Molar Absorption Coefficient

Introduction

Two solutions of the same concentration can have very different absorbances. A 1.0 × 10⁻⁴ mol dm⁻³ solution of a strong dye may be deeply coloured, while the same concentration of propanone barely absorbs at all. The difference lies in the molar absorption coefficient , ε, the term in the Beer–Lambert law that captures how strongly each substance absorbs. Its value tells us about the type of electronic transition, sets the sensitivity of an analysis and changes dramatically with wavelength.

Core explanation

Definition. Rearranging A = εcl gives ε = A/(cl). ε is numerically equal to the absorbance that a 1 mol dm⁻³ solution would have in a 1 cm path at the stated wavelength. It is a characteristic of the substance, the wavelength and, to a lesser extent, the solvent and temperature. The older name "molar extinction coefficient" is still common.

Units. Since A has no units:

ε = A/(cl) → 1 ÷ (mol dm⁻³ × cm) = dm³ mol⁻¹ cm⁻¹

In SI base units this is equivalent to 0.1 m² mol⁻¹, and some biochemistry sources quote ε in M⁻¹ cm⁻¹, which means the same as dm³ mol⁻¹ cm⁻¹ because M is mol dm⁻³. Always check that c and l are in units matching ε before calculating.

Magnitude and transition type. ε reflects the probability of the electronic transition:

Type of absorption Typical εmax / dm³ mol⁻¹ cm⁻¹ --- --- n → π (e.g. C=O in ketones) 10–100 d–d transitions in octahedral transition metal complexes 1–100 Benzene band near 254 nm about 200 π → π in conjugated dienes 10 000–30 000 Strongly conjugated dyes 50 000–150 000 Charge-transfer bands (e.g. MnO₄⁻) 1000–50 000

Values above about 10⁴ indicate allowed transitions with good orbital overlap; values below about 10³ point to forbidden transitions that occur only weakly. Because ε spans more than four orders of magnitude, spectra are often plotted as log ε against wavelength.

Dependence on wavelength. ε is not a single number for a compound; it is a function of wavelength. A plot of ε against λ has the same shape as the absorption spectrum, since A is proportional to ε at fixed c and l. At λmax, ε reaches its maximum value, εmax.

Why measure at λmax? Three reasons: - Sensitivity. The largest ε gives the largest absorbance for a given concentration, so low concentrations can be detected. - Robustness. At the top of a band the curve is flat, so a small error in setting the wavelength changes ε very little. - Linearity. Across the narrow range of wavelengths passed by the instrument, ε is nearly constant at the peak, which helps the Beer–Lambert law hold.

Sensitivity and detection. Because absorbance can be measured reliably down to about 0.01, the lowest usable concentration in a 1 cm cell is roughly 0.01/ε. For ε = 50 000 dm³ mol⁻¹ cm⁻¹ this is 2 × 10⁻⁷ mol dm⁻³, which is why intensely absorbing dyes and coloured derivatives are used in trace analysis.

Formulae

ε = A/(cl), units dm³ mol⁻¹ cm⁻¹. Approximate lowest measurable concentration ≈ 0.01/(εl).

Step-by-step reasoning

To assign a transition from its ε value:

1. Find εmax, calculating it from A, c and l if necessary. 2. Compare with typical ranges: below 10³ suggests forbidden (n → π or d–d); above 10⁴ suggests allowed (π → π or charge transfer). 3. Check λmax is consistent with the proposed chromophore. 4. State the assignment with both pieces of evidence.

Visual explanation

Plot ε against wavelength for propanone. A tall, narrow band rises below 200 nm (π → π , ε near 1000 or more), while a low, broad hump sits at about 280 nm (n → π , ε ≈ 15). Replot using log ε and both bands fit comfortably on the same axes.

Real-world analogy

ε is like the "catching skill" of a fielder in a cricket match. Two fielders standing in the same place may face the same number of balls, but a skilled fielder catches far more of them. Molecules with large ε are expert catchers of photons at that wavelength.

Real-world example

Nucleic acid concentrations are found from absorbance at 260 nm. Biochemists use average coefficients, expressed per mass rather than per mole, so that an absorbance of 1.0 in a 1 cm cell corresponds to about 50 μg cm⁻³ of double-stranded DNA. This quick measurement is performed thousands of times a day in research laboratories.

Why?

Why does ε vary so enormously between transitions? ε depends on the transition probability, which depends on how well the starting and finishing orbitals overlap and whether symmetry and spin rules allow the change. Well-overlapping π and π orbitals give intense bands; poorly overlapping n and π orbitals do not.

Common misconception

"ε is a constant for each compound, like its molar mass." ε is constant only at a given wavelength (and in a given solvent). Quoting ε without the wavelength is meaningless.

Worked example

Question: A 2.50 × 10⁻⁵ mol dm⁻³ solution in a 1.00 cm cuvette has A = 0.550 at λmax. Calculate ε and suggest the type of transition.

Reasoning: ε = A/(cl) = 0.550 ÷ (2.50 × 10⁻⁵ × 1.00) = 22 000 dm³ mol⁻¹ cm⁻¹. This large value indicates an allowed transition.

Answer: ε = 2.20 × 10⁴ dm³ mol⁻¹ cm⁻¹; a π → π transition in a conjugated system.

Quick check

1. What are the units of ε when concentration is in mol dm⁻³ and path length in cm? Answer: dm³ mol⁻¹ cm⁻¹, because ε = A/(cl) and absorbance itself has no units.

Exam focus

Show the unit derivation clearly. Remember that ε depends on wavelength, and justify measuring at λmax in terms of maximum sensitivity and minimum error. Large ε means strong absorption, so a lower concentration gives the same absorbance.

Advanced insight

The integrated area under an absorption band, rather than εmax alone, is proportional to the oscillator strength , a dimensionless measure of transition probability. A fully allowed transition has an oscillator strength near 1, corresponding to εmax around 10⁵ for a typical bandwidth, which explains the upper limit seen for organic dyes.

Summary

The molar absorption coefficient ε = A/(cl) measures how strongly a substance absorbs at a particular wavelength, in dm³ mol⁻¹ cm⁻¹. Values range from about 10 for forbidden n → π and d–d transitions to over 10⁵ for strongly allowed π → π transitions in dyes. ε varies with wavelength, peaking at λmax, where measurements are most sensitive and least affected by small wavelength errors.

Practice questions

1. Define the molar absorption coefficient. Answer: The constant ε in A = εcl, equal to the absorbance of a 1 mol dm⁻³ solution in a 1 cm path at a stated wavelength. 2. A band has εmax = 25 dm³ mol⁻¹ cm⁻¹ at 290 nm. Suggest the type of transition and explain. Answer: An n → π transition, because the small ε indicates a forbidden, low-probability transition typical of a carbonyl group. 3. Give two reasons for making quantitative measurements at λmax. Answer: ε is largest there, giving maximum sensitivity, and the band is flat at the peak, so small wavelength errors have little effect. 4. Estimate the lowest concentration measurable in a 1 cm cell for ε = 1.0 × 10⁴ dm³ mol⁻¹ cm⁻¹ if the smallest reliable absorbance is 0.01. Answer: c = 0.01 ÷ (1.0 × 10⁴ × 1) = 1.0 × 10⁻⁶ mol dm⁻³.