Calibration Curves in Colorimetry

Introduction

Calculating a concentration from A = εcl requires an accurate value of ε for exactly the instrument, wavelength and conditions in use. Often that value is not known, or a simple colorimeter does not produce perfectly monochromatic light. The practical solution is a calibration curve : measure the absorbance of several solutions of known concentration, plot a graph, and read the unknown from it. This approach is used in school laboratories and industrial quality control alike, because it builds in the behaviour of the real instrument.

Core explanation

The colorimeter. A colorimeter passes light from a lamp or LED through a coloured filter, through the sample in a cuvette and onto a photodetector. The filter is chosen to be the complementary colour of the solution, so that the light used is the light most strongly absorbed. A UV-visible spectrophotometer does the same job with a monochromator that selects a narrow band of wavelengths, usually set at λmax.

Standard solutions. A stock solution of accurately known concentration is prepared by weighing a pure solid and dissolving it in a volumetric flask. Accurate dilutions of this stock, using pipettes or burettes and volumetric flasks, give a series of standards, typically five or six, spanning the expected concentration of the unknown. If the analyte is colourless, a reagent is added to every standard and to the unknown to form a coloured product in the same way.

Zeroing with a blank. Before measuring, the instrument is set to A = 0 using a blank containing the solvent and any reagents but no analyte. This removes contributions from the cuvette, solvent and reagent colour.

Plotting the curve. Absorbance (y-axis) is plotted against concentration (x-axis) for each standard. If the Beer–Lambert law holds, the points lie on a straight line through the origin. A line of best fit is drawn, or a linear regression is calculated to give the equation A = mc + b, where m (the gradient) equals εl and b should be close to zero.

Reading the unknown. The unknown is treated exactly like the standards and its absorbance is measured. Its concentration is found by: - reading across from its absorbance to the line and down to the concentration axis; or - using the equation of the line: c = (A − b)/m.

If the unknown was diluted before measurement, the result is multiplied by the dilution factor.

Staying within range. The unknown's absorbance must lie between the lowest and highest standards. Reading within the range is interpolation and is reliable. Extending the line beyond the highest standard is extrapolation , which is unsafe because the line may curve at high concentration. An unknown that is too concentrated should be diluted and remeasured.

Quality of the calibration. Scattered points suggest errors in preparing standards or dirty or scratched cuvettes. Repeating measurements, handling cuvettes by their frosted sides, and orienting them the same way each time improve precision. A correlation coefficient close to 1 indicates a good linear fit.

Step-by-step reasoning

To find an unknown concentration using a calibration curve:

1. Choose the filter or wavelength that the solution absorbs most strongly. 2. Zero the instrument with a blank. 3. Measure the absorbance of each standard and plot A against c. 4. Draw the best-fit line and check it is linear through the range. 5. Measure the unknown, read its concentration from the line, and correct for dilution.

Visual explanation

Imagine a graph with five points rising in a straight line from the origin, labelled 0.2, 0.4, 0.6, 0.8 and 1.0 × 10⁻⁴ mol dm⁻³. A dashed horizontal line runs from A = 0.45 on the y-axis to meet the best-fit line, then a dashed vertical line drops to the concentration axis, showing the unknown's concentration.

Real-world analogy

A calibration curve is like marking a measuring jug by pouring in known volumes and drawing a line at each level. Once the jug is marked, you can read any unknown volume from it, but only between the lowest and highest marks you actually made.

Real-world example

Water companies monitor phosphate in rivers by adding a reagent that forms an intense blue complex with phosphate ions. A calibration curve from phosphate standards allows the analyst to read the concentration in each river sample, helping to detect pollution that could cause algal blooms.

Why?

Why use a calibration curve rather than a literature value of ε? A colorimeter's filter passes a band of wavelengths, and each instrument, reagent batch and temperature can differ slightly. Calibrating with standards measured on the same day under the same conditions automatically corrects for these effects.

Common misconception

"Any point on the extended line gives a reliable concentration." Beyond the highest standard the relationship may no longer be linear, so extrapolated readings can be seriously wrong. Dilute the unknown instead.

Worked example

Question: A calibration line has the equation A = 4850c + 0.004, where c is in mol dm⁻³. An unknown, diluted by a factor of 10, gives A = 0.392. Find the original concentration.

Reasoning: c = (0.392 − 0.004) ÷ 4850 = 8.00 × 10⁻⁵ mol dm⁻³ for the diluted solution. Multiply by 10.

Answer: 8.00 × 10⁻⁴ mol dm⁻³.

Quick check

1. Why should the absorbance of an unknown lie within the range of the standards used for calibration? Answer: Interpolation is reliable, but outside the range the line may curve, so extrapolated concentrations may be inaccurate.

Exam focus

Be able to describe preparing standards by accurate dilution, zeroing with a blank, choosing a complementary filter, plotting A against c and reading the unknown. Explain why the gradient equals εl, and why unknowns outside the range must be diluted.

Advanced insight

Where the sample contains other substances that affect the absorbance (a "matrix effect"), analysts use the standard addition method: known amounts of analyte are added to portions of the sample itself, and the line is extrapolated back to the x-axis. The negative intercept gives the original concentration while automatically correcting for the matrix.

Summary

A calibration curve is built by measuring the absorbances of standard solutions of known concentration and plotting A against c. After zeroing with a blank and choosing a complementary filter or λmax, the unknown's absorbance is read from the line or its equation, and corrected for any dilution. Readings should be interpolated within the calibrated range. The method corrects automatically for instrument and reagent effects.

Practice questions

1. Why is the filter in a colorimeter chosen to be the complementary colour of the solution? Answer: The solution absorbs its complementary colour most strongly, giving the largest absorbance change and highest sensitivity. 2. What is the purpose of the blank? Answer: It sets the absorbance to zero for everything except the analyte, cancelling absorption by the solvent, reagents and cuvette. 3. A calibration line passes through the origin with gradient 12 500 dm³ mol⁻¹. An unknown has A = 0.625. Find its concentration. Answer: c = 0.625 ÷ 12 500 = 5.00 × 10⁻⁵ mol dm⁻³. 4. An unknown gives an absorbance higher than the most concentrated standard. What should be done? Answer: Dilute it by a known factor, remeasure so the absorbance lies within the calibrated range, then multiply the result by the dilution factor.