AP Chemistry 3.13: Beer-Lambert Law

What does the Beer–Lambert law relate in a spectrophotometric measurement?

It relates the amount of light absorbed by a sample to the concentration of the absorbing species and the distance light travels through the sample.

Beer–Lambert law

For a uniform solution containing one absorbing species, absorbance is given by $A=\varepsilon c l$, where $A$ is absorbance, $\varepsilon$ is molar absorptivity, $c$ is molar concentration, and $l$ is path length.

What is absorbance in terms of incident and transmitted light intensity?

Absorbance is $A=\log_{10}(I_0/I)$, where $I_0$ is the intensity entering the sample and $I$ is the intensity leaving it. A larger absorbance means a smaller fraction of light is transmitted.

What is transmittance, and how is it related to absorbance?

Transmittance is the fraction of incident light that passes through a sample: $T=I/I_0$. The relationship to absorbance is $A=-\log_{10}T$ or $T=10^{-A}$.

How does doubling the concentration affect absorbance when Beer–Lambert conditions are met?

Absorbance doubles because $A$ is directly proportional to concentration: $A=\varepsilon c l$. Transmittance does not halve; instead, it changes exponentially according to $T=10^{-A}$.

How does doubling the path length affect absorbance under the Beer–Lambert law?

Absorbance doubles because it is directly proportional to path length: $A=\varepsilon c l$. The transmitted intensity decreases exponentially.

What does molar absorptivity $\varepsilon$ represent in $A=\varepsilon c l$?

Molar absorptivity measures how strongly a substance absorbs light at a specified wavelength. In common units, it is $\mathrm{L\,mol^{-1}\,cm^{-1}}$ when concentration is in $\mathrm{mol\,L^{-1}}$ and path length is in centimeters.

Why is absorbance unitless?

Absorbance is the base-10 logarithm of the ratio $I_0/I$, and a ratio of two intensities is dimensionless. Consequently, the units of $\varepsilon c l$ must cancel.

What graph is expected for a Beer–Lambert analysis of a single solute?

A plot of absorbance $A$ versus concentration $c$ should be linear, with slope $\varepsilon l$ and ideally a zero intercept. A calibration curve is commonly used to determine the concentration of an unknown.

Why is a calibration curve often preferred over directly calculating concentration from $A=\varepsilon c l$?

A calibration curve accounts for the actual instrument, cuvette, solvent, and experimental conditions. The unknown concentration is obtained from its absorbance using the measured linear relationship for standards.

What wavelength should generally be selected to measure the concentration of a colored analyte?

A wavelength near the analyte's maximum absorbance, $\lambda_{\max}$, is usually selected because it gives high sensitivity. The selected wavelength should also minimize interference from other substances.

Why should spectrophotometric measurements use nearly monochromatic light?

Molar absorptivity usually varies with wavelength, so a broad range of wavelengths can produce a non-linear or averaged response. Narrow-band light better supports the assumption of a single, well-defined $\varepsilon$.

How are absorbances combined for multiple independent absorbing species in the same solution?

Their absorbances are additive: $A=l\sum_i\varepsilon_i c_i$. At a given wavelength, each species contributes according to its concentration and molar absorptivity.

How can the concentrations of two absorbing species be determined in a mixture?

Measure absorbance at two wavelengths and use the simultaneous equations $A_{\lambda_1}=l(\varepsilon_{1,\lambda_1}c_1+\varepsilon_{2,\lambda_1}c_2)$ and $A_{\lambda_2}=l(\varepsilon_{1,\lambda_2}c_1+\varepsilon_{2,\lambda_2}c_2)$. The molar absorptivities must be known, and the equations must be sufficiently independent.

How does measuring a mixture at more than the minimum number of wavelengths improve analysis?

For a mixture with several absorbing species, measurements at multiple wavelengths can be fit using linear least squares. Extra wavelengths provide redundancy and can reduce the effect of random measurement error.

What is the distinction between absorption, scattering, and extinction in light attenuation?

Absorption removes light by converting its energy within the material, while scattering redirects light away from the detector. Extinction is the total attenuation caused by both absorption and scattering.

What assumptions are needed for a solution to follow the Beer–Lambert law?

The absorbing particles should act independently in a homogeneous, non-turbid sample; the beam should be parallel and traverse a consistent path length; and the light should be nearly monochromatic and sufficiently weak that it does not alter the species.

Why can high solute concentrations cause deviations from Beer–Lambert linearity?

At high concentrations, particles can interact, the solution may become turbid and scatter light, and the absorbing species may no longer behave independently. These effects make absorbance no longer proportional to concentration.

What types of deviations from the Beer–Lambert law can occur?

Real deviations arise from fundamental limitations of the law, chemical deviations arise from reactions or interactions that change the absorbing species, and instrumental deviations arise from the way light and absorbance are measured.

Why can intense incident radiation invalidate the Beer–Lambert law?

Very intense light can produce optical saturation, optical pumping, or other nonlinear effects. The light then changes the populations or behavior of the absorbing species instead of acting only as a probe.

What absorbance range is often preferred for maintaining good Beer–Lambert linearity?

An absorbance of approximately $0.2$ to $0.5$ is often considered a useful working range. Very low absorbance can make measurements sensitive to instrumental noise, while very high absorbance gives little transmitted light and may increase nonlinearity.

Why must the incident beam have a consistent path through the sample?

Because absorbance is proportional to path length: $A=\varepsilon c l$. If rays travel different distances or are not parallel, they experience different attenuation, which can distort the measured absorbance.

How does turbidity affect a spectrophotometric measurement based on Beer–Lambert law?

Suspended particles scatter light, causing the detector to receive less direct light and potentially reporting an apparent absorbance larger than that due to absorption alone. A clear, homogeneous sample is therefore preferred.

How can the Beer–Lambert law be used to determine an unknown concentration in spectrophotometry?

Measure the sample's absorbance at a chosen wavelength, then use either $c=A/(\varepsilon l)$ or a calibration curve prepared from standards of known concentration. A blank measurement is used to account for absorption by the solvent and cuvette.

What happens to transmittance when absorbance increases by one unit?

Because $T=10^{-A}$, increasing absorbance by 1 reduces transmittance by a factor of 10. For example, $A=1$ corresponds to $T=0.10$, or 10% transmission.