Topic 3.13 Beer-Lambert Law: use the Beer-Lambert law to relate the absorbance of a solution to its concentration, and apply a calibration to find an unknown concentration.

What this topic is asking

The College Board (Topic 3.13) wants you to use the Beer-Lambert law to relate the absorbance of a colored solution to its concentration, and to apply a calibration (a standard or a calibration curve) to find an unknown concentration. This is the quantitative payoff of the spectroscopy topics: because absorbance is proportional to concentration, a simple light measurement becomes an analytical tool.

The Beer-Lambert law

Absorbance measures how much light a solution removes from a beam passing through it. The law says that absorbance grows with three things: how strongly the species absorbs at that wavelength ($\varepsilon$), how far the light travels through the solution ($b$), and how concentrated the solution is ($c$). The more absorbing species in the path, the more light is absorbed.

Proportionality and the calibration curve

This is how the law is used in practice. You prepare several standards of known concentration, measure their absorbances, and plot $A$ against $c$. The points fall on a line through the origin with slope $\varepsilon b$. An unknown's concentration is then found either by reading its absorbance off the line or by dividing its absorbance by the slope. Choosing the wavelength of maximum absorbance makes the measurement most sensitive.

Why the wavelength matters

The molar absorptivity $\varepsilon$ depends on wavelength, because a species absorbs different wavelengths to different extents (its color). Measurements are made at the wavelength where the species absorbs most strongly, giving the largest $\varepsilon$ and so the steepest calibration line and the best precision. At a wavelength the species does not absorb, $\varepsilon$ and therefore $A$ would be near zero and the method would not work. This ties back to Topic 3.11: absorbance happens only where the photon energy matches an electronic transition.

Try this

Q1. State the Beer-Lambert law and the meaning of each term. [2 points]

• Cue. $A = \varepsilon b c$: absorbance equals molar absorptivity times path length times concentration.

Q2. A solution's absorbance is $0.600$ and the calibration slope is $30.0\ \text{M}^{-1}$. Calculate the concentration. [1 point]

• Cue. $c = \dfrac{A}{\varepsilon b} = \dfrac{0.600}{30.0} = 0.0200$ M.