How do you make a calibration curve with the Beer-Lambert law?

The Beer-Lambert law says absorbance is proportional to concentration: A = ε × l × c, where ε is the absorptivity and l the path length. A calibration curve turns that into a working tool by fitting a line through standards you measured.

What are the formulas?

For n standards with concentrations x and mean absorbances y:

• Slope = (n·Σxy − Σx·Σy) ÷ (n·Σx² − (Σx)²).
• Intercept = (Σy − slope·Σx) ÷ n.
• r² = (n·Σxy − Σx·Σy)² ÷ ((n·Σx² − (Σx)²)(n·Σy² − (Σy)²)).
• Residual standard deviation s_y = √(Σ(y − fitted y)² ÷ (n − 2)).
• LOD = 3.3 × s_y ÷ slope. LOQ = 10 × s_y ÷ slope.
• Unknown concentration = (absorbance − intercept) ÷ slope.

What does a worked example look like?

Six standards from 0 to 10 mg/L give mean absorbances of 0.002, 0.101, 0.205, 0.298, 0.402 and 0.497.

• Slope = 0.04959 absorbance units per mg/L, intercept = 0.0029, r² = 0.9998.
• s_y = 0.00278, so LOD = 3.3 × 0.00278 ÷ 0.04959 = 0.18 mg/L and LOQ = 10 × 0.00278 ÷ 0.04959 = 0.56 mg/L.
• A sample that reads 0.250 contains (0.250 − 0.0029) ÷ 0.04959 = 4.98 mg/L. If it was diluted 1 to 10 before reading, the original is 49.8 mg/L.

What are the limits of the method?

The fit assumes a linear response across the standards. Absorbance is only linear at low values, usually below about 1, and stray light and high concentrations bend the curve. A sample above the highest standard should be diluted and read again, not extrapolated. The LOD and LOQ here are estimates from one calibration, and a method validation study is needed for reported detection limits.

If you prepare the standards from a stock solution, how to calculate molarity covers the concentration arithmetic.

The Beer-Lambert calibration curve calculator does the SUMPRODUCT least squares, reports r², LOD and LOQ and converts absorbances to concentrations.