Experimental Accuracy and Error Analysis

Accuracy, Precision, and Concordancy

In laboratory chemistry, accuracy and precision describe two completely distinct qualities of experimental data.

• Accuracy reflects how close a measured experimental value is to the true or accepted value of the quantity.
• Precision reflects how close repeated experimental measurements are to one another, showing how consistent your technique is, whether or not the readings are close to the true value.

The target analogy illustrates how these two properties interact:

• High accuracy, high precision: Measurements cluster tightly together right in the centre of the bullseye.
• Low accuracy, high precision: Measurements group tightly together, but far from the central bullseye.
• High accuracy, low precision: Individual measurements are widely scattered, but their numerical average lands close to the bullseye.
• Low accuracy, low precision: Measurements scatter widely across the target and land far away from the centre.

Concordant Titres and the Trial Run

When carrying out a titration, you perform a first rough (trial) titration quickly to establish the approximate endpoint volume. You then perform subsequent accurate titrations, adding titrant dropwise near the endpoint until you obtain at least two concordant titres.

• Concordant titres are readings that agree with one another to within $0.1\text{ cm}^3$.
• Averaging rule: Average only the concordant titres. Never include the rough titration in your calculation, as doing so introduces a known positive error into the titre volume.

Obtaining concordant titres demonstrates that your laboratory handling is precise. It does not prove your work is accurate, because a consistent procedural mistake will simply repeat an inaccurate value with high precision.

Systematic versus Random Errors

Experimental error represents the numerical difference between a measured laboratory quantity and the accepted true value. Errors fall into two categories depending on their predictability and origin.

Systematic Errors

A systematic error shifts measured values in one predictable direction, making every reading consistently too high or consistently too low. Because every reading experiences the same shift, systematic errors degrade accuracy while leaving precision unaffected.

• Apparatus faults: An electronic balance that has not been tared (zeroed), or an air bubble trapped in the burette tip that gets dislodged during delivery.
• Procedural faults: Leaving a filter funnel in the top of the burette so that stray droplets fall into the solution during titration, or failing to mix a volumetric flask thoroughly (the solution remains denser near the bottom, so every aliquot taken is consistently too concentrated).
• Meniscus alignment: Reading the meniscus consistently from above or below horizontal eye level. Parallax error shifts values in one direction when viewed from the same angle throughout, though inconsistent viewing angles contribute random error.

Random Errors

A random error is an unpredictable fluctuation that causes experimental readings to scatter in both directions around the true value. Random errors undermine experimental precision.

• Surrounding variations: Air draughts disturbing an unshielded balance pan, or fluctuating laboratory temperatures altering solution volumes.
• Human reaction variations: Slight differences in reaction time when closing the burette stopcock at the first hint of an endpoint colour change.
• Delivery variations: Tiny differences in the volume of the final droplet delivered from the burette tip.

Correct Technique in Volumetric Analysis

Marking schemes award exact procedural descriptions for cleaning glassware, transferring solutions, and reading volumes.

The Four Glassware Rinsing Rules

Always use deionised water rather than tap water. Tap water contains dissolved mineral ions such as $\text{Ca}^{2+}$, $\text{Mg}^{2+}$, $\text{Cl}^-$, and $\text{HCO}_3^-$ that react with analytical reagents and spoil the titration.

Primary Standards and Preparing Standard Solutions

A primary standard is a substance obtainable in an exceptionally pure, stable state, allowing an analyst to produce a solution of accurately known concentration by direct weighing and dilution.

• Four mandatory requirements: High purity, complete stability in air (neither hygroscopic, deliquescent, nor efflorescent), good water solubility, and a reasonably high relative molecular mass ($M_r$) to keep weighing percentage errors small.
• Suitable primary standards: Anhydrous sodium carbonate ($\text{Na}_2\text{CO}_3$), ammonium iron(II) sulfate [Mohr's salt, $(\text{NH}_4)_2\text{Fe}(\text{SO}_4)_2\cdot 6\text{H}_2\text{O}$], and potassium dichromate ($\text{K}_2\text{Cr}_2\text{O}_7$).
• Substances that are not primary standards: Sodium hydroxide ($\text{NaOH}$) absorbs atmospheric moisture and carbon dioxide; potassium manganate(VII) ($\text{KMnO}_4$) decomposes in sunlight and undergoes reduction by trace organic matter. These secondary standards must be standardised against a primary standard by titration.

Standard Solution Preparation

1. Weigh the solid on an electronic balance by difference using a clean weighing bottle or clock glass.
2. Dissolve the solid completely in a beaker containing a small volume of deionised water, stirring with a clean glass rod.
3. Pour the solution through a funnel into the volumetric flask. Rinse the beaker, glass rod, and funnel with deionised water, adding all washings into the flask so no solute is lost.
4. Add deionised water until the level is just below the graduation mark on the neck, then use a dropper to bring the bottom of the meniscus onto the mark at eye level.
5. Stopper the volumetric flask and invert it at least 20 times so the solution becomes completely homogeneous.

Pipette Handling

Fill the pipette using a safety pipette filler. Draw liquid above the graduation mark, wipe the exterior of the tip with a tissue, and run the liquid down until the bottom of the meniscus rests on the mark. Allow the liquid to empty into the conical flask under gravity, touch the delivery tip against the inside wall of the flask, and do not blow out the final droplet, as the pipette calibration accounts for the retained liquid.

Direction of Error: 'Too High, Too Low, or No Effect?'

Higher Level papers regularly present an experimental slip and ask candidates to state the effect on the titre or calculated concentration, along with a justifying reason. To solve these, ask yourself whether the mistake alters the moles transferred to the flask or alters the concentration of liquid inside the burette.

Equivalence Point, Indicators, and Titration Setup

The equivalence point represents the stage where reactants have mixed in the precise stoichiometric ratio dictated by the balanced chemical equation. The endpoint is the stage where the chemical indicator alters colour to signal completion.

An indicator is suitable only when its colour-change pH interval lies completely within the steep (near-vertical) section of the titration pH curve.

Diagram Description: Standard Titration Setup

When sketching a titration apparatus, draw a tall burette held vertically in a retort stand with a burette clamp. The burette is graduated with zero at the top, calibrated to read to $0.05\text{ cm}^3$, and ends in a stopcock tap. Beneath the tip sits a conical flask containing the pipetted aliquot and two to three drops of indicator. Place the conical flask on a clean white tile to provide a bright background so the first permanent colour change can be detected clearly. The filter funnel used to fill the burette must be removed before setting the initial volume.

Percentage Error and Measurement Uncertainty

Quantifying error lets you evaluate whether a discrepancy stems from instrument limits or poor technique.

Burette Uncertainty

A standard class-A burette is read to the nearest $0.05\text{ cm}^3$. Because every titre requires two distinct readings (an initial and a final reading), the total reading uncertainty is $\pm 0.10\text{ cm}^3$.

• For a titre of $25.00\text{ cm}^3$: $\text{Percentage error} = \frac{0.10}{25.00} \times 100 = 0.40\%$
• For a titre of $5.00\text{ cm}^3$: $\text{Percentage error} = \frac{0.10}{5.00} \times 100 = 2.0\%$

Golden laboratory rule: Measuring larger quantities decreases the percentage error. This explains why we dilute concentrated solutions to give titres between $15\text{ cm}^3$ and $25\text{ cm}^3$, why we avoid tiny masses on the balance, and why larger temperature rises in calorimetry yield more accurate thermodynamic values.

Significant Figures and Instrument Precision

Always write instrument measurements to their correct physical decimal places:

• Burette: Record to two decimal places, ending in $.00$ or $.05$ (e.g., $21.45\text{ cm}^3$ or $19.00\text{ cm}^3$, never $19\text{ cm}^3$).
• Pipette: Quoted as a fixed volume to one decimal place (e.g., $25.0\text{ cm}^3$).
• Volumetric flask: $250.0\text{ cm}^3$.
• Balance: Two decimal places on standard school balances (e.g., $2.65\text{ g}$).

Measuring cylinders and beakers are designed only for approximate volumes; their wider bores give high percentage uncertainties. Final titration answers should be quoted to three significant figures with units.

Mandatory Experiment Error Protocols

Standard sources of error recur throughout the Leaving Certificate practical papers:

1. Redox Titrations ($\text{KMnO}_4$ / $\text{Fe}^{2+}$)

• Acidification: Acidify with excess dilute sulfuric acid ($\text{H}_2\text{SO}_4$). Never use hydrochloric acid, because $\text{MnO}_4^-$ oxidises chloride ions ($\text{Cl}^-$) to toxic chlorine gas, causing a falsely high titre. Never use nitric acid, as $\text{HNO}_3$ is an oxidising agent that reacts with $\text{Fe}^{2+}$.
• Insufficient acid: Causes a brown precipitate of manganese(IV) oxide ($\text{MnO}_2$) instead of colourless $\text{Mn}^{2+}$, masking the endpoint.
• Meniscus reading: Because potassium manganate(VII) is intensely dark, align the top of the meniscus with the scale lines rather than the bottom. Be consistent between initial and final readings.
• Mole ratio: $\text{MnO}_4^- : \text{Fe}^{2+} = 1 : 5$.

2. Thermochemistry (Heat of Neutralisation)

In calculating heat release ($q = mc\Delta T$), we assume the solution has the density ($1.0\text{ g cm}^{-3}$) and specific heat capacity of water ($4.18\text{ kJ kg}^{-1}\text{ K}^{-1}$), with negligible heat absorbed by the cup.

• Sources of error: Heat lost to the surroundings through conduction or air draughts, heat absorbed by the thermometer, and incomplete mixing.
• Controls: Use an expanded polystyrene cup fitted with a lid. Polystyrene is a poor thermal conductor, reducing heat transfer compared to glass. Stir continuously and record the absolute maximum temperature reached.

3. Rates of Reaction (Disappearing Cross)

• Error source: Deciding when the printed black cross is fully obscured by the yellow sulfur precipitate is inherently subjective.
• Controls: The same observer must view the cross from directly above for every trial, using the same reaction flask, viewing distance, and sheet of paper.
• Dilution and volume: Keep total reaction volume strictly constant by adding deionised water so that only the concentration under test varies.

4. Water Analysis and Hardness

• Sampling: Fill sample containers completely to exclude air bubbles (crucial when assessing dissolved oxygen), keep chilled, and analyse promptly.
• EDTA titration: Determine total hardness using standard $\text{EDTA}$ with Eriochrome Black T indicator. Maintain the mixture at pH 10 using an ammonia buffer, which is necessary for a crisp colour transition from wine-red to blue.
• Colorimetry: Prepare a series of standard solutions, plot an absorbance versus concentration calibration curve, zero the instrument using a deionised water blank, and wipe cuvette faces clean of fingerprints.

Key terms

Accuracy

How close a measured experimental value is to the true or accepted value of the quantity.

Precision

How close repeated experimental measurements are to one another, showing repeatability regardless of the true value.

Concordant Titres

Titre volumes that agree with each other to within 0.1 cm³.

Primary Standard

A substance obtainable in a pure, stable form that allows preparation of a standard solution by direct weighing and dilution.

Standard Solution

A solution whose concentration is accurately known.

Systematic Error

A repeating error that consistently shifts measurements in one direction, reducing accuracy while leaving precision unaffected.

Random Error

An unpredictable variation causing readings to scatter in both directions, reducing precision.

Equivalence Point

The point in a titration where reactants have been mixed in the exact mole ratio given by the balanced chemical equation.

Endpoint

The point in a titration where the indicator changes colour and delivery is stopped.

Parallax Error

An error introduced by observing a liquid meniscus from an improper angle rather than horizontally at eye level.