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Measurement and Data Processing - Uncertainties and errors in measurement

Grade 12IBChemistry

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Random errors occur due to unpredictable fluctuations in environmental conditions or difficulty in reading instruments. They affect the precision of a measurement and can be reduced by repeated trials and averaging.

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Systematic errors occur due to flawed experimental design or poorly calibrated instruments (e.g., a zero error on a balance). They shift all measurements in the same direction, affecting accuracy.

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Accuracy refers to how close a measured value is to the true or accepted value, while Precision refers to how close repeated measurements are to each other.

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The uncertainty of an analogue scale is typically ±\pm half of the smallest division, while for a digital scale, it is ±\pm the smallest scale division.

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Significant figures in a final result should reflect the precision of the least precise measurement used in calculations. In addition/subtraction, use the least number of decimal places; in multiplication/division, use the least number of significant figures.

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The Percentage Error is used to compare an experimental value to a theoretical/literature value: ∣Literature−Experimental∣Literature×100%\frac{|Literature - Experimental|}{Literature} \times 100\% .

📐Formulae

Absolute Uncertainty=Δx\text{Absolute Uncertainty} = \Delta x

Fractional Uncertainty=Δxx\text{Fractional Uncertainty} = \frac{\Delta x}{x}

Percentage Uncertainty=Δxx×100%\text{Percentage Uncertainty} = \frac{\Delta x}{x} \times 100\%

Propagation (Addition/Subtraction): y=a±b⇒Δy=Δa+Δb\text{Propagation (Addition/Subtraction): } y = a \pm b \Rightarrow \Delta y = \Delta a + \Delta b

Propagation (Multiplication/Division): y=a×bc⇒Δyy=Δaa+Δbb+Δcc\text{Propagation (Multiplication/Division): } y = \frac{a \times b}{c} \Rightarrow \frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}

Propagation (Powers): y=an⇒Δyy=∣n∣Δaa\text{Propagation (Powers): } y = a^n \Rightarrow \frac{\Delta y}{y} = |n| \frac{\Delta a}{a}

💡Examples

Problem 1:

A student measures the initial temperature of a reaction as 25.0 ∘C±0.5 ∘C25.0 \, ^\circ C \pm 0.5 \, ^\circ C and the final temperature as 40.0 ∘C±0.5 ∘C40.0 \, ^\circ C \pm 0.5 \, ^\circ C. Calculate the temperature change (ΔT\Delta T) and its absolute uncertainty.

Solution:

ΔT=Tfinal−Tinitial=40.0−25.0=15.0 ∘C\Delta T = T_{final} - T_{initial} = 40.0 - 25.0 = 15.0 \, ^\circ C. For subtraction, add absolute uncertainties: Δ(ΔT)=0.5+0.5=1.0 ∘C\Delta (\Delta T) = 0.5 + 0.5 = 1.0 \, ^\circ C. Final result: 15.0±1.0 ∘C15.0 \pm 1.0 \, ^\circ C.

Explanation:

When quantities are added or subtracted, their absolute uncertainties are summed to find the total uncertainty.

Problem 2:

The mass of a metal block is 20.00 g±0.01 g20.00 \, g \pm 0.01 \, g and its volume is 5.0 cm3±0.2 cm35.0 \, cm^3 \pm 0.2 \, cm^3. Calculate the density and its percentage uncertainty.

Solution:

Density ρ=mV=20.005.0=4.0 g⋅cm−3\rho = \frac{m}{V} = \frac{20.00}{5.0} = 4.0 \, g \cdot cm^{-3}. Sum of percentage uncertainties: (0.0120.00×100)+(0.25.0×100)=0.05%+4.0%=4.05%(\frac{0.01}{20.00} \times 100) + (\frac{0.2}{5.0} \times 100) = 0.05\% + 4.0\% = 4.05\%.

Explanation:

In multiplication and division, the percentage uncertainties (or fractional uncertainties) of the measurements are added together.

Problem 3:

If the experimental value for the enthalpy of combustion of ethanol is −1200 kJ⋅mol−1-1200 \, kJ \cdot mol^{-1} and the literature value is −1367 kJ⋅mol−1-1367 \, kJ \cdot mol^{-1}, calculate the percentage error.

Solution:

Percentage Error=∣−1367−(−1200)∣∣−1367∣×100%=1671367×100%≈12.2%\text{Percentage Error} = \frac{|-1367 - (-1200)|}{|-1367|} \times 100\% = \frac{167}{1367} \times 100\% \approx 12.2\%.

Explanation:

Percentage error measures the accuracy of the result compared to an established reference value.

Uncertainties and errors in measurement Revision - Grade 12 Chemistry IB