Tool 3: Mathematics — IB Chemistry SL Revision Notes & Topical Questions

Section 1 of 44Arithmetic in chemistry

1Decimals

• Decimals most calculations, e.g. concentrations.

2Fractions

• Fractions mainly in uncertainty calculations, and calculator fractions must be converted to decimals.

3Percentages

• Percentages percentage yield, atom economy, change, difference, error and uncertainty.

4Ratios

• Ratios mole ratios when the stoichiometry is not 1 : 1, e.g. H₂SO₄ + 2NaOH → Na₂SO₄ + 2H₂O.

5Reciprocals

• 1/V in gas laws and 1/T in rate work.

6Logarithms

• Logarithms pH and Arrhenius calculations.

7Data booklet

• Section 1 lists equations but is not exhaustive, e.g. percentage yield is not given.

8Final answers

• Final answers write as numbers, not fractions.

Section 2 of 44Mean and range

Mean

Mean the total of the values divided by the number of values, e.g. chlorine-35 and chlorine-37 in a 3 : 1 ratio give 35.5.

Chemistry uses

• Chemistry uses relative atomic mass from isotope abundances, and average titres.

Average titre

• Average titre exclude the rough titre and any non-concordant (anomalous) results.

Range

Range highest value minus lowest value ("Hi − Lo"), applied only to numerical data and affected by anomalous values.

Section 3 of 44Scientific notation and orders of magnitude

Form

Form a × 10ⁿ with 1 ≤ a < 10, n positive for large numbers and negative for small numbers.

Rounding

• Rounding only the value of a is rounded, so 4.37 × 10⁶ to 2 significant figures is 4.4 × 10⁶.

Order of magnitude

• Order of magnitude the power of ten, rounded up if a is large, e.g. 6 × 10⁸ has order 10⁹.

Comparison

• Comparison one order of magnitude is a factor of about ten, two is about 100.

Approximation

• Approximation a close but not exact value used when a quick answer is enough, such as taking a strong acid pH as about 1.0.

Estimation

• Estimation a reasoned guess when the true value is unknown or cannot be measured.

Weak acid assumption

• [H⁺] = [A⁻] and [HA] is taken as the initial concentration, so K_a = [H⁺]² ÷ [HA], but the full expression should be given first.

Section 4 of 44Significant figures

1Non-zero digits

• Non-zero digits always significant.

2Zeros between non-zero digits

• Zeros between non-zero digits significant, e.g. 4107 has 4 and 29.009 has 5.

3Leading zeros

• Leading zeros not significant, e.g. 0.00079 has 2.

4Trailing zeros

• Trailing zeros not significant in a whole number (57 000 has 2) but significant after a decimal point (689.0023 has 7).

1

Identify the significant figures.

2

Count to the specified number.

3

Look at the next digit as the rounding decider.

4

Round up if the decider is 5 or more.

Calculation rule

• Calculation rule the answer should not have more significant figures than the least precise value,
  • and addition and subtraction use the fewest decimal places.

Intermediate values

• Intermediate values keep the calculator value until the final answer and avoid rounding to 1 significant figure midway.

Constants

• Constants the number of significant figures of constants like N_A (6.02 × 10²³) does not limit the answer.

Section 5 of 44Units, prefixes and symbols

• SI base units
  • metre (m), kilogram (kg), second (s), kelvin (K), mole (mol), ampere (A) and candela (cd).

Prefix	Symbol	Power of ten
kilo	k	103
centi	c	10−2
milli	m	10−3
micro	µ	10−6
nano	n	10−9
pico	p	10−12

Common derived units

• Common derived units joule, pascal, coulomb, kJ mol⁻¹, J K⁻¹, volt and mol dm⁻³.

Symbols

• Symbols state symbols, chemical symbols, constants and terms in equations must be known,
  • and the same letter can mean different things, e.g. c for speed of light, specific heat capacity or concentration.

Conversion

• 184 pm = 184 × 10⁻¹² m = 1.84 × 10⁻¹⁰ m.

Section 6 of 44Uncertainty and its propagation

• Uncertainty a range around a measurement within which the true value is expected to lie, which is not the same as an error.

Source	Uncertainty
Analogue reading	± half the smallest division
Digital reading	± the last significant digit
Repeated data	± half the range

1Absolute uncertainty

• Absolute uncertainty the actual amount of uncertainty, e.g. ±0.05 cm³.

2Percentage uncertainty

(absolute uncertainty ÷ measured value) × 100.

3Addition and subtraction

• Addition and subtraction add the absolute uncertainties, so a difference of two readings has double the single uncertainty.

4Multiplication and division

• Multiplication and division add the percentage uncertainties and convert back to absolute.

5Reducing it

• Reducing it use apparatus with a smaller uncertainty or measure a larger quantity.

6Powers (HL)

• Powers (HL) multiply the percentage uncertainty by the exponent.

Section 7 of 44Percentage change, difference and error

Percentage change

(final − initial) ÷ initial × 100.

Percentage difference

(value 1 − value 2) ÷ (average of the two) × 100, ignoring the sign.

Percentage error

(accepted − experimental) ÷ accepted × 100, comparing a result with a literature value.

Distinction

• Distinction percentage error is about accuracy against a literature value, while percentage uncertainty is about the precision of a measurement.

Section 8 of 44Graphs

Graphs

Required types

• Required types bar charts, histograms, scatter graphs and line or curve graphs.

Features

• Features a title, labelled axes with units, an appropriate linear scale using at least half the grid,
  • and clear data points, plotted as small crosses in pencil.

Axes

• Axes the independent variable on the x-axis and the dependent variable on the y-axis.

Line of best fit

• Line of best fit can be straight or curved, shows the trend, passes through most points and has points evenly on either side.

Interpolation and extrapolation

• Interpolation and extrapolation interpolation reads within the data and extrapolation extends the line beyond it,
  • as in a temperature correction graph.

Sketch graphs

• Sketch graphs show trends without scales but must have labelled axes.

Uncertainty bars

• Uncertainty bars drawn above and below a point to show the absolute uncertainty.

Section 9 of 44Reading features of a graph

Reading features of a graph

Gradient

• Gradient for straight lines draw a triangle and use Δy/Δx, and for curves draw a tangent with a ruler using a large triangle.

Straight line equation

• Straight line equation y = mx + c, where m is the gradient and c the y-intercept.

Changing gradient

• Changing gradient a curve has changing gradients, as in the progress of a reaction rate.

Intercepts

• Intercepts where the line crosses an axis, e.g. free energy against temperature.

Maxima and minima

• Maxima and minima in energy profiles, maxima are transition states and minima are reactants and products.

Areas

• Areas such as under a Maxwell-Boltzmann curve, where the area to the right of E_a is the particles able to react.

Logarithmic forms

• Logarithmic forms pH = −log₁₀[H⁺],
  • and ln k against 1/T is linear with gradient −E_a/R (HL).
    What the symbols mean: pH has no unit · [H⁺] = concentration of hydrogen ions (mol dm⁻³); each 1 pH unit is a ten-fold change in [H⁺]

Section 10 of 44More from the specification

1

Direct proportion

• Direct proportion y = kx, so doubling one variable doubles the other;
  • a graph is a straight line through the origin.

2

Inverse proportion

• Inverse proportion y = k/x, so as one quantity increases the other decreases;
  • plotting y against 1/x gives a straight line.

3

Scientific notation

• Numbers are written as a x 10 to a power, which is helpful for very large or small quantities like Avogadro's constant.

4

Processing uncertainties

• Add absolute uncertainties when adding or subtracting, and add percentage uncertainties when multiplying or dividing.

5

Applying general mathematics

• Use relationships such as gradient and intercept to interpret data and present scientific information with correct significant figures.

6

Scientific notation

• A number written as a × 10ⁿ with 1 ≤ a < 10.

7

Directly proportional

• A relationship whose graph is a straight line through the origin.

8

Inversely proportional

• Relationship y = k / x giving a curve.

Section 11 of 44Percentage change and percentage difference

Percentage change

(final value − initial value) ÷ initial value × 100.

Percentage difference

(value 1 − value 2) ÷ (average of value 1 and value 2) × 100, ignoring the minus sign.

Purpose

• Purpose compare experimental results, reaction yields and other data.

Spread

• Spread percentage difference compares the spread of two results rather than the accuracy of one.

Section 12 of 44Percentage error

Percentage error

(accepted value − experimental value) ÷ accepted value × 100.

Denominator

• Denominator divide by the literature value, not the experimental value.

Meaning

• Meaning shows how accurate a result is against an accepted value.

Commenting

• Commenting be able to discuss differences between experimental and literature values.

Clarity

• Clarity state which value is the literature value so the sign of the error is meaningful.

Section 13 of 44Percentage uncertainty

Percentage uncertainty

(absolute uncertainty ÷ measured value) × 100.

Purpose

• Purpose compares the significance of an absolute uncertainty on a measurement.

Not error

• Not error it must not be confused with percentage error, which compares a result with a literature value.

Absolute uncertainty

• ± half a division for analogue and ± the last digit for digital readings.

Reducing it

• Reducing it use apparatus with a smaller uncertainty, or increase the measured value by using a larger mass or volume.

Section 14 of 44Combining uncertainties

1Addition and subtraction

• Addition and subtraction add the absolute uncertainties.

2Two readings

• Two readings a temperature change from two readings has double the uncertainty of one reading.

3Multiplication and division

• Multiplication and division add the percentage uncertainties and convert the total back to an absolute value.

4Dominant term

• Dominant term in q = mcΔT the temperature change usually has the largest percentage uncertainty.

5Improvement

• Improvement raise ΔT or use a more precise thermometer rather than a more precise balance.

6Powers (HL)

• Powers (HL) multiply the percentage uncertainty by the power.

Section 15 of 44Coefficient of determination

Symbol

• R², a measure of how well a trend line or curve fits the data.

R2

0 — the dependent variable cannot be predicted from the independent variable.

R2 between 0 and 1

• R2 between 0 and 1 the closer to 1, the better the fit.

R2

1 — a perfect fit, though it does not guarantee that the model is a good description of the relationship.

Section 16 of 44Describing trends in graphs and tables

Term	Meaning
Linear	A straight-line relationship
Proportional	Changes by the same factor
Increasing or decreasing	Direction of change
Plateau	Levels off to a constant value
Exponential	Rate of change depends on the current value

• Examples
  • temperature of an exothermic reaction against time, and volume of gas from the thermal decomposition of calcium carbonate.

1

Identify the pattern, e.g. rate rises as temperature increases.

2

Quote two data values with units.

3

Suggest an explanation based on chemistry.

4

Calculate a percentage change if asked about the size of the effect.

• Checks
  • use the same units throughout and note any anomalous value that breaks the trend.

Section 17 of 44Using formulae in an answer

Layout

• Layout write each formula, substitute values with units and give the result to an appropriate number of significant figures.

Precision

• Precision report percentages to the same level of precision as the data.

Section 18 of 44More from the specification

More from the specification

Scientific notation and significant figures

• In chemical calculations, values are expressed in scientific notation (A x 10^n,
• where 1 <= A < 10),
• and precision is governed by non-zero digits and significant figures derived from experimental measurements.

Graphical analysis and gradient determination

• When experimental data is plotted on linear axes (such as concentration versus time or ln(k) versus 1/T), the gradient of the line provides quantitative physical information such as rate or activation energy.

Logarithmic functions in chemical equations

• Logarithms are common in chemical relationships,
• such as the pH definition and Arrhenius equation (ln(k) = -Ea / RT + ln(A)), linearizing exponential curves for straightforward slope evaluation.
  • (pH = -log10[H+])
    What the symbols mean: pH has no unit · [H⁺] = concentration of hydrogen ions (mol dm⁻³); each 1 pH unit is a ten-fold change in [H⁺]

Propagation of uncertainties

• Combining quantitative values in stoichiometry and thermochemistry requires adding absolute uncertainties for sums/differences,
• and percentage uncertainties for multiplication, division,
• or powers.

Concentration

• Moles ÷ volume in dm³, or g/dm³.

Gradient

• The change in y divided by the change in x on a graph.

Scientific notation

• A number written as a × 10ⁿ with 1 ≤ a < 10.

Inversely proportional

• Relationship y = k / x giving a curve.

Sketch graph

• A labelled graph showing a qualitative relationship without scales.

Section 19 of 44Direct and inverse proportion

1Directly proportional

Directly proportional doubling one quantity doubles the other, y = kx, giving a straight line through the origin with a fixed gradient.

2Linear but not directly proportional

• Linear but not directly proportional a straight line that does not pass through the origin.

3Example

• Example the volume of a gas at constant pressure is directly proportional to its temperature in kelvin, but not in °C.

4Inversely proportional

Inversely proportional y = k/x, so doubling x halves y, giving a curve that is not a straight line and does not pass through the origin.

5Example

• Example the pressure and volume of a gas at constant temperature.

6Correlation

• Correlation positive means the gradient slopes upward, negative means it slopes downward.

Section 20 of 44Qualitative and quantitative data

Qualitative

• Qualitative described in words, e.g. a darker blue copper sulfate solution is more concentrated.

Quantitative

• Quantitative uses numbers, either discrete or continuous (measurements such as temperature or gas volume).
  • (countable values such as stoichiometric coefficients or energy levels)

Section 21 of 44Rates of change from tabulated data

Average rate of change

Average rate of change change in the dependent variable ÷ change in the independent variable.

Graph method

• Graph method the gradient of the plotted graph gives the rate.

Precision

• Precision choose points far apart on a straight section.

Unit conversion

• Unit conversion divide a rate in min⁻¹ by 60 to get s⁻¹.

Section 22 of 44SI units and prefixes

Quantity	SI unit	Symbol
Mass	Kilogram	kg
Amount of substance	Mole	mol
Temperature	Kelvin	K
Energy	Joule	J
Pressure	Pascal	Pa
Electrical charge	Coulomb	C
Enthalpy	Kilojoules per mole	kJ mol−1

Prefixes

• Prefixes nano (n) 10⁻⁹, pico (p) 10⁻¹², micro (µ) 10⁻⁶, milli (m) 10⁻³ and kilo (k) 10³.

Conversions

• 5.2 kg = 5.2 × 10³ g, and 184 pm = 184 × 10⁻¹² m = 1.84 × 10⁻¹⁰ m.

Units in equations

• Units in equations specific heat capacity in Q = mcΔT is J g⁻¹ K⁻¹, concentration in n = cV is mol dm⁻³,
  • and the subscript c in ΔH_c denotes combustion.

Section 23 of 44Significant figures

Significant figures

Identify the significant figures, remembering that zeros between non-zero digits count.

Count to the specified number; the next digit is the deciding digit.

Round up if the deciding digit is 5 or more, and down if it is less.

1Example of rounding

• 1.0478 to 3 significant figures is 1.05, because the deciding digit is 7.

2Multiplication and division

• Multiplication and division keep the same number of significant figures as the least precise value.

3Addition and subtraction

• Addition and subtraction keep the same number of decimal places as the least precise value.

4Leading and trailing zeros

• Leading and trailing zeros leading zeros are not significant and trailing zeros after a decimal point are.

Section 24 of 44Handling units in equations

Carrying units

• Carrying units write units in each line of a calculation and check them at the end.

q

q mcΔT — with m in g and c in J g⁻¹ K⁻¹, the energy is in J,

• and dividing by 1000 gives kJ mol⁻¹ for ΔH.

  What the symbols mean: E, Q or q = energy transferred (J) · m = mass · c = specific heat capacity (J kg⁻¹ °C⁻¹; water 4.18 J g⁻¹ °C⁻¹) · ΔT or Δθ = temperature change (°C or K)
  • (kg; g in chemistry calorimetry)

n

n cV — the volume must be in dm³.

What the symbols mean: c = concentration (mol dm⁻³) · n = amount (mol) · V = volume (dm³; divide cm³ by 1000)

pV

pV nRT — SI units are required: Pa, m³ and K.

What the symbols mean: p = pressure (Pa) · V = volume (m³) · n = amount of gas (mol) · R = 8.31 J mol⁻¹ K⁻¹ · T = temperature (K = °C + 273)

Ratios

• Ratios units may cancel, so no unit is needed.

Temperature

• Temperature kelvin is Celsius plus 273 (or 273.15), and a temperature difference is the same in K and °C.

Section 25 of 44More from the specification

More from the specification

Absolute vs. Percentage Uncertainty

• In chemical measurements, each reading carries an absolute uncertainty determined by instrument precision; the percentage uncertainty is calculated as × 100%.
  • (absolute uncertainty / measured value)

Propagating Uncertainties in Experimental Calculations

• When adding or subtracting measurements, absolute uncertainties are summed;
• when multiplying or dividing quantities in stoichiometry or kinetics, percentage uncertainties must be added together to find the overall error.

Uncertainty Bars on Experimental Graphs

• When plotting kinetics or equilibrium data on Cartesian graphs, experimental error is represented using uncertainty error bars;
  • trend lines or best-fit curves must pass through all uncertainty ranges.

Minimising Percentage Error in Laboratory Trials

• Percentage uncertainty increases when measured values are very small;
  • using higher masses of reactants or larger titration volumes effectively reduces relative experimental error.

Percentage error

• The difference between experimental and literature value as a percentage of the literature value.

Scientific notation

• A number written as a × 10ⁿ with 1 ≤ a < 10.

Percentage yield

• Actual yield divided by theoretical yield, times 100.

Sketch graph

• A labelled graph showing a qualitative relationship without scales.

Uncertainty

• The range within which the true value is expected to lie.

Percentage difference

• The difference between two values over their mean × 100.

Percentage uncertainty

• The absolute uncertainty divided by the measured value, multiplied by 100.

Percentage change

• ÷ initial × 100.
  • (final − initial)

Absolute uncertainty

• The actual amount by which a measurement is uncertain.

Section 26 of 44The mean average

Mean

• Mean the total of all the values divided by the number of values.

Anomalous results

• Anomalous results these must be left out of a mean because they distort it.
• A result is anomalous when it does not fit the pattern of the other repeats.
• The reason for leaving it out must be stated.

Weighted mean

• Weighted mean when quantities occur in a ratio, each value is counted according to its proportion.

Section 27 of 44Where means are used in chemistry

Relative atomic mass

• Relative atomic mass calculated as the mean mass of the isotopes weighted by their abundance.
• Isotopes present in a 3 : 1 ratio are counted as four atoms in the calculation.

Average titre

• Average titre calculated only from concordant titres.
• The rough titre is never included.
• Titres that are not concordant are treated as anomalous and excluded.

Reliability

• Reliability three concordant results make the average titre highly reliable.

Section 28 of 44Range of data

1Range

• Range the difference between the highest and lowest values, so it measures the spread of the data.

2Applies only to numerical data

• Applies only to numerical data it can be quoted as a span of values or as one number (the difference).
  • (for example 9.2 – 8.4)

3Memory aid

• Memory aid range is "hi minus lo".

4Weakness

• Weakness the range is changed if the highest or lowest value is itself anomalous.

Section 29 of 44Approximation and estimation

Approximation

• Approximation a value close to the true value but not exact, used when a precise calculation is unnecessary or a quick answer is enough.
• A strong acid is approximated as pH 1.0 even though its pH may range between 1.0 and 3.0.

Estimation

• Estimation a reasoned guess based on available data, used when the true value is unknown or cannot be measured directly.
• Percentage yield = actual yield ÷ theoretical yield × 100.

Ignoring small effects

• Ignoring small effects a small contribution may be neglected when it does not change the answer within the precision required.

Section 30 of 44Logarithms and pH

Definition

• Definition pH = −log₁₀[H⁺].
  What the symbols mean: pH has no unit · [H⁺] = concentration of hydrogen ions (mol dm⁻³); each 1 pH unit is a ten-fold change in [H⁺]

Inverse

• [H⁺] = 10^−pH, found with the 10^x key of a calculator.

Tenfold change

• Tenfold change a tenfold change in [H⁺] changes pH by 1.

Same pattern

• Same pattern pOH, pK_a and pK_w use the same log relationship.

Decimal places

• Decimal places the number of decimal places in a pH equals the number of significant figures in the concentration.

Logarithms are also used in Arrhenius calculations, and exponential functions appear in the Arrhenius equation (HL).

Section 31 of 44Assumptions in K_a calculations

Assumptions in K_a calculations

Weak acid dissociation

• HA ⇌ H⁺ + A⁻, with K_a = [H⁺][A⁻] / [HA].

All H⁺ comes from the acid

• All H⁺ comes from the acid water contributes a negligible number of ions, so [H⁺] = [A⁻].

Small dissociation

• [HA] at equilibrium is taken as the initial concentration of the acid.
• The error from this assumption is larger for acids with a larger K_a.

Simplified form

• K_a = [H⁺]² / [HA], so [H⁺] = √(K_a × [HA]).

Full expression

• Full expression always write the full K_a expression first when asked for it.

1

Write the dissociation equation and the K_a expression.

2

State the assumptions.

3

Rearrange to find [H⁺].

4

Calculate pH = −log[H⁺].

Section 32 of 44Rearranging formulae and checking answers

1Rearrange first

• Rearrange first rearrange the formula before substituting numbers.
• n = cV gives V = n / c and c = n / V.

2Check for sense

• Check for sense a weak acid solution has a higher pH than a strong acid of the same concentration, and a yield cannot exceed 100 %.

3Fractions

• Fractions give final answers as decimal numbers, not as fractions left by the calculator.

4Show steps

• Show steps each step shown can earn method marks even if the final answer is wrong.

Section 33 of 44Presenting and judging data

1Data tables

• Data tables put the quantity and unit in the column heading, not in each cell.

2Raw data

• Raw data record to the precision of the instrument.

3Uncertainty

• Uncertainty give it to the same number of decimal places as the measurement.

4Reliability

• Reliability judge it from the range of repeat results compared with the stated uncertainty.

5Validity

• Validity judge whether the method measured what it was intended to and whether variables were controlled.

6Precision versus accuracy

• Precision versus accuracy a precise set of results can still be inaccurate if a systematic error is present.

Section 34 of 44More from the specification

More from the specification

Plotting scientific graphs

• In scientific data analysis, the independent variable is plotted on the horizontal x-axis and the dependent variable on the vertical y-axis, with full axis labels including SI units.

Rate determination from gradient

• The instantaneous rate of a chemical reaction is evaluated by drawing a tangent to the concentration-time curve on a graph and calculating its gradient.
  • (gradient = delta_y / delta_x)

Absolute temperature on the Kelvin scale

• In kinetic and thermodynamic equations, temperature must be expressed in absolute temperature units,
  • (Kelvin: T = theta + 273.15)
• where absolute zero (0 K) represents zero thermal motion.

Common graphical transformations

• Common chemical relationships involve curve linearisation;
  • for instance, plotting rate against 1/T or taking natural logarithms transforms exponential curves into straight lines for straightforward parameter extraction.

Plotting scientific graphs

• In scientific data analysis, the independent variable is plotted on the horizontal x-axis and the dependent variable on the vertical y-axis, with full axis labels including SI units.

Rate determination from gradient

• The instantaneous rate of a chemical reaction is evaluated by drawing a tangent to the concentration-time curve on a graph and calculating its gradient.
  • (gradient = delta_y / delta_x)

Absolute temperature on the Kelvin scale

• In kinetic and thermodynamic equations, temperature must be expressed in absolute temperature units,
  • (Kelvin: T = theta + 273.15)
• where absolute zero (0 K) represents zero thermal motion.

Common graphical transformations

• Common chemical relationships involve curve linearisation;
  • for instance, plotting rate against 1/T or taking natural logarithms transforms exponential curves into straight lines for straightforward parameter extraction.

Dependent variable

• The variable that is measured.

Independent variable

• The variable that is changed.

Absolute uncertainty

• The actual amount by which a measurement is uncertain.

Scientific notation

• A number written as a × 10ⁿ with 1 ≤ a < 10.

Sketch graph

• A labelled graph showing a qualitative relationship without scales.

Gradient

• The change in y divided by the change in x on a graph.

Temperature

• A measure of the average kinetic energy of the particles.

Section 35 of 44Sketch graphs and proportionality

Sketch graph

• Sketch graph shows a qualitative trend with labelled axes but no scale or data points.

Directly proportional

• Directly proportional a straight line through the origin with fixed gradient, y = kx.
• If x doubles, y doubles.
• A straight line that does not pass through the origin is not directly proportional.

Inversely proportional

• Inversely proportional y = k / x, a curve that is not straight and does not pass through the origin.
• If x doubles, y halves.

Constant

• Constant a horizontal line means y does not change as x increases.

Positive correlation

• Positive correlation the gradient is positive, so y increases as x increases.

Negative correlation

• Negative correlation the gradient is negative, so y decreases as x increases.

Section 36 of 44Drawing a graph

Drawing a graph

Types

• Types bar charts, histograms, scatter graphs and line or curve graphs.

Axes

• Axes the independent variable goes on the x-axis and the dependent variable on the y-axis, each labelled with quantity and unit.

Title

• Title a clear title describing the relationship.

Scale

• Scale linear, without jumps, and plotted so that the data occupy at least half of the grid.

Data points

• Data points shown clearly, usually as small crosses, drawn in pencil.
• If the scale could be doubled and the points still fit, it is not appropriate.

Origin

• Origin decide whether (0,0) is a true data point, because it is the most accurate anchor if it is.

Section 37 of 44Best-fit lines, interpolation and extrapolation

Line of best fit

• Line of best fit shows the trend and may be a straight line or a curve.
• It need not pass through every point but passes through the majority.
• Scattered points lie evenly on both sides of it.
• It is smooth, not dot-to-dot.

Interpolation

• Interpolation using the best-fit line between plotted points.

Extrapolation

• Extrapolation extending the best-fit line beyond the plotted points.
• A temperature correction graph in calorimetry extrapolates two best-fit lines back to the time of mixing to find the maximum temperature rise.
• Extrapolated values are less reliable because the trend may not continue.

Anomalous points

• Anomalous points circle them and leave them out when drawing the line.

Section 38 of 44Gradients and tangents

Straight line

• Straight line y = mx + c, where m is the gradient and c the y-intercept.

Gradient

• Gradient m = Δy / Δx, with units.

Large triangle

• Large triangle the triangle drawn on the line should be as large as possible to reduce precision errors.

Curve

• Curve the gradient at a point is found from a tangent drawn with a ruler lined up against the curve at that point.

Changing gradient

• Changing gradient a curve has different gradients at different points,
  • which show the progress of a rate or the effect of a factor such as concentration.

Unit conversion

• Unit conversion rates may need converting, for example dividing by 60 changes min⁻¹ to s⁻¹.

1

Line up a ruler so the straight edge touches the curve at one point.

2

Draw a long straight tangent.

3

Build a large right-angled triangle on the tangent.

4

Calculate Δy / Δx and quote the unit.

Section 39 of 44Intercepts, maxima, minima and areas

Intercept

• Intercept a point where a line or curve crosses an axis.
• On a graph of free energy against temperature, the x-intercept is the temperature at which the reaction stops being feasible.
• The y-intercept gives the value when x is zero, such as the sign of the enthalpy term.

Maximum

• Maximum the highest point, where the gradient goes from positive to zero to negative.

Minimum

• Minimum the lowest point, where the gradient goes from negative to zero to positive.

Energy profile

• Energy profile maxima represent transition states and minima represent reactants and products.

Area under a curve

• Area under a curve on a Maxwell–Boltzmann curve it represents the total number of particles.
• The E_a line divides it into particles with and without the minimum energy to react.

Section 40 of 44Uncertainty bars and R²

Uncertainty bar

• Uncertainty bar a bar drawn above and below a point to show the absolute uncertainty of that value.
  • (or each side of it)

Direction

• Direction usually vertical for y-values but can be horizontal for x-values.

Size

• Size a larger bar indicates a larger uncertainty and more random error.

Good fit

• Good fit the best-fit line passes through the bars of all the points.

Gradient uncertainty

• Gradient uncertainty the steepest and shallowest lines that pass through the bars show the uncertainty in the gradient.

R2

• R2 the coefficient of determination measures how well a trend line fits, from 0 to 1 (a perfect fit).
  • (no prediction possible)

Limit of R2

• Limit of R2 a value close to 1 does not by itself prove the model is a good description of the relationship.

Section 41 of 44Linearising a curve

Purpose

• Purpose plotting against a transformed quantity turns a curve into a straight line so the gradient gives a constant.

Examples

• Examples graphs against 1/T or against the logarithm of a quantity.

Reciprocals

• Reciprocals appear in gas laws as 1/V and in rate graphs as 1/time.

Section 42 of 44More from the specification

1Significant figures in analytical calculations

• Significant figures in analytical calculations rounding numerical data to appropriate significant figures prevents overstating measurement precision in final reported quantities.

2Percentage yield and measurement uncertainty

• Percentage yield and measurement uncertainty determining experimental percentage yield requires comparing actual product mass against theoretical yield derived from stoichiometric calculations.

3Average calculations from replicate measurements

• Average calculations from replicate measurements calculating the arithmetic mean of concordant experimental trials minimizes random error across repetitive laboratory measurements.

4Significant figures and precision in calculations

• Experimental data must be quoted to consistent significant figures reflecting measurement precision, propagating lowest significant figures in multiplication and division.

5Calculating percentage uncertainties

• Percentage uncertainty is calculated as * 100, combining additively when physical quantities are multiplied.
  • (absolute uncertainty / measured quantity)

6Linear regression and gradient determination

• Line-of-best-fit gradients on experimental Cartesian plots quantify rate constants or physical constants via linear regression: y = mx + c.

7Scientific notation

• A number written as a × 10ⁿ with 1 ≤ a < 10.

8Percentage yield

• Actual yield divided by theoretical yield, times 100.

9Symbol

• Shorthand for an element, e.g. Na.

10Significant figures

• The digits in a value that are reliable and carry meaning.

11Percentage difference

• The difference between two values over their mean × 100.

12Percentage uncertainty

• The absolute uncertainty divided by the measured value, multiplied by 100.

Section 43 of 44Key terms

Term	Meaning
Significant figures	The digits in a value that are reliable and carry meaning.
Absolute uncertainty	The actual amount by which a measurement is uncertain.
Percentage uncertainty	The absolute uncertainty as a percentage of the measured value.
Scientific notation	A number written as a × 10n with 1 ≤ a < 10.
Interpolation	Reading values from a line of best fit within the plotted data.
Extrapolation	Extending a line of best fit beyond the plotted data.

Term	Meaning
Percentage error	The difference between experimental and accepted value as a percentage of the accepted value.
Percentage uncertainty	The absolute uncertainty as a percentage of the measurement.
Percentage difference	The difference between two values over their mean × 100.
Literature value	An accepted value from a reference source.
Coefficient of determination	R2, a measure of the fit of a trend line to data.

Term	Meaning
Directly proportional	A relationship whose graph is a straight line through the origin.
Gradient	The change in y divided by the change in x.
SI unit	A unit from the International System of Units.
Significant figures	The digits in a value that carry meaning.
Prefix	A letter denoting a power of ten.

Term	Meaning
Mean	Sum of the values divided by the number of values.
Range	Difference between the highest and lowest values.
Anomalous result	A result that does not fit the pattern of the other repeats.
Concordant titres	Titres that agree closely, usually within 0.10 cm3.
Logarithm	The power to which 10 is raised to give a number.
Ka	Acid dissociation constant of a weak acid.
Approximation	A value close to but not exactly the true value.
Estimation	A reasoned guess based on available data.

Term	Meaning
Interpolation	Reading values between plotted data points.
Extrapolation	Extending a best-fit line beyond the data.
Tangent	A straight line touching a curve at one point; its gradient is the gradient at that point.
Intercept	The point where a line or curve crosses an axis.
Uncertainty bar	A bar on a data point showing the absolute uncertainty.
Directly proportional	Linear relationship through the origin, y = kx.
Inversely proportional	Relationship y = k / x giving a curve.
Coefficient of determination	R2, a measure of how well a line fits the data.

Section 44 of 44Exam technique

1

Rounding in the middle of a multi-step calculation — keep full calculator values until the end.

2

Leaving the rough titre or an anomalous result in the mean — exclude them.

3

Confusing percentage error with percentage uncertainty — one compares to a literature value, the other compares an uncertainty to a measurement.

4

Using the same uncertainty for a difference as for a single reading — add the uncertainties of both readings.

1

Confusing percentage error with percentage uncertainty — one compares with a literature value and the other with a measurement.

2

Dividing by the experimental value instead of the literature value in percentage error.

3

Adding percentage uncertainties for sums and differences — add absolute uncertainties for those.

4

Forgetting units for the uncertainty — quote it with the same unit as the measurement.

1

Calling a graph with an intercept directly proportional — it must pass through the origin.

2

Rounding in the middle of a multi-step calculation — round only at the end.

3

Mixing prefixes such as cm³ and dm³ — convert before calculating.

4

Reporting too many significant figures from a calculator — match the least precise data.

1

Averaging all titres including the outlier — only concordant titres are averaged, and the rough titre is never used.

2

Using the initial acid concentration in place of [H⁺] in the pH calculation — [H⁺] comes from K_a.

3

Giving pH to the wrong number of decimal places — match it to the significant figures of the concentration.

4

Omitting the stated assumptions in a K_a calculation — both assumptions must be named.

5

Leaving answers as fractions — write them as decimal numbers.

1

Joining the points dot-to-dot instead of drawing a best-fit line or curve.

2

Describing any upward diagonal line as directly proportional — it must pass through the origin.

3

Drawing too small a triangle for a gradient, or leaving out the units of the gradient.

4

Treating extrapolated data as certain — the trend may not continue.