Making and Using a Calibration Curve

Most analytical instruments don’t measure concentration directly. A colorimeter measures how much light is absorbed. A chromatograph measures a peak area. A flame photometer measures light intensity. To turn those signals into concentrations, chemists use a calibration curve: a graph that links the instrument’s response to known concentrations, so an unknown’s concentration can be read straight off it.

Calibration curves are used everywhere, from school colorimetry practicals to hospital blood tests, drinking water analysis and pharmaceutical quality control.

The idea

1. Make a series of standard solutions with accurately known concentrations of the substance you want to measure.
2. Measure the instrument’s signal for each standard.
3. Plot signal (y-axis) against concentration (x-axis) and draw the best-fit line.
4. Measure the signal for the unknown sample.
5. Read off the unknown’s concentration from the graph, or calculate it from the line’s equation.

Step 1: prepare standard solutions

Start with a stock solution of accurately known concentration, made by weighing a pure solid (see using a balance) and dissolving it in a volumetric flask.

Make standards by dilution. For example, from a 100 mg/dm³ stock, prepare 10.0 cm³ of each standard in volumetric flasks or with accurate pipettes and burettes:

Standard	Stock volume (cm³)	Water volume (cm³)	Concentration (mg/dm³)
Blank	0.00	10.00	0
1	1.00	9.00	10
2	2.00	8.00	20
3	4.00	6.00	40
4	6.00	4.00	60
5	8.00	2.00	80

Using c₁V₁ = c₂V₂: for standard 3, (100 × 4.00) ÷ 10.00 = 40 mg/dm³.

Tips:

• Use at least five standards plus a blank.
• Choose concentrations that bracket the expected unknown: some lower, some higher.
• Use accurate volumetric glassware, not measuring cylinders. See measuring volume accurately.
• Make standards in the same matrix (same solvent, pH and other ingredients) as the samples, as far as possible.

Step 2: measure the blank and the standards

The blank contains everything except the substance being measured (for example, just the solvent and any reagents). It’s used to zero the instrument, so that the signal from the solvent, the cuvette or background reagents isn’t mistaken for the analyte.

Then measure each standard, ideally from lowest to highest concentration, rinsing the cuvette or sample line between measurements. Repeating each measurement and averaging reduces random error.

Step 3: plot and fit

Plot signal on the y-axis against concentration on the x-axis.

For many techniques, the relationship is a straight line over a certain range. In spectrophotometry, this is described by the Beer–Lambert law: absorbance is proportional to concentration.

Draw a line of best fit. Spreadsheets and graphing calculators can calculate it by linear regression, giving:

• the equation y = mx + c (m is the gradient, c the intercept)
• the R² value (coefficient of determination), which measures how well the points fit a straight line. R² = 1.000 is a perfect fit; well-behaved calibrations typically give R² above 0.995.

Ideally the line passes through (or very close to) the origin, since zero concentration should give zero signal after blanking.

Step 4: find the unknown

Measure the unknown under exactly the same conditions. Then either:

• Read from the graph: draw a horizontal line from the unknown’s signal to the calibration line, then down to the x-axis; or
• Calculate from the equation: x = (y − c) ÷ m.

Worked example

Standards of a blue dye give these absorbances at 630 nm:

Concentration (mg/dm³)	Absorbance
0	0.000
10	0.121
20	0.238
40	0.482
60	0.719
80	0.958

Linear regression gives: absorbance = 0.01197 × concentration + 0.0009, with R² = 0.9999. A sports drink sample gives an absorbance of 0.350. Find the dye concentration.

concentration = (0.350 − 0.0009) ÷ 0.01197 = 29.2 mg/dm³

The result lies within the calibrated range (0–80 mg/dm³), so it’s reliable.

If the sample had been diluted, for example 5.00 cm³ of drink diluted to 25.00 cm³, multiply by the dilution factor (5) to get the concentration in the original drink: 146 mg/dm³.

Stay within the linear range

Calibration lines are only valid between the lowest and highest standards. Never extrapolate beyond them.

At high concentrations, many calibrations curve and flatten out. In colorimetry, this happens because the Beer–Lambert law breaks down at high absorbance (and because very little light reaches the detector). If an unknown gives a signal above the top standard:

• dilute the sample by a known factor and measure again, then multiply back up; or
• prepare higher standards to extend the range, checking linearity.

At very low concentrations, the signal becomes lost in noise. The lowest concentration that can be reliably detected is the limit of detection; the lowest that can be reliably measured is the limit of quantification.

Checking the calibration is reliable

• Look at the residuals. Points should scatter randomly above and below the line. A systematic pattern (for example, the middle points all above the line) suggests the relationship isn’t truly linear.
• Check R², but don’t rely on it alone; a curved calibration can still give a high R².
• Run a quality control standard of known concentration, made separately from the calibration standards, and check that the calibration reads it correctly.
• Recalibrate regularly, because instrument response can drift over time.

Internal standards and standard addition

Two refinements are used when samples are complicated:

Internal standard. A fixed amount of a different, similar compound is added to every standard and sample. The ratio of analyte signal to internal standard signal is plotted instead of the raw signal. This corrects for variations in injection volume or instrument sensitivity and is common in chromatography. See gas chromatography.

Standard addition. Known amounts of the analyte are added to portions of the sample itself, and the signal is plotted against the amount added. The line is extrapolated back to where the signal is zero; the distance along the negative x-axis gives the original concentration. This corrects for matrix effects, where other substances in the sample (such as salts in seawater or proteins in blood) change the signal.

Where calibration curves are used

• Colorimetry and UV–visible spectroscopy: concentrations of coloured species, iron in tablets, nitrate and phosphate in water. See colorimetry.
• Chromatography: drug content in tablets, caffeine in drinks, pollutants in water.
• Atomic spectroscopy: lead, sodium and other metals.
• Clinical analysers: glucose, cholesterol and many other blood tests.
• pH and ion-selective electrodes: calibrated with standard buffers and solutions.

Common mistakes

• Using too few standards or only one point.
• Standards that don’t cover the unknown’s concentration.
• Forgetting to blank the instrument.
• Extrapolating beyond the highest standard.
• Forgetting to multiply by the dilution factor.
• Preparing standards with measuring cylinders instead of volumetric glassware.

Key takeaways

• A calibration curve links an instrument’s signal to known concentrations so unknowns can be measured.
• Use at least five accurately prepared standards plus a blank, bracketing the unknown’s concentration.
• Fit a line of best fit (y = mx + c) and calculate unknowns with x = (y − c) ÷ m.
• Only trust results within the linear, calibrated range; dilute samples that read too high.
• Internal standards and standard addition correct for instrument variation and matrix effects.