Spectrophotometry
Beer–Lambert Law

Learn how light absorption reveals concentration, understand the equation $ A = \varepsilon \ell c $, and master quantitative analysis in the lab.

Biochemistry

What Is Spectrophotometry?

Spectrophotometry measures how much light a substance absorbs at a specific wavelength. A beam of monochromatic light passes through a sample, and a detector records the light that makes it through. The decrease in intensity tells us about the sample's composition.

Transmittance and Absorbance

Transmittance $ T $

The fraction of light that passes through: $$ T = \frac{I}{I_0} $$ Ranges from 0 to 1 (or 0% to 100%).

Absorbance $ A $

A logarithmic measure of how much light is absorbed: $$ A = -\log_{10}(T) = \log_{10}\!\left(\frac{I_0}{I}\right) $$ Unitless: ranges from 0 to about 2 in practice.

Why Does This Matter?

Spectrophotometry is one of the most widely used analytical techniques in biochemistry, clinical chemistry, and environmental science. It allows researchers to measure the concentration of a substance without destroying it.

Protein Assays

Bradford assay measures protein at 595 nm using Coomassie dye.

DNA Purity

$ A_{260}/A_{280} $ ratio reveals nucleic acid purity.

Enzyme Kinetics

Track reaction rate by monitoring absorbance change over time.

The Beer–Lambert Law

$ A $

Absorbance: unitless, measured by the spectrophotometer.

$ \varepsilon $

Molar absorptivity (extinction coefficient): units $ \text{L mol}^{-1}\text{cm}^{-1} $. A property of the substance at a given $ \lambda $.

$ \ell $

Path length: distance light travels through the sample, usually 1 cm.

$ c $

Concentration: in $ \text{mol/L} $ (M). The unknown you often solve for.

Visualizing $ A = \varepsilon \ell c $

The Standard Curve

Because $ A = \varepsilon \ell c $ is linear (at a fixed $ \varepsilon $ and $ \ell $), plotting absorbance vs. concentration gives a straight line through the origin. The slope equals $ \varepsilon \ell $.

Absorbance vs. Transmittance

	Absorbance $ A $	Transmittance $ T $
Definition	$ \log_{10}(I_0/I) $	$ I / I_0 $
Units	Unitless	Unitless (or %)
Range (practical)	0 to ~2	1 to ~0.01
Relationship to $ c $	Directly proportional	Exponentially decreasing
Preferred for analysis?	✓ Yes: linear with $ c $	Less useful: nonlinear

Why Is Absorbance Linear?

The Beer–Lambert Law derives from exponential attenuation: each thin layer of solution absorbs the same fraction of incident light. The transmitted intensity decays exponentially:

Taking $ -\log_{10} $ of both sides turns this exponential into a linear equation: $$ A = -\log_{10}\!\left(\frac{I}{I_0}\right) = \varepsilon \ell c $$

How to Apply Beer–Lambert

1

Choose the wavelength $ \lambda_{\max} $: use the wavelength of maximum absorbance for your analyte.

2

Blank the instrument: zero the spectrophotometer with a cuvette of pure solvent.

3

Build a standard curve: measure $ A $ for at least 5 solutions of known $ c $. Plot $ A $ vs. $ c $.

4

Measure the unknown: record its absorbance and use the best-fit line (or $ A = \varepsilon \ell c $) to find $ c $.

5

Check your answer: ensure $ A $ is between 0.1 and 1.0 for reliable results. Dilute if needed.

Example 1: Finding Concentration

A dye solution has $ \varepsilon = 12{,}000 \;\text{L mol}^{-1}\text{cm}^{-1} $ at 540 nm. Using a 1 cm cuvette, the spectrophotometer reads $ A = 0.60 $. What is the concentration?

1

Write the law: $ A = \varepsilon \ell c $

2

Solve for $ c $: $$ c = \frac{A}{\varepsilon \ell} $$

3

Substitute: $$ c = \frac{0.60}{12{,}000 \times 1} = 5.0 \times 10^{-5}\;\text{M} $$

Example 2: Using a Standard Curve

You measured these standards of NADH at 340 nm and got the data below. An unknown sample reads $ A = 0.45 $. Find its concentration.

$ c $ (mM)	$ A $
0.05	0.15
0.10	0.31
0.15	0.46
0.20	0.62
0.25	0.78

Example 3: Finding $ \varepsilon $

A researcher dissolves a new compound at $ c = 2.5 \times 10^{-4}\;\text{M} $ and measures $ A = 0.875 $ in a 1 cm cuvette. What is its molar absorptivity?

1

Rearrange: $ \varepsilon = \frac{A}{\ell \, c} $

2

Substitute: $$ \varepsilon = \frac{0.875}{1 \times 2.5 \times 10^{-4}} = 3{,}500\;\text{L mol}^{-1}\text{cm}^{-1} $$

When the Law Breaks Down

The Beer–Lambert Law assumes ideal conditions. Several factors can cause deviations from linearity.

High Concentration

At concentrations above $ \approx 0.01\,\text{M} $, intermolecular interactions (aggregation, refractive index changes) distort the relationship.

Non-monochromatic Light

If the light source has a broad bandwidth, the measured $ \varepsilon $ varies across wavelengths, flattening the curve.

Scattering

Turbid or particulate samples scatter light, causing apparent absorbance that does not follow $ A = \varepsilon \ell c $.

Common Mistakes

✕ Wrong Units

Mixing up $ c $ in mM vs. M, or using a path length in mm instead of cm. Always convert to base units before substituting.

✕ Forgetting the Blank

Not zeroing the spectrophotometer with a blank cuvette. This adds a constant offset to every reading.

✕ Absorbance Too High

Using $ A > 1.0 $ readings directly. At high absorbance, stray light causes nonlinearity. Dilute your sample first.

Summary: Key Relationships

Quantity	Symbol	Units	Depends On
Absorbance	$ A $	Unitless	$ \varepsilon, \ell, c $
Transmittance	$ T $	Unitless	$ A $ (inverse log)
Molar absorptivity	$ \varepsilon $	$ \text{L mol}^{-1}\text{cm}^{-1} $	Substance, $ \lambda $
Path length	$ \ell $	cm	Cuvette size
Concentration	$ c $	$ \text{mol/L (M)} $	Sample preparation

Question 1 of 5

If the absorbance of a solution doubles, what happens to the concentration (assuming $ \varepsilon $ and $ \ell $ are constant)?

Since $ A = \varepsilon \ell c $ is linear, doubling $ A $ means $ c $ also doubles. This direct proportionality is the central feature of the Beer–Lambert Law.

Question 2 of 5

A solution has a transmittance of $ T = 0.10 $ (10%). What is its absorbance?

$ A = -\log_{10}(T) = -\log_{10}(0.10) = 1.0 $. Remember: $ \log_{10}(0.1) = -1 $, so the negative sign gives $ A = 1.0 $.

Question 3 of 5

A compound has $ \varepsilon = 6{,}200\;\text{L mol}^{-1}\text{cm}^{-1} $. In a 1 cm cuvette, you measure $ A = 0.31 $. What is the concentration?

$ c = A / (\varepsilon \ell) = 0.31 / (6{,}200 \times 1) = 5.0 \times 10^{-5}\;\text{M} $. Always check your exponent: dividing a number less than 1 by thousands gives a very small result.

Question 4 of 5

Which of the following would cause the Beer–Lambert Law to deviate from linearity?

At very high concentrations, molecular interactions (like aggregation) and refractive index changes cause deviations from the ideal linear relationship. Options A, B, and C all improve accuracy.

Question 5 of 5

In the equation $ A = \varepsilon \ell c $, the quantity $ \varepsilon $ depends on which of the following?

$ \varepsilon $ (molar absorptivity) is an intrinsic property of the substance at a specific wavelength. It does not change with path length, concentration, or light intensity. Different molecules absorb different wavelengths differently.

Spectrophotometry
Beer–Lambert Law

What Is Spectrophotometry?

Transmittance and Absorbance

Why Does This Matter?

The Beer–Lambert Law

Visualizing \( A = \varepsilon \ell c \)

The Standard Curve

Absorbance vs. Transmittance

Why Is Absorbance Linear?

How to Apply Beer–Lambert

Example 1: Finding Concentration

Example 2: Using a Standard Curve

Example 3: Finding \( \varepsilon \)

When the Law Breaks Down

Common Mistakes

Summary: Key Relationships

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

	Absorbance \( A \)	Transmittance \( T \)
Definition	\( \log_{10}(I_0/I) \)	\( I / I_0 \)
Units	Unitless	Unitless (or %)
Range (practical)	0 to ~2	1 to ~0.01
Relationship to \( c \)	Directly proportional	Exponentially decreasing
Preferred for analysis?	✓ Yes: linear with \( c \)	Less useful: nonlinear

Quantity	Symbol	Units	Depends On
Absorbance	\( A \)	Unitless	\( \varepsilon, \ell, c \)
Transmittance	\( T \)	Unitless	\( A \) (inverse log)
Molar absorptivity	\( \varepsilon \)	\( \text{L mol}^{-1}\text{cm}^{-1} \)	Substance, \( \lambda \)
Path length	\( \ell \)	cm	Cuvette size
Concentration	\( c \)	\( \text{mol/L (M)} \)	Sample preparation

SpectrophotometryBeer–Lambert Law

What Is Spectrophotometry?

Transmittance and Absorbance

Why Does This Matter?

The Beer–Lambert Law

Visualizing \( A = \varepsilon \ell c \)

The Standard Curve

Absorbance vs. Transmittance

Why Is Absorbance Linear?

How to Apply Beer–Lambert

Example 1: Finding Concentration

Example 2: Using a Standard Curve

Example 3: Finding \( \varepsilon \)

When the Law Breaks Down

Common Mistakes

Summary: Key Relationships

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Spectrophotometry
Beer–Lambert Law