially conjugation length, to absorption wavelength — the Woodward-Fieser rules for conjugated dienes and enones — were developed by Robert Burns Woodward and later extended by Louis Fieser and Mary Fieser through the mid-20th century, and remain a standard structure-elucidation tool from a UV-vis spectrum alone.

Solvent choice also shifts a chromophore's absorption maximum, an effect called solvatochromism: a polar solvent stabilises a polar excited state relative to the ground state (or vice versa) to a different degree than it stabilises the ground state alone, shifting \(\lambda_{\max}\) measurably depending on solvent polarity; this is a routine, practical complication whenever comparing literature UV-vis data recorded in different solvents.

Common misconception: that the colour an object appears is the colour of light it absorbs. As with spectrochemical-colour's coordination-complex case, the observed colour of a transmitted or reflected sample is the complement of the absorbed wavelength, not the absorbed wavelength itself.

Worked examples

\[ \text{Ethylene (isolated C=C): }\lambda_{\max}\approx170\,\text{nm}; \quad \text{buta-1,3-diene (conjugated): }\lambda_{\max}\approx217\,\text{nm} \]

Extending conjugation from one isolated double bond to two conjugated double bonds shifts the \(\pi\to\pi^*\) absorption maximum from the far-UV, invisible to a standard instrument, to a longer, still-UV but more accessible wavelength, illustrating Step 3's red shift directly. A

\[ \beta\text{-carotene (extensively conjugated polyene): }\lambda_{\max}\approx450\,\text{nm, appears orange} \]

Continuing to extend conjugation shifts absorption all the way into the visible region; \(\beta\)-carotene absorbs strongly in the blue-violet region and, by Step 4's complementary-colour rule, appears strongly orange-coloured as a result. A

\[ \text{Ethylene (170 nm)} \to \text{butadiene (217 nm)} \to \beta\text{-carotene (450 nm, visible, orange)} \]

Reading. Progressively extending conjugation red-shifts absorption from the far UV, through the near UV, and eventually into the visible range, exactly the mechanism (Step 3) behind why extensively conjugated natural pigments are visibly coloured.

Scope. The same trend, extending conjugation red-shifts absorption, applies generally across essentially all classes of conjugated organic chromophores.

Further example. A solution of a conjugated organic dye is prepared in a cuvette with a path length of \(1.00\,\text{cm}\). At the absorption maximum, the measured absorbance is \(A = 0.450\). The molar extinction coefficient for this dye at this wavelength is known to be \(\varepsilon = 2.50 \times 10^4\,\text{M}^{-1}\text{cm}^{-1}\). Calculate the molar concentration of the dye solution.

\[ A = \varepsilon l c \]

This is the Beer-Lambert Law, which states that absorbance is directly proportional to the concentration of the absorbing species, the path length, and the molar extinction coefficient. A

\[ c = \frac{A}{\varepsilon l} \]

Rearranging the equation to solve for the unknown concentration \(c\). A

\[ c = \frac{0.450}{(2.50 \times 10^4\,\text{M}^{-1}\text{cm}^{-1})(1.00\,\text{cm})} \]

Substituting the given values for absorbance, molar extinction coefficient, and path length into the rearranged equation. A

\[ c = \frac{0.450}{25000}\,\text{M} \]

Performing the division to find the numerical value of the concentration. A

\[ c = 1.80 \times 10^{-5}\,\text{M} \]

Expressing the result in scientific notation with three significant figures, consistent with the precision of the input data. A

\[ c = 1.80 \times 10^{-5}\,\text{mol L}^{-1} \]

Reading. The concentration of the dye solution is \(1.80 \times 10^{-5}\,\text{M}\). This calculation demonstrates how UV-visible spectroscopy can be used quantitatively to determine the concentration of a chromophore in a solution.

Scope. This method is valid as long as the solution obeys the Beer-Lambert Law, which typically holds for dilute solutions where intermolecular interactions do not significantly alter the absorption properties of the individual molecules.

Problems

Rank hexane, hexa-1,3-diene, and hexa-1,3,5-triene by expected order of increasing \(\lambda_{\max}\).
Solution
Hexane (no conjugation, absorbs only in the far, vacuum UV, effectively transparent to a standard instrument) \(\lt\) hexa-1,3-diene (two conjugated double bonds) \(\lt\) hexa-1,3,5-triene (three conjugated double bonds, the most extended conjugation of the three, hence the longest \(\lambda_{\max}\)), following Step 3's conjugation-length trend directly.
Explain why a saturated hydrocarbon is essentially invisible to a standard UV-visible instrument, referencing the Hypotheses.
Solution
A saturated hydrocarbon has no chromophore beyond isolated \(\sigma\) bonds, whose \(\sigma\to\sigma^*\) transitions require far more energy than any transition available to a conjugated or heteroatom-containing system; these transitions fall in the vacuum-UV region, well outside the roughly \(200\)-\(800\,\text{nm}\) range a standard instrument covers, so no absorption is observed in the ordinary measurable range at all.
A solution shows absorbance \(A=0.60\) at its \(\lambda_{\max}\), with \(\varepsilon=1.2\times10^4\,\text{M}^{-1}\text{cm}^{-1}\) and a \(1.00\,\text{cm}\) pathlength. Find the concentration.
Solution
By Step 4, \(c=A/(\varepsilon l)=0.60/[(1.2\times10^4)(1.00)]=5.0\times10^{-5}\,\text{M}\).
A student measures the absorbance of a solution of a chromophore in a 1.00 cm cuvette. At \(\lambda_{\max}\), the absorbance is \(A = 0.85\). The molar extinction coefficient is known to be \(\varepsilon = 1.5 \times 10^4\,\text{M}^{-1}\text{cm}^{-1}\). Calculate the concentration of the solution in mol L\(^{-1}\).
Solution
Using the Beer-Lambert Law \(A = \varepsilon l c\): \[ c = \frac{A}{\varepsilon l} \] Substitute the given values: \[ c = \frac{0.85}{(1.5 \times 10^4\,\text{M}^{-1}\text{cm}^{-1})(1.00\,\text{cm})} \] \[ c = \frac{0.85}{15000}\,\text{M} \] \[ c = 5.67 \times 10^{-5}\,\text{M} \] The concentration is \(5.67 \times 10^{-5}\,\text{mol L}^{-1}\).
Compare the expected \(\lambda_{\max}\) for the following three compounds: (i) hexane, (ii) hexa-1,3-diene, (iii) hexa-1,3,5-triene. Explain the trend based on the extent of conjugation and the resulting HOMO-LUMO gap.
Solution
(i) Hexane: Contains only \(\sigma\) bonds. The \(\sigma \to \sigma^*\) transition requires high energy, corresponding to a very short wavelength (vacuum UV, < 150 nm). It is essentially transparent in the standard UV-Vis range (200-800 nm). (ii) Hexa-1,3-diene: Contains two conjugated double bonds. The \(\pi \to \pi^*\) transition energy is lower than in isolated alkenes due to conjugation, shifting \(\lambda_{\max}\) to the near-UV region (typically ~210-220 nm). (iii) Hexa-1,3,5-triene: Contains three conjugated double bonds. Increased conjugation further reduces the HOMO-LUMO gap, shifting \(\lambda_{\max}\) to a longer wavelength (typically ~260-270 nm). Trend: \(\lambda_{\max}(\text{hexane}) < \lambda_{\max}(\text{hexa-1,3-diene}) < \lambda_{\max}(\text{hexa-1,3,5-triene})\). Explanation: As the length of the conjugated \(\pi\)-system increases, the energy gap between the HOMO and LUMO decreases. Since \(E = hc/\lambda\), a smaller energy gap corresponds to a longer wavelength of absorption (red shift).