The Beer-Lambert law

Result

The Beer-Lambert law

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Statement

Absorbance proportional to concentration and path length.

Why it matters

rotational-spectroscopy, vibrational-ir-spectroscopy, nmr-chemical-shift, and uv-vis-electronic each explain what determines the position (frequency, chemical shift) of a spectroscopic signal; the Beer-Lambert law instead explains what determines its intensity, and specifically how that intensity relates to how much absorbing species is present. It is the quantitative foundation that turns spectroscopy from a purely qualitative, structure-identifying technique into a quantitative analytical one, letting an absorbance measurement be converted directly into a concentration.

This makes it one of the most practically used relationships in the whole of chemistry: essentially every UV-visible or infrared spectrophotometer in routine laboratory use is applying the Beer-Lambert law, explicitly or implicitly, every time a concentration is read off an absorbance value, and calibration-standard-addition's calibration-curve method is a direct, immediate application of the linear relationship this result establishes.

Hypotheses

The incident light is monochromatic (a single wavelength, or a narrow enough band that the absorbing species' molar absorptivity \(\varepsilon\) is effectively constant across it).Since \(\varepsilon\) itself generally varies with wavelength (it is exactly what a UV-visible absorption spectrum plots), using a wavelength band wide enough for \(\varepsilon\) to vary meaningfully within it would average over different absorptivities and break the simple linear relationship derived below. The sample is dilute enough that absorbing molecules act independently, with no significant intermolecular interaction, aggregation, or concentration-dependent equilibrium shift affecting \(\varepsilon\).At high concentration, absorbing species can interact with one another (electronically or via aggregation) in ways that change their effective absorptivity, and stray light and instrumental effects also become proportionally more significant; both effects cause real, systematic deviation from strict linearity at high absorbance. For a solution containing several non-interacting absorbing species at the same wavelength, absorbances are additive, \(A_{\text{total}}=\sum_i\varepsilon_ic_il\); this additivity, itself a direct consequence of the same underlying probabilistic, independent-absorption-event derivation given below, is the basis of simultaneous multi-component spectrophotometric analysis.

Proof

-\,dI = \alpha\,c\,I\,dx

Consider a thin slab of solution of thickness \(dx\) along the light path: the fraction of light intensity \(I\) absorbed while crossing it is proportional both to the concentration \(c\) of absorbing species present and to the thickness \(dx\) itself, with \(\alpha\) a proportionality constant characteristic of the absorbing species and wavelength. A

\int_{I_0}^{I} \frac{dI}{I} = -\alpha c \int_0^l dx \quad\Rightarrow\quad \ln\frac{I_0}{I} = \alpha c l

Integrating Step 1's differential relationship over the full path length \(l\) through a sample of uniform concentration \(c\) (the constant \(\alpha\) and constant \(c\) both come out of the integral) gives an exponential (Beer's-law) attenuation of intensity with path length and concentration. A

A \equiv \log_{10}\frac{I_0}{I} = \frac{\alpha}{2.303}\,c\,l \equiv \varepsilon\, c\, l

Defining absorbance \(A\) using a base-\(10\) logarithm (the historical, still near-universal spectroscopic convention) rather than the natural logarithm of Step 2 simply rescales the proportionality constant; the rescaled constant is defined as the molar absorptivity \(\varepsilon\) (units \(\text{L mol}^{-1}\text{cm}^{-1}\) when \(c\) is in \(\text{mol/L}\) and \(l\) in \(\text{cm}\)), a fixed, wavelength-specific property of the absorbing species. A

Result

A = \varepsilon\,c\,l

Reading. Absorbance is directly proportional to both the concentration of the absorbing species and the path length the light travels through it, with the molar absorptivity \(\varepsilon\) as the proportionality constant characteristic of that species at that wavelength.

Scope. Holds for monochromatic light and dilute, non-interacting solutions (Hypotheses); deviates at high concentration, in the presence of chemical equilibria that shift with concentration (e.g. dimerisation, protonation), or with stray light and detector non-linearity at very high absorbance.

Corollaries & converses

Because \(A\) is linear in \(c\), a calibration curve of measured \(A\) against known standard concentrations (calibration-standard-addition) has slope \(\varepsilon l\) and, ideally, a zero intercept; an unknown's concentration is read directly from its measured \(A\) using this line.
Doubling the path length \(l\) (e.g. using a longer cuvette) doubles the absorbance for a fixed concentration exactly as doubling the concentration does, since \(c\) and \(l\) enter the Result identically and multiplicatively.
Converse: a measured deviation from strict linearity of \(A\) against \(c\), particularly a levelling-off at high concentration, is itself diagnostic evidence that one of the Hypotheses (typically dilute, non-interacting behaviour) is failing, rather than evidence against the law itself.

Fails without

Use a wavelength band wide enough that \(\varepsilon\) varies appreciably across it: averaging over different absorptivities breaks the simple linear proportionality of the Result, producing systematic deviations even in an otherwise ideal, dilute sample (Hypotheses, first point).
Push concentration high enough that absorbing molecules interact or aggregate: the independent-absorption-event assumption underlying Step 1 fails, and measured absorbance rises less than proportionally with concentration, curving the calibration line downward at high concentration.

Common errors

Assuming percentage transmittance itself is linear in concentration; it is absorbance, \(A=-\log_{10}(I/I_0)\), that is linear in \(c\), not transmittance \(T=I/I_0\), which is exponential in \(c\).
Mixing base-\(10\) absorbance (the spectroscopic convention, Step 3) with the natural-log form of Step 2 when quoting or converting \(\varepsilon\) values, without adjusting by the factor of \(2.303\).
Extrapolating a calibration curve confidently into a concentration range well outside where it was actually measured, where deviations from linearity (Hypotheses) are more likely.
Forgetting that \(\varepsilon\) is wavelength-dependent, and applying an \(\varepsilon\) value measured at one wavelength to an absorbance measured at a different one.

Discussion

The relationship is named for two separate, sequential contributions: Pierre Bouguer (and, independently, Johann Heinrich Lambert) established the exponential dependence of absorption on path length in the eighteenth century, while August Beer added the analogous dependence on concentration in 1852, giving the combined, now-standard form.

Common misconception: that the Beer-Lambert law is an exact, universally valid physical law with no domain of applicability, rather than an idealisation that holds well specifically for dilute, non-interacting, monochromatically illuminated samples (Hypotheses). Real spectrophotometric measurements are always checked against, and typically confined to, an absorbance range (commonly quoted as roughly \(0.1\) to \(1\)) where linearity and instrumental accuracy are both reliably good.

Worked examples

A = 0.532, \qquad \varepsilon = 1.5\times10^{4}\,\text{L mol}^{-1}\text{cm}^{-1}, \qquad l = 1.00\,\text{cm}

A solution's absorbance is measured in a standard \(1\,\text{cm}\) cuvette at a wavelength where the analyte's molar absorptivity is known from a reference value or a prior calibration. A

c = \frac{A}{\varepsilon l} = \frac{0.532}{1.5\times10^{4}\times1.00} = 3.55\times10^{-5}\,\text{mol/L}

Rearranging the Result directly for concentration and substituting the measured absorbance and known \(\varepsilon\) and \(l\) gives the unknown concentration, the single most common practical use of this law. A

c \approx 3.6\times10^{-5}\,\text{mol/L}

Reading. A single absorbance reading, combined with a known molar absorptivity and path length, gives the unknown concentration directly, with no need to prepare a full series of standards if \(\varepsilon\) is already reliably known.

Scope. Where \(\varepsilon\) is not independently known, or for greater reliability, the same relationship instead underlies constructing and reading a full calibration curve (calibration-standard-addition).

Problems

A \(2.0\times10^{-4}\,\text{mol/L}\) solution has an absorbance of \(0.60\) in a \(1.00\,\text{cm}\) cuvette. Find \(\varepsilon\).
Solution
\(\varepsilon = A/(cl) = 0.60/(2.0\times10^{-4}\times1.00) = 3.0\times10^{3}\,\text{L mol}^{-1}\text{cm}^{-1}\).
Using the \(\varepsilon\) found in Problem 1, predict the absorbance of the same solution diluted to half its concentration and measured in a \(2.00\,\text{cm}\) cuvette instead.
Solution
New concentration \(=1.0\times10^{-4}\,\text{mol/L}\); \(A=\varepsilon c l = 3.0\times10^{3}\times1.0\times10^{-4}\times2.00 = 0.60\). The absorbance is unchanged, because halving \(c\) and doubling \(l\) exactly offset one another in the Result's product \(cl\).
A student measures a series of standards and finds the calibration curve of \(A\) against \(c\) is linear at low concentration but curves downward (absorbance rises less than proportionally) at the highest standard concentrations. Suggest a likely cause, referencing the Hypotheses.
Solution
This is a classic signature of departure from the dilute, non-interacting-species assumption: at sufficiently high concentration the absorbing species may begin to aggregate, undergo a concentration-dependent equilibrium shift, or the instrument may suffer from stray light or detector non-linearity at high absorbance, any of which breaks the strict proportionality of the Result and produces the characteristic downward-curving high-concentration deviation.