Beer–Lambert Law — Solve Any Variable (A = εlc)

About the Beer–Lambert law

The Beer–Lambert law relates absorbance \(A\) to molar absorptivity \(\varepsilon\), path length \(l\), and concentration \(c\): \[ A = \varepsilon \, l \, c. \] Provide any three, and the fourth is computed. If you enter percent transmittance, the tool uses \(A = -\log_{10}(T)\) with \(T = \%T/100\).

• \(\varepsilon\): often tabulated in L·mol⁻¹·cm⁻¹. SI m²·mol⁻¹ is supported (1 L·mol⁻¹·cm⁻¹ = 0.1 m²·mol⁻¹).
• l: typical cuvette path length is 1.00 cm (10 mm). You can pick cm, mm, or m.
• c: pick M, mM, or µM; conversions are handled automatically.

Worked example

NADH at 340 nm. Given \(\varepsilon = 6220\ \mathrm{L\,mol^{-1}\,cm^{-1}}\), \(l = 1.00\ \mathrm{cm}\), and \(c = 0.10\ \mathrm{mM}\):

1. Convert \(c = 0.10\,\mathrm{mM} = 1.0\times10^{-4}\,\mathrm{mol\,L^{-1}}\).
2. \(A = \varepsilon l c = 6220 \times 1.00 \times 1.0\times10^{-4} \approx 0.622\).

Understanding the Beer–Lambert Law (Absorbance Spectroscopy)

The Beer–Lambert law, sometimes called the Beer–Lambert–Bouguer law, is a cornerstone of modern analytical chemistry and biochemistry. It provides a simple mathematical relationship between the absorbance of light by a sample and three key physical quantities: the molar absorptivity (\(\varepsilon\)), the path length of the light through the sample (\(l\)), and the concentration of the absorbing species (\(c\)). The equation is written as:

\( A = \varepsilon \, l \, c \)

Key terms explained

• Absorbance (A): A logarithmic measure of how much light is absorbed. It is dimensionless and directly proportional to concentration.
• Molar absorptivity (ε): Also called the extinction coefficient. It reflects how strongly a substance absorbs light at a given wavelength, usually in units of L·mol⁻¹·cm⁻¹.
• Path length (l): The distance the light travels through the sample. Standard cuvettes are typically 1.00 cm, but other sizes exist.
• Concentration (c): The amount of absorbing species, usually expressed in molarity (mol·L⁻¹, or M). Dilute samples often use mM or µM.

Absorbance and percent transmittance

Many spectrophotometers measure % transmittance (%T), the fraction of light that passes through the sample compared to a blank. Absorbance is related by:

\( A = -\log_{10}(T) \), where \( T = \dfrac{\%T}{100} \).

For example, if a sample transmits 25% of the incident light, then \( A = -\log_{10}(0.25) = 0.602 \).

Why it matters

The Beer–Lambert law is widely used in UV-Vis spectroscopy, biological assays (such as protein concentration by absorbance at 280 nm), enzyme kinetics, and even environmental monitoring (detecting pollutants in water). Because absorbance is linear with concentration under ideal conditions, it enables precise and reproducible quantification without destroying the sample.

Limitations and best practice

The relationship holds best for solutions that are dilute, homogeneous, and measured with monochromatic light. At very high absorbance values (A > 2), too little light passes through, leading to inaccuracies. Scattering, fluorescence, or chemical interactions can also cause deviations. For reliable results, always use an appropriate blank, keep within the linear range of your instrument, and report the wavelength used along with ε values.

In summary, the Beer–Lambert law bridges light and matter: by measuring how much light a solution absorbs, you can back-calculate concentration or other variables. This makes it one of the most versatile and important equations in laboratory science.