Empirical data collection and analysis

Class 12 Chemistry, unit 16, book pages 256–266

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Every result in chemistry comes from measurements, and every measurement carries some error. This unit explains the two kinds of data (qualitative and quantitative), the two main kinds of error (random and systematic), how random errors propagate into calculated results, and why repeating an experiment reduces random but not systematic errors. It ends with graphs: sketched graphs that show trends, drawn graphs that give numbers, and how to find a slope.

Learning outcomes (7)

Differentiate between qualitative data (all non-numerical information obtained from observation, not from measurement) and quantitative data (obtained from measurements and always associated with random errors or uncertainties caused by the apparatus and by human limitations such as reaction time).
Justify that the propagation of random errors in data processing shows the impact of the uncertainties on the final result: when data containing random errors are processed, the errors can propagate or accumulate through the calculation, giving a large uncertainty in the final result (for example, errors in the length and width of a rectangle propagate into its calculated area). This helps to assess the reliability of data and to draw valid conclusions.
Analyse the concept that experimental design and procedure usually lead to systematic errors in measurements, which cause deviation in a particular direction.
Justify that repeat trials and measurements will reduce random errors but not systematic errors.
Explain that graphical techniques are an effective means of communicating the effect of an independent variable on a dependent variable, and can lead to the determination of physical quantities.
Discuss that sketched graphs have labelled but unscaled axes, and are used to show qualitative trends such as variables that are proportional or inversely proportional.
Discuss that drawn graphs have labelled and scaled axes, and are used in quantitative measurements.

Analytical chemistry and why data matter

Analytical chemistry is mainly related to the chemical characterisation of materials: finding out what a substance is made of and how much of each part is present. Everything we use or consume is made of chemicals, so knowing the chemical composition of a substance is important in daily life.

Analytical chemistry plays a part in nearly all branches of chemistry: agricultural, clinical, environmental, forensic, manufacturing, metallurgical and pharmaceutical chemistry. Its results are usually reported as numbers, tables, charts and graphs (the book's Fig 16.1 shows a collage of such charts).

Analytical chemistry in every field of life
Field	What is analysed	Why it matters
Agriculture	Percentage of nitrogen in a fertilizer	Decides the value (quality and price) of the fertilizer
Food	Essential nutrients, minerals and vitamins	Decides the quality of the food
Environment	Pollutants in the air	Decides the air quality
Clinical	Glucose in the blood of diabetics	Monitored regularly to control the disease
Forensic	Trace elements from gunpowder on a suspect's hand	Proves that a gun was fired
Pharmaceutical	Purity of a drug	Decides how effective (efficacious) the drug is

Qualitative analysis: Analysis that finds which substances (elements, ions or functional groups) are present. Example: a flame test, or a precipitation test such as silver nitrate giving a white precipitate with chloride ions.
Quantitative analysis: Analysis that determines how much of one or more constituents is present. Examples: volumetric analysis (titration), gravimetric analysis (weighing a precipitate) and spectrophotometry.

16.1Types of dataOften examined

Chemical characterisation data are of two types. Both are important for understanding chemical phenomena.

Qualitative data: Non-numerical information obtained from observation, not from measurement. It describes the properties and behaviour of substances and reactions.
Quantitative data: Numerical information obtained from measurements with instruments. It is about measuring and calculating numerical values, and it always carries some random error or uncertainty.

Qualitative and quantitative data compared
Point	Qualitative data	Quantitative data
Nature	Words that describe	Numbers with units
Obtained by	Observation with the senses	Measurement with instruments
Examples	Colour change of a reagent showing specific ions; type of reaction (exothermic, endothermic, absorption); odour, colour and state (solid, liquid or gas) of a sample	Mass, volume, concentration, temperature, weight, pH
Uncertainty	No numerical uncertainty, but can depend on the observer's judgement	Always has random error or uncertainty from the apparatus and from human limits such as reaction time
Sample statement	'Adding silver nitrate gives a white precipitate, so chloride ions are present.'	' $24.10 c m^{3}$ of acid neutralised $25.00 c m^{3}$ of the alkali.'

Uncertainty in quantitative data

Measurements of mass, volume, concentration and temperature always contain some random error and uncertainty. These arise from the limitations of the apparatus and from human factors. The uncertainty of a measurement is the range within which the true value is expected to lie. It is written with a $\pm$ sign.

burette reading = 24.10 \pm 0.05 c m^{3} ⟹ 24.05 c m^{3} \leq true value \leq 24.15 c m^{3}

The book gives two everyday examples from the laboratory:

Titration glassware. A burette, a pipette and a volumetric flask are used to find the concentration of a substance by titration. Inaccurate calibration of this glassware, improper cleaning, and human error in reading the meniscus are sources of error. In a titration to find the concentration of an acid, the volume of titrant is read from the burette; small errors in reading the meniscus at the initial and final readings cause uncertainty in the calculated concentration.
Analytical balance. Improper calibration of the balance, air currents in the laboratory, and a dirty balance pan cause errors in mass. Even a slight draught or some dust on the pan changes the mass recorded for a sample.

Typical uncertainties of common apparatus (the exact value is printed on the glassware or taken as half the smallest division)
Apparatus	Smallest reading	Typical uncertainty
Analytical balance	$0.0001 g$	$\pm 0.0001 g$
Top-pan balance	$0.01 g$	$\pm 0.01 g$
Burette ( $50 c m^{3}$ , divisions of $0.1 c m^{3}$ )	$0.05 c m^{3}$ (estimated)	$\pm 0.05 c m^{3}$ per reading
Pipette ( $25 c m^{3}$ )	one fixed volume	about $\pm 0.06 c m^{3}$
Volumetric flask ( $250 c m^{3}$ )	one fixed volume	about $\pm 0.3 c m^{3}$
Measuring cylinder ( $100 c m^{3}$ )	$1 c m^{3}$	$\pm 0.5 c m^{3}$
Thermometer (divisions of $1^{\circ} C$ )	$0.5^{\circ} C$	$\pm 0.5^{\circ} C$

16.2Types of errorsOften examined

Error is the difference between the value obtained in an experiment and the value accepted for it (the true value, or the value given in the literature).

error percentage error = x_{measured} - x_{accepted} = \frac{∣ x _{measured} - x _{accepted} ∣}{x _{accepted}} \times 100%

Example: a student finds the boiling point of pure water to be $99.2^{\circ} C$ instead of $100.0^{\circ} C$ . The error is $99.2 - 100.0 = - 0.8^{\circ} C$ , and the percentage error is $\frac{0.8}{100.0} \times 100% = 0.8%$ .

Primarily there are two types of errors in experiments: random errors and systematic errors. Careless mistakes, called blunders, are a third kind (Fig 16.2).

1. Random errors

A random error is the random difference between the observed value and the true value. Random errors change from one reading to the next in an unpredictable way. They are also called indeterminate errors, because their exact cause in any one reading cannot be found.

Random errors can be affected by:

how easy the instrument or scale is to touch and read, for example a person reads the scale incorrectly or estimates the last digit between two marks slightly differently each time;
changes in the environment, such as temperature fluctuations in the laboratory or air currents in the room.

Random errors make the result deviate from the accepted value in either direction (sometimes too high, sometimes too low). Repeating the experiment and working with the average of the results reduces their effect, because the high and low readings partly cancel.

2. Systematic errors

A systematic error is a consistent, repeatable error due to faulty equipment, a flawed experimental design or procedure, or human mistakes made the same way every time. Systematic errors always pull the result away from the accepted value in the same direction (always too high or always too low). They are also called determinate errors, because their cause can be found and removed.

If an electronic balance is not zeroed (using the tare button) and it already shows a reading with the pan empty, every mass recorded is greater than it should be.
If a burette is not read at eye level, every volume is wrong in the same direction because of a parallax error. With the eye above the meniscus, all readings are lower than they should be.
If the cap is not kept on a spirit burner during a calorimetry experiment, alcohol evaporates. The loss in mass is then larger than the mass actually burned, so the calculated heat released per gram (or per mole) of fuel always comes out too small.

Repeating the test and working with the mean will not eliminate systematic errors, because every repeat carries the same error.

Parallax error in reading a burette. Only the line of sight level with the bottom of the meniscus gives the true reading. The scale of a burette increases downwards, so a mark seen higher up is a smaller number.

3. Blunders

Blunders (gross errors) are errors caused by carelessness; they are typically accidents. Examples are spilling liquid before it can be measured, writing $24.10$ as $21.40$ , or using the wrong solution. A blunder is not an error to be averaged: the reading is discarded and the measurement is repeated.

Types of error (redrawn from the book's Fig 16.2). Systematic errors can be instrumental, observational, environmental or theoretical; random errors can be environmental or observational.

Random and systematic errors compared
Point	Random error	Systematic error
Definition	Random difference between observed and true value	Consistent, repeatable error
Cause	Unpredictable changes; difficulty in reading the scale; draughts and temperature changes	Faulty or uncalibrated equipment, flawed design or procedure, impure reagents, the same human mistake every time
Direction	Either direction (too high or too low)	Always the same direction
Other name	Indeterminate error	Determinate error
Mainly affects	Precision (readings scatter)	Accuracy (all readings shifted)
Shown by	Spread in repeated readings	Comparing with a standard or another method
Reduced by	Repeating and averaging	Finding and removing the cause (calibration, pure reagents, correct technique); repeating does not help
Examples	Fluctuating balance reading in a draught; judging the end-point colour slightly differently each time	Balance not zeroed; burette always read from above; wrongly calibrated pipette

Accuracy and precision

Accuracy: How close a measured value (or the mean of several values) is to the true or accepted value. Small error means high accuracy.
Precision: How close repeated measurements are to one another, whatever their closeness to the true value. A small spread means high precision.

Random errors scatter the readings, so they spoil precision. Systematic errors shift all readings in one direction, so they spoil accuracy, even when the readings agree well with each other.

Accuracy and precision shown as shots at a target whose centre is the true value.

Classifying results by accuracy and precision

A titration should need exactly

25.00 c m^{3}

of titrant. Four students each record three readings (in

c m^{3}

): A:

24.98, 25.02, 25.00

; B:

24.50, 24.52, 24.51

; C:

24.20, 25.80, 25.10

; D:

24.10, 24.90, 23.70

. Say whether each set is accurate and whether it is precise, and name the main type of error.

Work out the mean (for accuracy) and the range (largest minus smallest reading, for precision) of each set.

\overset{x}{ˉ}_{A} \overset{x}{ˉ}_{B} = \frac{24.98 + 25.02 + 25.00}{3} = 25.00 = \frac{24.50 + 24.52 + 24.51}{3} = 24.51

\overset{x}{ˉ}_{C} \overset{x}{ˉ}_{D} = \frac{24.20 + 25.80 + 25.10}{3} = 25.03 = \frac{24.10 + 24.90 + 23.70}{3} = 24.23

Student	Mean ( $c m^{3}$ )	Range ( $c m^{3}$ )	Accurate	Precise	Main error
A	25.00	0.04	yes	yes	only small random errors
B	24.51	0.02	no (error $- 0.49$ , about $- 2%$ )	yes	systematic
C	25.03	1.60	yes, on average	no	random
D	24.23	1.20	no	no	both

Student B's readings agree within $0.02 c m^{3}$ , yet all are about $0.5 c m^{3}$ too low, which points to a systematic error such as reading the burette from above or using a wrongly calibrated pipette.

16.2.1Propagation of random errorsOften examined

When we process data that contain random errors, these errors propagate (carry forward and accumulate) through the calculation, and the final result has a larger uncertainty than the individual measurements. For example, if we measure the length and width of a rectangle to calculate its area, any small error in the length and width propagates through the area calculation, giving a larger uncertainty in the final area.

Derivation Uncertainty in the area of a rectangle

Aim. Find how the uncertainties $Δ L$ in the length and $Δ W$ in the width combine into an uncertainty $Δ A$ in the area $A = L W$ .

Assumptions.

Only random errors are present, and each uncertainty is small compared with the quantity measured.

L

and

W

are each a little too large, the area gains two strips,

L Δ W

and

W Δ L

, and a tiny corner

Δ L Δ W

The measured (central) area is length times width.
$A = L \times W$
The largest possible area uses the largest possible length and width.
$A_{m a x} = (L + Δ L) (W + Δ W)$
Multiply out the brackets. The four terms are the four regions in the figure.
$A_{m a x} = L W + L Δ W + W Δ L + Δ L Δ W$
The uncertainty is the extra area above $L W$ .
$Δ A = A_{m a x} - L W = W Δ L + L Δ W + Δ L Δ W$
$Δ L Δ W$ is the product of two small numbers (here $0.1 \times 0.1 = 0.01 c m^{2}$ ), so it is negligible and is dropped.
$Δ A \approx W Δ L + L Δ W$
Divide both sides by $A = L W$ to get the rule in terms of relative (fractional) uncertainties.
$\frac{Δ A}{A} = \frac{W Δ L}{L W} + \frac{L Δ W}{L W} = \frac{Δ L}{L} + \frac{Δ W}{W}$

Result

Δ A or \frac{Δ A}{A} \times 100% = W Δ L + L Δ W = \frac{Δ L}{L} \times 100% + \frac{Δ W}{W} \times 100%

Each measurement passes its uncertainty on to the result. When quantities are multiplied, their percentage uncertainties add, so the area is always less certain than either measurement on its own.

The book's rectangle

The length of a rectangle is measured with a scale whose smallest division is

1 cm

, so each measurement is taken to the nearest

0.1 cm

L = 10.0 \pm 0.1 cm

and

W = 5.0 \pm 0.1 cm

. Find the area and its uncertainty.

Given

$L = 10.0 cm$
$Δ L = 0.1 cm$
$W = 5.0 cm$
$Δ W = 0.1 cm$

To find

$A$ and $Δ A$

Calculate the area.
$A = L \times W = 10.0 cm \times 5.0 cm = 50 c m^{2}$
Propagate the uncertainty: each side's uncertainty is multiplied by the other side.
$Δ A = W Δ L + L Δ W = (5.0 \times 0.1) + (10.0 \times 0.1) = 0.5 + 1.0 = 1.5 c m^{2}$
Check with percentages: $1%$ from the length and $2%$ from the width give $3%$ of the area.
$\frac{0.1}{10.0} \times 100% + \frac{0.1}{5.0} \times 100% = 1% + 2% = 3%, 3% of 50 = 1.5 c m^{2}$
Check with the extreme values.
$10.1 \times 5.1 = 51.51 c m^{2}, 9.9 \times 4.9 = 48.51 c m^{2}$

So the area can be reported as $50.0 \pm 1.5 c m^{2}$ : it could be as low as $48.5 c m^{2}$ or as high as $51.5 c m^{2}$ . This range shows the practical variability in our measurements and helps us judge the reliability of the data and draw valid conclusions.

A = 50.0 \pm 1.5 c m^{2}

Rules for propagating uncertainties

How random uncertainties combine
Calculation	Rule	Example
Addition or subtraction, $Q = a + b$ or $Q = a - b$	Add the absolute uncertainties: $Δ Q = Δ a + Δ b$	Titre = final reading minus initial reading
Multiplication or division, $Q = ab$ or $Q = a / b$	Add the percentage (relative) uncertainties	Area $= L W$ ; density $= m / V$ ; concentration $= n / V$
Power, $Q = a^{n}$	Multiply the percentage uncertainty by $n$	Area of a circle $π r^{2}$ : twice the percentage uncertainty of $r$
Exact number times a quantity, $Q = k a$	$Δ Q = k Δ a$ (percentage unchanged)	Mass of 3 identical coins from the mass of one

Q = a \pm b : Q = ab or \frac{a}{b} : Q = a^{n} : Δ Q = Δ a + Δ b \frac{Δ Q}{Q} = \frac{Δ a}{a} + \frac{Δ b}{b} \frac{Δ Q}{Q} = n \frac{Δ a}{a}

Uncertainty in a titre

The initial burette reading is

0.50 \pm 0.05 c m^{3}

and the final reading is

24.60 \pm 0.05 c m^{3}

. Find the titre with its uncertainty.

The titre is a difference of two readings.
$V = 24.60 - 0.50 = 24.10 c m^{3}$
For a difference, the absolute uncertainties add.
$Δ V = 0.05 + 0.05 = 0.10 c m^{3}$
Percentage uncertainty of the titre.
$\frac{0.10}{24.10} \times 100% = 0.41%$

V = 24.10 \pm 0.10 c m^{3}

Uncertainty in a density

A solid has mass

12.50 \pm 0.01 g

(top-pan balance) and volume

5.0 \pm 0.1 c m^{3}

(measuring cylinder). Find its density and the uncertainty.

Density is mass divided by volume.
$ρ = \frac{m}{V} = \frac{12.50}{5.0} = 2.50 g c m^{- 3}$
For a quotient, the percentage uncertainties add.
$\frac{0.01}{12.50} \times 100% + \frac{0.1}{5.0} \times 100% = 0.08% + 2.0% = 2.08%$
Convert back to an absolute uncertainty and round it to 1 significant figure.
$Δ ρ = 2.08% \times 2.50 = 0.052 \approx 0.05 g c m^{- 3}$

Almost all of the uncertainty comes from the volume. To improve the result, measure the volume more precisely (for example with a burette), not the mass.

ρ = 2.50 \pm 0.05 g c m^{- 3}

Uncertainty in the concentration of a standard solution

2.650 \pm 0.001 g

of anhydrous sodium carbonate (

M = 106 g mo l^{- 1}

) is dissolved and made up to

250.0 \pm 0.3 c m^{3}

in a volumetric flask. Find the concentration and its uncertainty.

Moles of solute. The molar mass is taken as exact.
$n = \frac{2.650}{106} = 0.02500 mol$
Concentration in $mol d m^{- 3}$ ( $250.0 c m^{3} = 0.2500 d m^{3}$ ).
$c = \frac{0.02500}{0.2500} = 0.1000 mol d m^{- 3}$
Add the percentage uncertainties of mass and volume.
$\frac{0.001}{2.650} \times 100% + \frac{0.3}{250.0} \times 100% = 0.04% + 0.12% = 0.16%$
Absolute uncertainty.
$Δ c = 0.16% \times 0.1000 = 0.00016 \approx 0.0002 mol d m^{- 3}$

c = 0.1000 \pm 0.0002 mol d m^{- 3}

Significant figures and reporting results

Significant figures are all the digits of a measured value that are known reliably, plus the last digit, which is estimated (uncertain). The number of significant figures shows the precision of the measuring instrument. A burette reading of 24.15\,\mathrm{cm^3} has 4 significant figures: 2, 4 and 1 are certain and the final 5 is estimated.

Rules for counting significant figures

All non-zero digits are significant: 245 has 3.
Zeros between non-zero digits are significant: 1.005 has 4.
Leading zeros (to the left of the first non-zero digit) are not significant; they only fix the decimal point: 0.0025 has 2.
Trailing zeros after a decimal point are significant: 2.50 has 3 and 24.10 has 4.
Trailing zeros in a whole number without a decimal point are ambiguous: 2500 may have 2, 3 or 4. Scientific notation removes the doubt: 2.5\times10^{3} has 2, 2.50\times10^{3} has 3.
Exact numbers (counted objects, defined conversions such as 1\,\mathrm{dm^3}=1000\,\mathrm{cm^3}, and coefficients in equations) have unlimited significant figures.

Counting significant figures
Quantity	Significant figures	Reason
0.00250\,\mathrm{g}	3	leading zeros do not count; the final zero does
1.0050\,\mathrm{g}	5	zeros between digits and trailing zeros after the decimal point count
24.10\,\mathrm{cm^3}	4	the last zero shows the burette was read to 0.01\,\mathrm{cm^3} steps
100.0\,^\circ\mathrm{C}	4	trailing zero after the decimal point
6.022\times10^{23}	4	only the digits before the power of ten count
5000\,\mathrm{m}	1 to 4	ambiguous; write 5.0\times10^{3}\,\mathrm{m} for 2

Rounding off

If the first digit to be dropped is less than 5, leave the last kept digit unchanged (2.543\to2.54). If it is 5 or more, increase the last kept digit by 1 (2.546\to2.55, 2.545\to2.55). Some textbooks round an exact 5 to the nearest even digit (2.545\to2.54); use one rule consistently.

Significant figures in calculations

Multiplication and division: the answer has as many significant figures as the value with the fewest significant figures. Example: 12.50\,\mathrm{g}\div5.0\,\mathrm{cm^3}=2.5\,\mathrm{g\,cm^{-3}} (2 significant figures, because of 5.0).
Addition and subtraction: the answer has as many decimal places as the value with the fewest decimal places. Example: 12.1+0.345+3.00=15.445\to15.4\,\mathrm{g} (one decimal place, because of 12.1).
Keep extra digits in the middle of a calculation and round only the final answer.

Reporting a result with its uncertainty

Round the uncertainty to 1 (at most 2) significant figures, then round the value to the same decimal place as the uncertainty. So 2.5006\pm0.052\,\mathrm{g\,cm^{-3}} is reported as 2.50\pm0.05\,\mathrm{g\,cm^{-3}}, and the book's rectangle as 50.0\pm1.5\,\mathrm{cm^2}.

Practice with significant figures

(a) How many significant figures are in 0.04060? (b) Find the moles of \ce{NaOH} in 25.00\,\mathrm{cm^3} of 0.1000\,\mathrm{mol\,dm^{-3}} solution. (c) Round 0.078451 to 3 significant figures.

(a) Leading zeros do not count; the zero between 4 and 6 and the trailing zero after the decimal point do.
0.04060\ \Rightarrow\ 4,\,0,\,6,\,0\ \Rightarrow\ 4\ \text{significant figures}
(b) Convert the volume to \mathrm{dm^3} and multiply. Both data have 4 significant figures, so the answer has 4.
n=cV=0.1000\times0.02500=2.500\times10^{-3}\,\mathrm{mol}
(c) The first three significant figures are 7, 8, 4; the next digit is 5, so round up.
0.078451\to0.0785

16.2.2Systematic errors and experimental design and procedure

Experiments are carefully planned to reduce errors, but their design and procedure can inadvertently introduce systematic errors that consistently skew measurements away from the true value. These errors can come from equipment calibration, environmental conditions and human biases, and they reduce the accuracy and reliability of the results. The book gives two examples.

Example 1: finding a concentration with a spectrophotometer

A spectrophotometer measures how much light of one wavelength a solution absorbs. White light from the source passes through the monochromator (entrance slit, a dispersive element such as a prism or grating, and an exit slit), which lets only one wavelength through. This light passes through the sample in a small glass or quartz container called a cuvette, and the detector measures how much light gets through. The instrument displays the absorbance (optical density). A more concentrated coloured solution absorbs more light.

Main parts of a spectrophotometer (redrawn from the book's Fig 16.3).

To find the concentration of a dye such as alizarin, the absorbances of several standard solutions of known concentration are measured and plotted against concentration. This graph is the calibration curve. The absorbance of the unknown solution is then measured and its concentration is read from the curve (a worked example is in section 16.5.3). The following systematic errors can occur:

Systematic errors in spectrophotometry
Source of systematic error	What goes wrong	How to prevent it
Error in calibration	The standard solutions are prepared wrongly, or the calibration curve is inaccurate. Every result read from the curve is then wrong in the same way, so all final results are doubtful.	Prepare standards carefully with calibrated glassware; recalibrate regularly
Sample preparation	The sample is prepared the same wrong way each time, for example always pipetting slightly more stock solution when diluting. Every sample is then more concentrated than intended, so every absorbance and every calculated concentration is too high.	Use calibrated pipettes and follow one fixed procedure
Instrument performance	Over time the instrument drifts: ageing of the lamp or detector, or damage from shifting and misuse, gives consistent errors in absorbance.	Warm up the lamp; set zero with a blank and check a standard before each batch; service the instrument
Impurity (matrix) effect	Other substances in the sample (the matrix) also absorb light or interfere with the absorption, giving an error in the concentration.	Use a blank containing the same matrix, or remove the impurities first

Example 2: titrating iron(II) sulfate with acidified potassium permanganate

The concentration of the reducing agent iron(II) sulfate, \ce{FeSO4}, is found by titration with acidified \ce{KMnO4}. Potassium permanganate acts as its own indicator: the end point is the first permanent pale pink colour.

\ce{MnO4- + 5Fe^2+ + 8H+ -> Mn^2+ + 5Fe^3+ + 4H2O}

The following systematic errors can occur:

Error in volume measurements. If the student uses a burette or pipette that is not properly calibrated, or reads the meniscus inaccurately (in the same way every time), the result deviates from the true value.
Impurity in analytical reagents. If the \ce{KMnO4} or the sulfuric acid contains impurities, the measured concentration deviates from the true concentration of the sample. \ce{KMnO4} solution also decomposes slowly, so it must be standardised before use.
Effect of temperature fluctuation. The rate of the reaction (and the position of equilibrium) is greatly affected by temperature. For example, warming the mixture of sulfuric acid and iron(II) sulfate during the titration increases the rate of reaction, and the concentration measured deviates from the true value.
Incomplete redox reaction. If the reaction between the oxidising and reducing agents is not complete, the end point is wrong and so is the calculated concentration. This may be due to inhibitors in the reaction mixture or inadequate stirring (swirling).

To avoid such systematic (and random) errors in an experiment, we should revise the experimental design and procedure.

16.3Repeat trials and the elimination of errorsOften examined

Random errors can be reduced by repeat trials and measurements. Random errors are unpredictable: some readings come out too high and some too low. When several readings are averaged, the highs and lows partly cancel, so the mean is closer to the true value than a single reading is likely to be. The spread of the repeated readings also shows the uncertainty (variability) of the analysis. That is why repeated and concordant readings are taken to estimate the uncertainty along with the true value.

Concordant readings: Repeated readings that agree closely with each other, within the uncertainty of the apparatus. In titrations, titres within 0.10\,\mathrm{cm^3} of each other are usually taken as concordant.
Mean (average): The sum of the readings divided by the number of readings, \bar{x}=\dfrac{\sum x_i}{n}.

Systematic errors cannot be eliminated by repeat trials, because systematic errors are consistent. They are determinate errors: they are predictable and happen for known reasons, such as incorrect calibration of instruments, impurity in analytical reagents or temperature fluctuations. These reasons shift every measurement in the same way (either positive or negative), so the mean is shifted too.

Ruling out systematic errors

To rule out systematic errors, first identify the source of the error and then control it:

proper and correct calibration of instruments and glassware (for example, check a pipette by weighing the water it delivers);
using high-quality, purest reagents (analytical grade) and standardising solutions;
ensuring consistent conditions in the laboratory, such as a steady temperature;
correct technique: zero (tare) the balance, read the meniscus at eye level, keep burners capped;
running a blank (the same procedure without the sample) and analysing a standard sample of known composition to reveal any bias.

This practice minimises the impact of systematic errors on accuracy.

Choosing concordant titres

In a titration a student records a rough titre of 24.60\,\mathrm{cm^3} and then accurate titres of 24.20, 24.10, 24.15 and 24.30\,\mathrm{cm^3}. Which titres should be averaged, and what is the mean titre?

The rough titre is only a quick trial to locate the end point, so it is not used.
Pick the titres that agree within 0.10\,\mathrm{cm^3}. 24.20, 24.10 and 24.15 are concordant (range 0.10\,\mathrm{cm^3}). 24.30 differs from 24.10 by 0.20\,\mathrm{cm^3}, so it is left out.
Average the concordant titres.
\bar{V}=\dfrac{24.20+24.10+24.15}{3}=\dfrac{72.45}{3}=24.15\,\mathrm{cm^3}

\bar{V}=24.15\,\mathrm{cm^3}

Why repeat trials help with random but not systematic errors
	Random errors	Systematic errors
Effect of repeating and averaging	Reduced, because highs and lows cancel	No effect; every repeat has the same error
Elimination	Only reduced, never removed completely	Yes, by finding and removing the cause
Method	Repeat trials; concordant readings; mean	Calibration; pure reagents; consistent conditions; correct technique

Analysing repeated readings

Mean: \bar{x}=\dfrac{x_1+x_2+\dots+x_n}{n}, the best estimate of the true value when only random errors are present.
Deviation: d_i=x_i-\bar{x}, how far one reading is from the mean.
Average (mean) deviation: The mean of the sizes of the deviations, \dfrac{\sum|d_i|}{n}. It measures precision: the smaller it is, the more precise the readings.
Relative average deviation: The average deviation as a percentage of the mean.
Standard deviation: s=\sqrt{\dfrac{\sum (x_i-\bar{x})^2}{n-1}}, a more detailed measure of spread used in research laboratories.
Absolute error and percentage error: E=\bar{x}-x_{\text{accepted}} and \dfrac{|E|}{x_{\text{accepted}}}\times100\%. They measure accuracy.

Mean, deviation and percentage error of titres

The concordant titres from section 16.3 are 24.20, 24.10 and 24.15\,\mathrm{cm^3}. The accepted titre (found with a standard solution) is 24.00\,\mathrm{cm^3}. Find the mean, the average deviation, the relative average deviation and the percentage error.

Mean.
\bar{x}=\dfrac{24.20+24.10+24.15}{3}=24.15\,\mathrm{cm^3}
Deviation of each reading from the mean.
d=+0.05,\ -0.05,\ 0.00\,\mathrm{cm^3}
Average deviation (ignore the signs).
\dfrac{0.05+0.05+0.00}{3}=0.033\approx0.03\,\mathrm{cm^3}
Relative average deviation, a measure of precision.
\dfrac{0.033}{24.15}\times100\%=0.14\%
Error and percentage error, a measure of accuracy.
E=24.15-24.00=+0.15\,\mathrm{cm^3},\qquad \dfrac{0.15}{24.00}\times100\%=0.63\%

The readings are very precise (spread about 0.1\%), but the error (0.6\%) is several times larger than the spread. This suggests a small systematic error, for example in the calibration of the burette or in the concentration of the solution used.

\bar{x}=24.15\pm0.03\,\mathrm{cm^3},\qquad \text{percentage error}=0.63\%

16.4Graphical techniques and data of variablesOften examined

Drawing a graph is an effective way to show the relationship between two variables, that is, the effect of an independent variable on a dependent variable.

Independent variable: The variable that is manipulated or controlled by the person doing the experiment. It is plotted on the x-axis.
Dependent variable: The variable that is affected by changes in the independent variable. It is usually plotted on the y-axis.
Controlled variables: Quantities kept constant so that they do not affect the result, for example temperature and the amount of gas in a Boyle's law experiment.

For example, in the pressure-volume graph of Boyle's law, the pressure is the independent variable and the volume is the dependent variable. The pressure on a fixed amount of gas is changed at constant temperature, and the resulting volume is measured.

Boyle's law apparatus (after the book's Fig 16.5): adding weight to the piston increases the pressure, and the volume of the gas decreases at constant temperature.

Volume of 1 mol of an ideal gas at 0 °C (273 K) at different pressures
P (\mathrm{atm})	V (\mathrm{dm^3})	PV (\mathrm{dm^3\,atm})	1/P (\mathrm{atm^{-1}})
0.5	44.8	22.4	2.0
1.0	22.4	22.4	1.0
2.0	11.2	22.4	0.50
4.0	5.6	22.4	0.25

Boyle's law isotherm at 0 °C

P in atm, V in \mathrm{dm^3}. A drawn version of the book's Graph 16.4. The volume falls as the pressure rises: doubling P halves V. The curve is a hyperbola (inverse proportion).

Making a graph makes the relationship visible: we can see at a glance how the independent variable affects the dependent variable. Graphs show whether a relationship is direct, inverse or more complex.

Common relationships and their graphs
Relationship	Equation	Shape of graph	Chemistry example
Direct proportion	y=kx	Straight line through the origin	V against T (in kelvin) at constant pressure (Charles's law)
Linear	y=mx+c	Straight line with intercept c	V against temperature in ^\circ\mathrm{C} (the line meets V=0 at -273\,^\circ\mathrm{C})
Inverse proportion	y=k/x	Hyperbola (curve that falls ever more slowly)	V against P (Boyle's law)
Complex	none simple	Curve with a maximum or minimum	Density of water against temperature (maximum at 4\,^\circ\mathrm{C})

A graph can also be used to determine physical quantities. The rate of change of one variable with the other is the slope of the graph, \text{slope}=\tan\theta. The slope can represent an important physical property, such as acceleration (slope of a velocity-time graph), the rate of a reaction (slope of a concentration-time graph), or the constant in a law.

Finding Boyle's law constant from a straight-line graph

Use the data in the table to plot V against 1/P and find the constant k in PV=k.

Boyle's law, PV=k, can be written as V=k\left(\dfrac{1}{P}\right). This has the form y=mx, so a graph of V against 1/P is a straight line through the origin whose slope is k.

1/P in \mathrm{atm^{-1}}, V in \mathrm{dm^3}. Plotting V against 1/P turns the Boyle's law curve into a straight line.

Choose two points far apart on the line.
(x_1,y_1)=(0.5,\ 11.2),\qquad (x_2,y_2)=(2.0,\ 44.8)
Slope is rise divided by run; its unit is the unit of y divided by the unit of x.
k=\dfrac{44.8-11.2}{2.0-0.5}=\dfrac{33.6}{1.5}=22.4\,\mathrm{dm^3\,atm}
Check: for 1 mol of gas at 273\,\mathrm{K}, k=nRT.
nRT=1\times0.0821\times273=22.4\,\mathrm{dm^3\,atm}

k=PV=22.4\,\mathrm{dm^3\,atm}

Thus the graphical method has greater visual appeal than a table. It shows relationships clearly and helps in drawing further plots, such as turning a curve into a straight line.

16.5.1Sketched graphs: unscaled but labelled axes

A sketched graph is a visual representation whose purpose is to show qualitative trends rather than precise mathematical values. It has the following features.

Labelled axes

The variables on the x- and y-axes are labelled, but numerical values are missing: the axes are not scaled. For example, the book's Graph 16.4 (Boyle's law) is a sketched graph: its axes are labelled volume and pressure, but there are no values of pressure and volume along the axes.

Unscaled axes

Because the graph is unscaled, there are no grid lines or tick marks that correspond to specific numerical increments. The main focus of the graph is the shape and direction of the line or curve. Sketched graphs are used to show trends such as variables that are directly proportional or inversely proportional.

Sketched graphs: labelled axes, no scale. A straight line through the origin shows direct proportion; a hyperbola shows inverse proportion; plotting V against 1/P turns Boyle's law into a straight line.

Uses and advantages

Conceptual comprehension: Sketched graphs help to visualise and explain how variables behave, which aids understanding of physical phenomena.
Preliminary assessment: They are valuable in the first look at data, helping to spot possible patterns and correlations before a detailed quantitative analysis.
Educational applications: They are widely used in teaching to introduce principles and concepts without the complexity of precise numbers.
Instant visualisation: They are a quick and effective way to illustrate relationships, useful for brainstorming, presentations and rapid analyses.

16.5.2Drawn graphs with labelled and scaled axes

Drawn graphs are quantitative graphical representations of the relationship between two variables. They have labelled and scaled axes and give important quantitative information about the quantities plotted. The scale gives the interval of the quantity on each axis in a suitable unit; it decides the gaps at which values are placed along each axis, so that the data points are correctly placed relative to one another. This allows precise measurements and comparisons between data points. For example, on a temperature-density graph each mark on the x-axis may stand for one degree Celsius, so temperature gaps are shown accurately.

The density of water at different temperatures (book table)
Temperature (^\circ\mathrm{C})	Density (\mathrm{g\,cm^{-3}})
100	0.9584
50	0.9880
25	0.9970
10	0.9997
6	0.9999
4	1.0000
0 (water)	0.9998
0 (ice)	0.9167

Density of water from 0 to 10 °C

T in ^\circ\mathrm{C}. A scaled (drawn) graph, like the book's Graph 16.6. The very fine scale on the y-axis shows the maximum density at 4\,^\circ\mathrm{C} clearly.

Reading the graph: the density rises from 0\,^\circ\mathrm{C} to a maximum of 1.0000\,\mathrm{g\,cm^{-3}} at 4\,^\circ\mathrm{C} and then falls. So when water is cooled from 4\,^\circ\mathrm{C} to 0\,^\circ\mathrm{C} its density decreases. Below 4\,^\circ\mathrm{C} the molecules begin to form the open, hydrogen-bonded arrangement found in ice, so the same mass takes up more volume.

Density of water from 0 to 100 °C

T in ^\circ\mathrm{C}. The book's table drawn to scale. Liquid water becomes less dense as it is heated above 4\,^\circ\mathrm{C}; ice (open circle) is far less dense than liquid water, so it floats.

It is more convenient to read patterns of increase or decrease from a scaled graph than from a table. From the full graph we can also interpolate (read a value between plotted points): at 20\,^\circ\mathrm{C} the density is about 0.998\,\mathrm{g\,cm^{-3}}.

Drawn graphs are useful for representing quantitative measurements, where numerical data are plotted: density, distance, mass, temperature or any other measurable quantity. When data points are plotted, patterns, trends and anomalies (odd points) become visually clear, which helps in interpreting data. Drawn graphs are also an effective way to communicate results to a wide audience, including the general public, because they convey complex information in a visually accessible form.

Sketched and drawn graphs compared
Feature	Sketched graph	Drawn graph
Axes labelled	Yes	Yes, with units
Axes scaled	No	Yes
Grid lines and tick marks	None	Present, at regular numerical steps
Purpose	Qualitative trend: shape and direction	Quantitative values, slopes and intercepts
Data points	Not needed	Plotted accurately
Typical use	Concepts, preliminary assessment, teaching, quick ideas	Measurements, calculations and reporting results
Example	Shape of V against P for Boyle's law	Density of water against temperature

16.5.3How to calculate the slope of a graphOften examined

The slope (gradient) of a straight line tells us how steep it is: how much y changes for each unit change in x. The book gives four steps:

Pick any two points on the line and label them point 1 (x_1,y_1) and point 2 (x_2,y_2).
Find the vertical change (rise): subtract the y-values, y_2-y_1.
Find the horizontal change (run): subtract the x-values, x_2-x_1.
Divide the rise by the run to get the slope.

m=\tan\theta=\dfrac{\text{perpendicular}}{\text{base}}=\dfrac{y_2-y_1}{x_2-x_1}

Here m is the slope and \theta is the angle the line makes with the x-axis; the rise is the perpendicular and the run is the base of the right-angled triangle drawn on the line. The unit of the slope is the unit of y divided by the unit of x.

Positive slope: the line goes uphill (y increases as x increases).
Negative slope: the line goes downhill (y decreases as x increases).
Zero slope: the line is flat (horizontal).
Undefined slope: the line is vertical (the run is zero).

Slope of a calibration curve (spectrophotometry)

Standard solutions of a dye give these absorbances. Concentration (10^{-5}\,\mathrm{mol\,dm^{-3}}): 0, 2.0, 4.0, 6.0, 8.0. Absorbance: 0.00, 0.15, 0.30, 0.45, 0.60. Find the slope of the calibration line, and the concentration of a sample whose absorbance is 0.36.

c in 10^{-5}\,\mathrm{mol\,dm^{-3}}; A has no unit. Calibration curve: the slope is found from the dashed triangle; the dotted lines read the sample's concentration.

Pick two points far apart on the line.
(x_1,y_1)=(2.0,\ 0.15),\qquad (x_2,y_2)=(6.0,\ 0.45)
Rise and run.
\begin{aligned}y_2-y_1&=0.45-0.15=0.30\\x_2-x_1&=6.0-2.0=4.0\ (\times10^{-5}\,\mathrm{mol\,dm^{-3}})\end{aligned}
Divide rise by run.
m=\dfrac{0.30}{4.0\times10^{-5}\,\mathrm{mol\,dm^{-3}}}=7.5\times10^{3}\,\mathrm{dm^{3}\,mol^{-1}}
The line passes through the origin, so A=mc. Solve for c of the sample.
c=\dfrac{A}{m}=\dfrac{0.36}{7.5\times10^{3}}=4.8\times10^{-5}\,\mathrm{mol\,dm^{-3}}

Absorbance is directly proportional to concentration (the Beer-Lambert law, A=\varepsilon lc), which is why the calibration line is straight; its slope equals \varepsilon l, the molar absorptivity times the path length.

m=7.5\times10^{3}\,\mathrm{dm^3\,mol^{-1}},\qquad c=4.8\times10^{-5}\,\mathrm{mol\,dm^{-3}}

A negative slope: rate of a reaction

In a zero-order reaction the concentration of reactant A falls in a straight line with time: [\ce{A}]=0.70\,\mathrm{mol\,dm^{-3}} at t=10\,\mathrm{s} and 0.30\,\mathrm{mol\,dm^{-3}} at t=50\,\mathrm{s}. Find the slope and the rate of reaction.

t in s, [\mathrm{A}] in \mathrm{mol\,dm^{-3}}. The line falls, so its slope is negative.

Rise (a fall here, so it is negative).
y_2-y_1=0.30-0.70=-0.40\,\mathrm{mol\,dm^{-3}}
Run.
x_2-x_1=50-10=40\,\mathrm{s}
Slope; the negative sign means the line goes downhill.
m=\dfrac{-0.40}{40}=-0.010\,\mathrm{mol\,dm^{-3}\,s^{-1}}
The rate of reaction is the decrease in concentration per second, so it is the size of the slope.
\text{rate}=-m=0.010\,\mathrm{mol\,dm^{-3}\,s^{-1}}

m=-0.010\,\mathrm{mol\,dm^{-3}\,s^{-1}},\qquad \text{rate}=0.010\,\mathrm{mol\,dm^{-3}\,s^{-1}}

Formula sheet

Error

E=x_{\text{measured}}-x_{\text{accepted}}

x_{\text{measured}}: value obtained in the experiment; x_{\text{accepted}}: true or literature value

same unit as x

Percentage error

\text{percentage error}=\dfrac{|x_{\text{measured}}-x_{\text{accepted}}|}{x_{\text{accepted}}}\times100\%

measures accuracy

Unit: \%

Percentage uncertainty

\dfrac{\Delta x}{x}\times100\%

\Delta x: absolute uncertainty of the measurement x

Unit: \%

Mean

\bar{x}=\dfrac{\sum x_i}{n}

x_i: individual readings; n: number of readings

same unit as x

Average deviation

\bar{d}=\dfrac{\sum |x_i-\bar{x}|}{n}

measures precision; relative average deviation =\bar{d}/\bar{x}\times100\%

same unit as x

Standard deviation

s=\sqrt{\dfrac{\sum (x_i-\bar{x})^2}{n-1}}

a detailed measure of the spread of repeated readings

same unit as x

Uncertainty of a sum or difference

Q=a\pm b\ \Rightarrow\ \Delta Q=\Delta a+\Delta b

absolute uncertainties add, even for a difference

same unit as Q

Uncertainty in the area of a rectangle

\Delta A=W\,\Delta L+L\,\Delta W

L,W: length and width; \Delta L,\Delta W: their uncertainties

Units: \mathrm{cm^2} or \mathrm{m^2}

Uncertainty of a product or quotient

Q=ab\ \text{or}\ \dfrac{a}{b}\ \Rightarrow\ \dfrac{\Delta Q}{Q}=\dfrac{\Delta a}{a}+\dfrac{\Delta b}{b}

percentage (relative) uncertainties add

no unit (ratio) or \%

Uncertainty of a power

Q=a^{n}\ \Rightarrow\ \dfrac{\Delta Q}{Q}=n\,\dfrac{\Delta a}{a}

n: the power

no unit (ratio) or \%

Slope of a straight line

m=\tan\theta=\dfrac{y_2-y_1}{x_2-x_1}

(x_1,y_1), (x_2,y_2): two points on the line; \theta: angle with the x-axis

unit of y divided by unit of x

Boyle's law as a straight line

PV=k\ \Rightarrow\ V=k\left(\dfrac{1}{P}\right)

slope of V against 1/P is k (at constant T and amount of gas)

Unit: k in \mathrm{dm^3\,atm}

Calibration line (Beer-Lambert law)

A=\varepsilon l c

A: absorbance; \varepsilon: molar absorptivity; l: path length; c: concentration; slope of A against c is \varepsilon l

A has no unit; \varepsilon l in \mathrm{dm^3\,mol^{-1}}

Key terms

Analytical chemistry: The branch of chemistry mainly related to the chemical characterisation of materials: what is present and how much.
Qualitative analysis: Analysis that identifies which substances, ions or groups are present, for example a precipitation or flame test.
Quantitative analysis: Analysis that determines how much of one or more constituents is present, for example titration, gravimetry or spectrophotometry.
Qualitative data: Non-numerical information obtained from observation, not measurement, such as colour, odour, state or type of reaction.
Quantitative data: Numerical information obtained from measurements with instruments; always associated with random error or uncertainty.
Uncertainty: The range within which the true value of a measurement is expected to lie, written as \pm a value.
Error: The difference between the value obtained in an experiment and the accepted (true or literature) value.
Random error: The random difference between the observed value and the true value; it varies unpredictably in either direction (indeterminate error).
Systematic error: A consistent, repeatable error due to faulty equipment, flawed design or procedure, or human mistakes; always in the same direction (determinate error).
Blunder: An error caused by carelessness, typically an accident such as spilling a sample; the reading is discarded and repeated.
Determinate error: Another name for a systematic error: it has a known, predictable cause.
Indeterminate error: Another name for a random error: its cause in any single reading cannot be found.
Parallax error: An error in reading a scale caused by viewing it from an angle instead of at eye level.
Tare (zeroing): Setting a balance to read zero before weighing, to remove a zero error.
Calibration: Checking or adjusting an instrument or glassware against standards of known value.
Calibration curve: A graph of an instrument's response (such as absorbance) against known concentrations of standards, used to find unknown concentrations.
Matrix (impurity) effect: Interference by other substances in a sample that changes the measured signal, giving an error in concentration.
Drift: A slow change in an instrument's readings over time, for example due to ageing components.
Spectrophotometer: An instrument that measures the absorbance of light of one selected wavelength by a solution.
Monochromator: The part of a spectrophotometer (entrance slit, dispersive element and exit slit) that selects a single wavelength.
Propagation of errors: The carrying forward and accumulation of the uncertainties of measured quantities into a calculated result.
Absolute uncertainty: The uncertainty written in the same unit as the quantity, for example \pm0.05\,\mathrm{cm^3}.
Percentage uncertainty: The absolute uncertainty divided by the measured value, times 100.
Accuracy: How close a measured value or mean is to the true or accepted value.
Precision: How close repeated measurements are to each other.
Concordant readings: Repeated readings that agree closely, for example titres within 0.10\,\mathrm{cm^3}.
Mean: The sum of a set of readings divided by the number of readings.
Significant figures: The reliably known digits of a measurement plus the first uncertain (estimated) digit.
Independent variable: The variable manipulated or controlled by the experimenter; plotted on the x-axis.
Dependent variable: The variable that changes in response to the independent variable; plotted on the y-axis.
Controlled variable: A quantity kept constant during an experiment so it does not affect the result.
Sketched graph: A graph with labelled but unscaled axes that shows a qualitative trend (shape and direction).
Drawn graph: A graph with labelled and scaled axes on which data are plotted for quantitative measurement.
Scale: The numerical interval represented by each division on a graph axis.
Slope (gradient): The rise divided by the run between two points on a line, m=\tan\theta=(y_2-y_1)/(x_2-x_1).
Isotherm: A graph of pressure and volume of a gas at one constant temperature, such as the Boyle's law curve at 0\,^\circ\mathrm{C}.
Interpolation: Reading a value from a graph between plotted points.
Anomalous result: A reading that does not fit the pattern of the others; it is checked and usually repeated.

Multiple choice questionsp. 263–264

Which of the following is the example of qualitative analysis?

AVolumetry
BSpectrophotometry
CGravimetry
DPrecipitationAnswer

Why

A precipitation test tells which ion is present (for example, silver nitrate gives a white precipitate with chloride ions), so it is qualitative. Volumetry, spectrophotometry and gravimetry all measure how much is present, so they are quantitative. Gravimetry does use a precipitate, but it weighs it to find an amount.

During titration the burette is not correctly calibrated. The error that will generate due to this is categorised as:

ARandom error
BSystematic errorAnswer
CIndeterminate error
DRough error

Why

A wrongly calibrated burette gives the same error in every reading, always in the same direction, so it is a systematic (determinate) error. An indeterminate error (C) is another name for a random error.

iii

Which error can be reduced or eliminated by taking repeat trials?

ARandom errorsAnswer
BSystematic error
CDeterminate error
DRough error

Why

Random errors scatter readings above and below the true value, so averaging repeat trials reduces them. Systematic (determinate) errors shift every reading the same way and are not removed by repetition.

Which one of the following gives more information?

AInformation on x-axis
BInformation on y-axis
CInformation in un-scaled graph
DInformation in scaled graphAnswer

Why

A scaled (drawn) graph has labelled and scaled axes, so exact values, slopes and intercepts can be read from it. An un-scaled (sketched) graph shows only the shape of a trend, and one axis alone gives no relationship.

Note. The book prints option (d) as 'Information in scales graph'; 'scaled graph' is meant.

The density of water ______ from 4\,^\circ\mathrm{C} to 0\,^\circ\mathrm{C}.

AIncreases
BDecreasesAnswer
CRemains constant
DFluctuates irregularly

Why

Water is densest at 4\,^\circ\mathrm{C} (1.0000\,\mathrm{g\,cm^{-3}}). On cooling to 0\,^\circ\mathrm{C} its density falls to 0.9998\,\mathrm{g\,cm^{-3}} (book table), because the molecules begin to form the open, hydrogen-bonded arrangement of ice.

Short answer questionsp. 264

i3 marks

Explain qualitative and quantitative analysis with examples.

Solution

Qualitative analysis finds which substances, elements, ions or functional groups are present in a sample. It gives qualitative data: non-numerical information from observation, such as a colour change or a precipitate. Examples: a flame test (sodium gives a golden-yellow flame); adding silver nitrate solution to a solution gives a white precipitate if chloride ions are present.

Quantitative analysis determines how much of one or more constituents is present. It gives quantitative data: numbers measured with instruments, which always carry some uncertainty. Examples: titration (volumetric analysis) to find the concentration of an acid, gravimetric analysis, spectrophotometry, and finding the percentage of nitrogen in a fertilizer.

ii3 marks

What are random errors, write few examples.

Solution

A random error is the random difference between the observed value and the true value. It varies unpredictably from one reading to the next, making the result sometimes too high and sometimes too low. Random errors arise from difficulty in reading a scale and from changes in the environment, and they are reduced by repeating the measurement and taking the mean.

Examples:

estimating the last digit of a burette reading slightly differently each time;
judging the colour change at the end point of a titration slightly differently in each trial;
a fluctuating balance reading caused by air currents or vibration;
temperature fluctuations in the laboratory, and the observer's reaction time in starting and stopping a stopwatch.

iii3 marks

How does the error propagate? Give an example.

Solution

When a result is calculated from measured quantities, the random error (uncertainty) of each measurement is carried into the result and they accumulate, so the final result has a larger uncertainty than any single measurement. For sums and differences the absolute uncertainties add; for products and quotients the percentage uncertainties add.

Example: a rectangle has L=10.0\pm0.1\,\mathrm{cm} and W=5.0\pm0.1\,\mathrm{cm}, so A=LW=50\,\mathrm{cm^2}.

\Delta A=W\,\Delta L+L\,\Delta W=(5.0)(0.1)+(10.0)(0.1)=1.5\,\mathrm{cm^2}

So A=50.0\pm1.5\,\mathrm{cm^2}: the uncertainties of 1\% in L and 2\% in W have combined into 3\% in the area.

iv3 marks

Explain with one example each of dependent variable and independent variable.

Solution

The independent variable is the variable that the experimenter manipulates or controls; it is plotted on the x-axis. The dependent variable is the variable that changes as a result; it is measured and plotted on the y-axis.

Example 1 (Boyle's law): the pressure on a fixed amount of gas is changed at constant temperature and the volume is measured. Pressure is the independent variable and volume is the dependent variable.

Example 2 (rate of reaction): the temperature of a reaction mixture is set at different values and the rate is measured. Temperature is the independent variable and the rate of reaction is the dependent variable.

Long answer questionsp. 264

i7 marks

Explain errors and their types in detail.

Solution

Definition of error

An error is the difference between the value obtained in an experiment and the accepted (true or literature) value. Errors decide how reliable a result is.

\begin{aligned}\text{error}&=x_{\text{measured}}-x_{\text{accepted}}\\\text{percentage error}&=\dfrac{|x_{\text{measured}}-x_{\text{accepted}}|}{x_{\text{accepted}}}\times100\%\end{aligned}

There are two main types of error, random and systematic, and careless mistakes called blunders.

Types of error and their sub-types.

1. Random errors

A random error is the random difference between the observed value and the true value. Its size and sign change unpredictably from reading to reading, so results deviate in either direction (too high or too low). They are also called indeterminate errors.

Causes: difficulty in reading an instrument or scale (estimating between marks, judging an end-point colour), and changes in the environment such as temperature fluctuations and air currents.
Sub-types: environmental (unpredictable changes in the surroundings) and observational (variation in the observer's judgement).
Examples: slightly different burette readings of the same volume; a fluctuating balance reading in a draught.
Effect: they reduce precision (readings scatter).
Reduction: repeat the measurement and take the mean of concordant readings.

2. Systematic errors

A systematic error is a consistent, repeatable error due to faulty equipment, a flawed design or procedure, or human mistakes made the same way each time. It always shifts results in the same direction. Systematic errors are also called determinate errors because their cause can be found.

Instrumental: a faulty or uncalibrated instrument, for example a balance that is not zeroed (tare), so every mass is too large.
Observational: the observer reads wrongly in the same way each time, for example a parallax error from reading a burette from above.
Environmental: surroundings that affect every measurement the same way, for example a warm laboratory.
Theoretical (methodological): a flawed procedure, for example leaving a spirit burner uncapped in calorimetry so alcohol evaporates, or using \ce{HCl} instead of \ce{H2SO4} in a permanganate titration.
Effect: they reduce accuracy; results can be precise but wrong.
Removal: repeating does not help; find the cause and remove it by calibration, pure reagents, correct technique and constant conditions.

3. Blunders

Blunders (gross errors) are caused by carelessness, such as spilling liquid before it is measured or writing a reading wrongly. The result is discarded and the measurement is repeated.

Comparison

Random error	Systematic error
Unpredictable, either direction	Consistent, same direction
Indeterminate	Determinate
Reduces precision	Reduces accuracy
Reduced by repeat trials and averaging	Not reduced by repeats; removed by eliminating the cause

Conclusion: random errors can be reduced but never removed completely, while systematic errors can be eliminated once their source is identified. A good experiment controls both.

ii6 marks

What are few methods to eliminate errors in analysis.

Solution

Introduction

Errors in chemical analysis cannot all be removed, but they can be minimised. Random errors are reduced by repetition and better technique, while systematic errors are eliminated by finding and removing their cause. Blunders are avoided by care.

1. Methods to reduce random errors

Repeat trials: take several readings and use the mean of the concordant readings (titres within 0.10\,\mathrm{cm^3}); highs and lows cancel.
More precise instruments: use an analytical balance (\pm0.0001\,\mathrm{g}) instead of a top-pan balance, or a burette instead of a measuring cylinder.
Larger quantities: a larger mass or titre gives a smaller percentage uncertainty.
Controlled surroundings: avoid draughts and vibration near the balance; keep the temperature steady.
Good technique: read the bottom of the meniscus at eye level against a white card; judge the end point against the same background each time.

2. Methods to eliminate systematic errors

Calibration of instruments and glassware against standards (for example, weigh the water delivered by a pipette), and recalibration of instruments such as spectrophotometers.
Zeroing: tare the balance before weighing; set the zero of a spectrophotometer with a blank.
Pure reagents: use analytical-grade chemicals and standardise solutions such as \ce{KMnO4} before use.
Blank determination: carry out the procedure without the sample and subtract the blank reading.
Analysis of a standard sample of known composition to reveal any bias in the method.
Independent method: check the result by a different technique; agreement shows no systematic error.
Correct experimental design and procedure: for example, use dilute \ce{H2SO4} (not \ce{HCl}) in permanganate titrations, keep the spirit burner capped in calorimetry, and stir so that the reaction is complete.

3. Avoiding blunders

Work carefully, record readings immediately in ink, check calculations, label solutions, and repeat any reading that looks wrong.

Summary of methods
Type of error	Main method
Random	Repeat trials and average; precise instruments; controlled conditions
Systematic	Calibration; pure, standardised reagents; blanks and standards; correct design
Blunder	Care, checking, and repeating the measurement

Conclusion: by combining repetition (for random errors) with calibration, pure reagents and a sound procedure (for systematic errors), a chemist obtains results that are both precise and accurate.

Exam focus

Explain errors and their types (random, systematic, blunders) with examples. (long)
The main long question of the unit; learn the definitions, causes, direction and removal of each type.
Go to answer
Methods to eliminate or reduce errors in analysis. (long)
The second long question; separate the methods for random and for systematic errors.
Go to answer
Why do repeat trials reduce random errors but not systematic errors? (short)
A direct SLO of section 16.3 and the basis of MCQs (ii) and (iii).
Go to answer
Explain qualitative and quantitative analysis (or data) with examples. (short)
Short question (i) and MCQ (i); remember that precipitation tests are qualitative.
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Define random error and give examples. (short)
Short question (ii); give at least three laboratory examples.
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How do errors propagate? Calculate the area of a rectangle with its uncertainty. (numerical)
Short question (iii); the book's example 50.0\pm1.5\,\mathrm{cm^2} must be learned with the correct formula \Delta A=W\Delta L+L\Delta W.
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Distinguish accuracy from precision and link them to systematic and random errors. (short)
Needed to explain every error question; a common conceptual question.
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Define dependent and independent variables with examples. (short)
Short question (iv); Boyle's law is the book's example.
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Compare sketched and drawn graphs. (short)
Two SLOs of the unit; learn the comparison table and MCQ (iv).
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Calculate the slope of a straight-line graph and give its unit. (numerical)
Section 16.5.3 lists the steps; slope problems link to rates and calibration curves.
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Explain the systematic errors in a spectrophotometric or permanganate titration experiment. (short)
The two examples of section 16.2.2 show how design and procedure cause systematic errors.
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