Topic 3.13 Notes – Beer-Lambert Law

1. How Light Absorption Is Measured

When a beam of light passes through a solution, some photons are absorbed by molecules or ions and the rest pass through.

• Incident light, $I_{0}$ = light entering the sample
• Transmitted light, $I$ = light that exits
• Absorbance, $A$ = how much light is absorbed (unitless)

A spectrophotometer shines a selected wavelength through a cuvette and measures transmitted light. It then converts that into absorbance.

Here’s the basic idea visually:

Basic components of a spectrophotometer

Light from the source is narrowed and filtered so only one wavelength reaches the cuvette. The detector compares the light entering the sample ($I_{0}$) to the light that exits ($I$) and the instrument reports absorbance.

Absorbance increases when:

• Concentration increases → more absorbing particles in the path
• Path length increases → light travels through more solution
• Molar absorptivity is large → each particle absorbs strongly at that wavelength

The key physical idea is particle count in the light’s path. More particles means more chances for photons to be absorbed.

2. The Beer-Lambert Law

The quantitative relationship is:

$$A = \varepsilon b c$$

Where:

• $A$ = absorbance (unitless)
• $\varepsilon$ = molar absorptivity in $\text{L}\\cdot\\text{mol}^{-1}\text{}\\cdot\\text{cm}^{-1}$
  • Measures how strongly a species absorbs light at a specific wavelength
• $b$ = path length in cm
  • Usually 1.0 cm cuvette
• $c$ = concentration in mol/L

What each variable means physically

• Increasing $c$ increases the number of absorbing particles per volume.
• Increasing $b$ increases how many particles the light encounters.
• Increasing $\varepsilon$ means each particle is more effective at absorbing that wavelength.

All three are directly proportional to $A$. If any one doubles (with the others constant), absorbance doubles.

On AP questions, they often change one variable and ask how $A$ changes. Think proportional reasoning, not memorization.

3. Why Beer’s Law Is Linear

If wavelength and path length are held constant, then $\varepsilon$ and $b$ are constants. The equation becomes:

$$A = (\varepsilon b)c$$

This has the form $A = mc$, where slope $m = \varepsilon b$.

That means a graph of absorbance vs. concentration is a straight line through the origin in the ideal case.

Ideal Beer’s Law calibration curve

This linearity is why spectrophotometry works for finding unknown concentrations. You measure the absorbance, use the straight-line calibration curve, and read back to the concentration.

If concentration doubles, absorbance doubles. If concentration is zero, absorbance should be zero. On real data, slight deviations happen, but AP problems usually assume ideal behavior.

4. Choosing the Wavelength

Absorbance depends on wavelength because $\varepsilon$ depends on wavelength.

A solution appears a certain color because it transmits that color and absorbs its complement.

Examples:

• Blue solution → absorbs orange light
• Yellow solution → absorbs violet light

Chemists typically set the spectrophotometer to $\lambda_{\text{max}}$, the wavelength of maximum absorbance.

Why?

• $\varepsilon$ is largest at $\lambda_{\text{max}}$
• Absorbance is highest for a given concentration
• Small concentration changes produce noticeable changes in $A$

In most AP-level experiments:

• $b$ is fixed (same cuvette)
• Wavelength is fixed (at $\lambda_{\text{max}}$)

So absorbance depends only on concentration.

That’s a favorite test move. They’ll describe constant path length and wavelength, then expect you to conclude $A \propto c$.

5. Using Beer’s Law to Determine Concentration

Direct calculation

If $\varepsilon$, $b$, and $A$ are known:

$$c = \frac{A}{\varepsilon b}$$

Example idea: If absorbance is 0.800, $\varepsilon = 4.00 \times 10^{3}$, and $b = 1.0\text{ cm}$, then
$$c = \frac{0.800}{(4.00 \times 10^{3})(1.0)} = 2.00 \times 10^{-4}\text{ mol/L}$$

Check units cancel properly. They love unit consistency on FRQs.

Calibration curve method

More common in labs:

1. Prepare standard solutions of known concentrations
2. Measure absorbance of each
3. Plot $A$ vs. $c$
4. Draw best-fit line
5. Measure absorbance of unknown
6. Use line to determine its concentration

This shows up in free-response questions where they give you a table of data and expect you to interpret slope or use the equation of the line.

Higher absorbance always corresponds to higher concentration under constant conditions.