Colorimetry and Beer-Lambert Law

Spec Mapping — OCR H432 Module 5.3.1 — Transition elements (colorimetry), covering the use of a colorimeter to measure the absorbance of light by coloured transition-metal complexes, the choice of filter (complementary colour to the absorbing species), the Beer-Lambert law $A = \\varepsilon c l$A=varepsiloncl relating absorbance to molar absorptivity, concentration and path length, the construction and use of a calibration curve (absorbance vs concentration), and the practical determination of transition-metal concentrations including the use of ligand exchange (NH3 for Cu2+, SCN- for Fe3+) to intensify weak colours before measurement (refer to the official OCR H432 specification document for exact wording).

Colorimetry is the quantitative companion to the qualitative tests in the previous lesson: once you know which transition-metal cation is in your unknown sample, colorimetry lets you measure how much using nothing more than a simple light source, a filter, a sample cuvette and a photodetector. The underlying physics — the Beer-Lambert law — was established by Pierre Bouguer (1729), Johann Lambert (1760) and August Beer (1852) and remains the foundation of every modern UV-Vis spectrophotometer, fluorimeter and many environmental-monitoring instruments. For OCR A-Level the law is examined as a single linear equation $A = \\varepsilon c l$A=varepsiloncl, with the practical context being the determination of Cu2+ (as the deep blue tetraamine), Fe3+ (as the red thiocyanate), or Cr3+ (after oxidation to the orange dichromate). This lesson teaches the equipment, the algebra, the calibration-curve method and the experimental craft (cuvette handling, blanking, dilution into the linear range) needed for a full-marks Paper 1 or Paper 3 colorimetry question.

Key Definition — The Beer-Lambert Law states that the absorbance $A$A of a dilute coloured solution is directly proportional to its molar concentration $c$c and to the path length $l$l of the light through the sample: $A = \\varepsilon c l$A=varepsiloncl where $\\varepsilon$varepsilon is the molar absorptivity (or molar extinction coefficient) — a property of the absorbing species at the chosen wavelength — with units of mol$^{-1}$ dm$^3$ cm$^{-1}$. For a given solution and cuvette, $\\varepsilon$varepsilon and $l$l are constants, so $A \\propto c$Aproptoc and a graph of $A$A against $c$c is a straight line through the origin (the calibration curve). The unknown concentration is then read from the graph.

By the end of this lesson you should be able to:

• Describe the principle of a colorimeter and state the four components (light source, filter, cuvette, photodetector)
• Explain how wavelength and complementary colour are chosen for a given complex (a blue solution absorbs orange light; a green solution absorbs red light)
• State and apply the Beer-Lambert Law $A = \\varepsilon c l$A=varepsiloncl with correct units
• Use a calibration curve of absorbance vs concentration to determine the concentration of an unknown coloured solution
• Explain why a ligand exchange (Cu2+ → [Cu(NH3)4(H2O)2]2+, Fe3+ → [Fe(SCN)(H2O)5]2+, Cr3+ → Cr2O7 2-) is sometimes used to intensify the colour before measurement
• Describe the practical procedure for a colorimetric determination including blanking, cuvette handling, and dilution into the linear range
• Identify and explain deviations from linearity at high concentration (Beer-Lambert assumes dilute, non-aggregating, monochromatic conditions)

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Why Colorimetry? From Qualitative to Quantitative

Lesson 8 introduced the qualitative-analysis flowchart for identifying which transition-metal cation is in an unknown sample using NaOH and NH₃ reagents. Colorimetry is the complementary quantitative technique that tells you how much of that cation is present. The underlying physical observation is simple:

More concentrated solution → more intense colour → more light absorbed at the appropriate wavelength.

A colorimeter measures the fraction of incident light absorbed by a coloured solution at a specific wavelength (selected by a filter). It is cheaper and far easier to use than a full UV-Vis spectrophotometer (which uses a monochromator and scans a continuous wavelength range), but it relies on the same physical principle: the Beer-Lambert law, which states that absorbance is directly proportional to concentration in dilute solution. Modern colorimeters are pocket-sized, battery-powered, and routinely used by water-quality analysts, environmental monitors, and clinical laboratories for rapid metal-ion determinations in the field.

The historical roots of the Beer-Lambert law go back nearly 300 years. Pierre Bouguer published Essai d'optique sur la gradation de la lumière in 1729, establishing that absorbance scales linearly with path length (the "Lambert" part — though Bouguer arguably deserves more credit than the law-name suggests). Johann Heinrich Lambert refined this in 1760 in Photometria. August Beer extended the relationship in 1852 to include concentration linearity, giving the modern A = εcl form. The "molar absorptivity" ε (also called the "molar extinction coefficient" in older literature) is the unique fingerprint of the absorbing species at the chosen wavelength and characteristic of how strongly that species couples to photons of that energy.

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How a Colorimeter Works

A simple colorimeter has four main components, arranged so the light from the source passes sequentially through filter, sample, and detector:

1. Light source — usually a tungsten filament lamp or a white LED. It emits a broad spectrum of visible light (roughly 400–700 nm). Some instruments include an additional UV LED for measurements below 400 nm.
2. Filter — selects a narrow wavelength range (typically ±20 nm bandwidth) from the source. The filter is chosen to match the complementary colour of the solution (the colour the solution absorbs, not the colour it appears). Better instruments use a grating monochromator that can be tuned continuously to any wavelength.
3. Cuvette (sample holder) — a rectangular plastic or glass vessel of known path length (usually exactly 1.00 cm, sometimes 0.5 cm or 2 cm for special applications) containing the solution.
4. Photodetector — typically a photodiode or photomultiplier tube that converts transmitted light intensity to an electrical signal, which is displayed as absorbance.

The colorimeter displays the absorbance A, calculated from the ratio of incident to transmitted intensities:

$A = \\log_{10}\\left(\\frac{I_0}{I}\\right)$A=log10​left(fracI0​Iright)

where I₀ is the incident light intensity (with the blank cuvette in place) and I is the transmitted intensity (with the sample cuvette). Three reference points to anchor the scale: A = 0 means no absorption (I = I₀, 100% transmission); A = 1 means the solution absorbs 90% of the incident light (I = I₀/10, 10% transmission); A = 2 means 99% absorption; A = 3 means 99.9% absorption. The dynamic range of most colorimeters is approximately 0 to 2 absorbance units, with best precision in the 0.1–1.0 range.

graph LR
  A["Light source<br/>white light"] --> B["Filter<br/>selects complementary colour"]
  B --> C["Cuvette<br/>sample"]
  C --> D["Photodetector<br/>measures intensity"]
  D --> E["Display<br/>absorbance A"]

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Choosing the Right Filter: The Complementary-Colour Rule

The single most-examined principle in OCR colorimetry is the complementary-colour rule: the filter must select the colour that is absorbed by the solution, not the colour the solution appears to be. These two colours are complementary — they sit opposite each other on the colour wheel. A blue solution looks blue because it has absorbed the orange light from the white-light spectrum, leaving the remaining (mostly blue) light to be transmitted to the eye. To measure the concentration of that blue solution, you need to use an orange filter so the colorimeter measures the absorbed wavelengths (where the signal is strongest), not the transmitted ones (where the signal is at its weakest and the absorbance reading would be near zero).

Solution colour (transmitted)	Absorbed colour (use this filter)	Approx. wavelength of absorbed light
Red	Green	500–570 nm
Orange	Blue	430–490 nm
Yellow	Violet	400–440 nm
Green	Red	620–700 nm
Blue	Orange	580–620 nm
Violet	Yellow	560–590 nm

So to measure the concentration of [Cu(NH₃)₄(H₂O)₂]²⁺ (deep royal blue) we use an orange filter (around 600 nm); the deep blue solution strongly absorbs orange light, giving high absorbance and good sensitivity. To measure the concentration of [Fe(SCN)(H₂O)₅]²⁺ (intense red), we use a green or blue-green filter (around 470–520 nm). To measure the concentration of MnO₄⁻ (deep purple), we use a yellow-green filter around 525 nm.

The optimal wavelength is the wavelength of maximum absorbance, denoted λ_max. It is determined experimentally by recording the absorbance of a single standard solution across a range of available filters and plotting A against wavelength to locate the peak. For most A-level coloured complexes the λ_max sits comfortably in the visible range and can be matched by a standard set of six filters that come with school colorimeters (red, orange, yellow, green, blue, violet). Picking a filter close to λ_max maximises the sensitivity (the gradient of the calibration curve), minimises interference from other absorbing species, and gives the best signal-to-noise ratio at low concentrations.

Complex	Colour of solution	λ_max approx.	Filter colour
[Cu(H₂O)₆]²⁺	pale blue	800 nm	red/orange
[Cu(NH₃)₄(H₂O)₂]²⁺	deep royal blue	610 nm	orange
[Fe(SCN)(H₂O)₅]²⁺	intense red	480 nm	blue-green
[Ni(H₂O)₆]²⁺	pale green	720 nm	red
[Co(H₂O)₆]²⁺	pink	510 nm	green
MnO₄⁻	deep purple	525 nm	yellow-green
Cr₂O₇²⁻	orange	440 nm	violet/blue
CrO₄²⁻	yellow	370 nm (UV-near)	violet

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The Beer-Lambert Law

The Beer-Lambert law relates absorbance to concentration in a single linear equation:

$A = \\varepsilon \\, c \\, l$A=varepsilon,c,l

where:

• A = absorbance (dimensionless)
• ε (epsilon) = molar absorptivity (or molar extinction coefficient), units of mol⁻¹ dm³ cm⁻¹ (often written as M⁻¹ cm⁻¹)
• c = concentration of the absorbing species (mol dm⁻³)
• l = path length of light through the cuvette (cm)

For a given absorbing species at a fixed wavelength in a standard cuvette, ε and l are constants of the experiment, so A is directly proportional to c. A graph of absorbance vs concentration is therefore a straight line passing through the origin, with gradient = εl. This is the calibration curve, and it is the central experimental tool of colorimetry.

The molar absorptivity ε characterises how strongly a particular species absorbs light at a particular wavelength. Strong-field complexes with allowed d-d transitions (like [Cu(NH₃)₄(H₂O)₂]²⁺ at 610 nm) typically have ε ≈ 50–150 mol⁻¹ dm³ cm⁻¹; weak-field complexes (like [Cu(H₂O)₆]²⁺ at 800 nm) have ε ≈ 10–20 mol⁻¹ dm³ cm⁻¹. Charge-transfer complexes (like [Fe(SCN)(H₂O)₅]²⁺ at 480 nm, which involves an LMCT transition) can have ε values of 1,000–10,000 mol⁻¹ dm³ cm⁻¹, hundreds of times stronger than ordinary d-d transitions. The very large ε of MnO₄⁻ at 525 nm (~2,400 mol⁻¹ dm³ cm⁻¹) is what makes the permanganate ion such an intensely coloured species — even a 10⁻⁴ mol dm⁻³ solution gives a visible deep purple.

Assumptions of the Beer-Lambert Law

The law holds only when all four of the following experimental conditions are met:

1. The solution is dilute. At high concentrations, absorbing molecules interact electronically (aggregation, dimer formation, refractive-index changes) and the linear relationship breaks down. For most coloured transition-metal complexes, "dilute" means concentrations below about 10⁻² mol dm⁻³.
2. Monochromatic light. The law assumes a single wavelength. Real filters have finite bandwidth (~20 nm), and if the absorbing species' absorption spectrum varies significantly across this bandwidth, deviations occur. This is why narrow-bandwidth filters give better adherence to linearity.
3. No chemical change with concentration. The absorbing species must not associate (dimerise) or dissociate as the concentration changes. For example, dichromate ⇌ chromate equilibrium shifts with pH, so absorbance at a fixed wavelength is not a pure function of total Cr concentration unless pH is controlled.
4. Constant cuvette path length and orientation. Cuvettes are made to a tolerance of ~0.01 mm; the four faces are not equivalent (only two are optical-quality). Inserting the cuvette in the wrong orientation can introduce systematic errors.

When any of these assumptions fails, the calibration curve curves at high concentration (typically convex downward, with absorbance levelling off rather than continuing to rise linearly). The practical remedy is to stay in the dilute, linear region of the calibration curve — typically absorbance values between 0.1 and 1.0. Outside this range, dilute the sample and re-measure, then multiply back to recover the original concentration.

The Calibration Curve Method

The standard procedure for determining an unknown concentration:

Step 1: Prepare Standards

Make up a series of standard solutions of known concentration of the metal ion (e.g. 0.000, 0.020, 0.040, 0.060, 0.080, 0.100 mol dm-3). Use a volumetric flask and an accurate stock solution.

If the ion is weakly coloured, convert it to a more intensely coloured complex first by adding an excess of a strong-field ligand. For example:

• Cu2+ -> [Cu(NH3)4(H2O)2]2+ by adding excess NH3
• Fe3+ -> [Fe(SCN)(H2O)5]2+ by adding excess KSCN solution
• Cr3+ -> Cr2O7^2- by oxidising with an excess of acidified peroxide

Step 2: Measure Absorbance of Each Standard

Place each standard solution in a cuvette and measure its absorbance in the colorimeter. Use the filter with wavelength closest to lambda_max. Remember to "zero" the colorimeter with a blank cuvette containing distilled water (or the solvent without analyte).

Step 3: Plot Calibration Curve

Plot A on the y-axis and c on the x-axis. Draw a line of best fit through the points (it should pass through the origin if Beer-Lambert is obeyed).

Step 4: Measure Absorbance of the Unknown

Prepare the unknown solution in the same way as the standards (same ligand added, same volume of reagent, same dilution). Measure its absorbance.

Step 5: Read Concentration from the Graph

Look up the absorbance on the y-axis and read across to the line, then down to the concentration on the x-axis. Alternatively, use the gradient of the line (which equals eps x l) to calculate c directly from A.

Worked Example

Suppose a calibration with 0.0100 mol dm-3 Cu(NH3)4^2+ standards gives the following data:

Concentration (mol dm-3)	Absorbance
0.000	0.00
0.0020	0.40
0.0040	0.80
0.0060	1.20
0.0080	1.60
0.0100	2.00

The gradient is A/c = 2.00 / 0.0100 = 200 mol-1 dm3. In a 1 cm cuvette, this means eps = 200 mol-1 dm3 cm-1.

An unknown Cu2+ solution (after adding excess NH3) gives A = 1.00. From the graph or from A = eps c l:

c = A / (eps l) = 1.00 / (200 x 1) = 0.00500 mol dm-3 = 5.00 x 10-3 mol dm-3

So the unknown is 5.00 x 10-3 mol dm-3 Cu2+. If the sample was diluted before analysis, back-calculate the original concentration.

Why Ligand Exchange Before Measurement?

Sometimes the aqua complex is too weakly coloured to measure reliably:

• [Cu(H2O)6]2+ is pale blue (eps ~12 mol-1 dm3 cm-1)
• [Cu(NH3)4(H2O)2]2+ is deep royal blue (eps ~60 mol-1 dm3 cm-1)

Adding excess ammonia converts all the Cu2+ to the more intense complex, giving a five-fold increase in absorbance and hence five-fold better sensitivity. This is standard practice for Cu, Ni, Co, and other weakly coloured aqua complexes. You must add exactly the same amount of ligand to every standard and the unknown to ensure the complex is consistent.

Similarly for Fe3+, adding thiocyanate (SCN-) converts the pale yellow Fe3+ to an intensely red [Fe(SCN)(H2O)5]2+ complex. This is the standard method for determining trace iron in water samples.

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Common Practical Considerations

A full-marks practical answer must address every one of the following sources of error: