1.5.3. Worked Example: Concentration from Calibration Curve

Suppose you measure absorbance A of a solution using UV‐Vis spectroscopy. You use a calibration curve with slope m and intercept b (from a linear fit to standards), so that A = mC + b. You measure A sample = 0.450 ± 0.005 (absorbance units), and you know from calibration that m = 1.234 ± 0.010 (absorbance per concentration unit) and b = 0.012 ± 0.002 (absorbance units). You want concentration C = (A – b) ÷ m.

In this example, we are determining the concentration of a solution based on its absorbance measured in UV-Vis spectroscopy. The absorbance (A) is linked to concentration (C) via the calibration equation, which is defined by the slope (m) and intercept (b). The provided measurements come with uncertainties, highlighting their inherent variability.

Imagine you're cooking and using a recipe that tells you the amount of an ingredient based on the sweetness of a mixture you've tasted (absorbance). Just as the recipe depends on how much of each ingredient you put in, the concentration of your solution depends on the measured absorbance and calibration curve.

Step 1: Compute nominal C: C = (0.450 – 0.012) ÷ 1.234 = 0.438 ÷ 1.234 ≈ 0.355 (concentration units)

Here, we are calculating the nominal concentration (C) using the absorbance value (0.450) and the intercept (b = 0.012) subtracted from the absorbance. We then divide by the slope (m = 1.234). This gives us the concentration in concentration units, which in this case is about 0.355.

Think of this as adjusting a drink recipe. You start with the total taste (absorbance), subtract the non-recipe part (the intercept), and then relate what’s left to the main ingredient (the concentration) using your recipe’s strength (the slope).

Step 2: Propagate uncertainty using partial derivatives: ● f(A, b, m) = (A – b) ÷ m. ● ∂f/∂A = 1 ÷ m. ● ∂f/∂b = –1 ÷ m. ● ∂f/∂m = – (A – b) ÷ m².

To find the uncertainty in the calculated concentration, we use partial derivatives to understand how the uncertainty in absorbance (A), the intercept (b), and the slope (m) affect the final computation. We calculate how much each of these variables influences our concentration result by evaluating their derivatives at the nominal values.

It's like adjusting your recipe for a cake. If you alter the amount of sugar, it changes the final taste (concentration). If you add too much flour (representing uncertainty in slope), that also alters how the cake turns out. We assess the impacts of these possible changes on the cake's sweetness as we check our ingredients.

Now multiply each derivative by the uncertainty in its variable, square, and sum: ● (∂f/∂A × δA)² = (0.8106 × 0.005)² = (0.004053)² = 1.643 × 10⁻⁵ ● (∂f/∂b × δb)² = (–0.8106 × 0.002)² = (–0.001621)² = 2.627 × 10⁻⁶ ● (∂f/∂m × δm)² = (–0.2878 × 0.010)² = (–0.002878)² = 8.283 × 10⁻⁶.

Here we are calculating the individual contributions to the uncertainty in the concentration. Each term results from multiplying the derivative by the uncertainty of its respective measurement variable (A, b, m), squaring it, and then summing these values to get the overall uncertainty contribution.

This is similar to calculating the total calories in a complex dish. Each ingredient contributes a portion (uncertainty) to the total count. By knowing how much each ingredient might vary, we can estimate the overall calorie uncertainty for the dish.

Sum = 1.643 × 10⁻⁵ + 2.627 × 10⁻⁶ + 8.283 × 10⁻⁶ = 2.734 × 10⁻⁵ Take square root to get δC: δC = sqrt(2.734 × 10⁻⁵) = 0.00523 Thus C = 0.355 ± 0.005 (concentration units). Report final result with two significant figures in the uncertainty: C = 0.355 ± 0.005.

After summing the contributions to uncertainty, we take the square root of that sum to get the final uncertainty in concentration (δC). The final concentration result is then reported in a format that clearly indicates the uncertainty, aligning with significance rules.

Imagine you finished baking a cake and want to present it. You note not only how sweet it is (the concentration) but also that it could be a bit sweeter or less sweet (the uncertainty) due to how you measured ingredients. That way, everyone knows the range they can expect when they taste it.

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