DP IB Chemistry: HL

DP IB Chemistry: HL Your notes Tool 3: Mathematics Contents Applying General Mathematics in Chemistry Using Units, Symbols & Numerical Values in Chemistry Processing Uncertainties in Chemistry Graphing in Chemistry © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 1 Applying General Mathematics in Chemistry Applying general mathematics in Chemistry Your notes Using arithmetic and algebraic calculations in chemistry Chemistry often requires the use of calculations, which can include: Decimals Most chemical calculations use decimals, e.g. the concentration of a chemical Fractions These are most commonly used in uncertainty calculations Most scientific calculators will initially give answers as fractions Make sure you know where the S⇔D button is so that you convert the fraction into a decimal Percentages There are many percentage calculations, including percentage yield, percentage atom economy, percentage change, percentage difference, percentage error and percentage uncertainty Ratios These are commonly used in moles calculations where the stoichiometry of the balanced chemical equation is not 1 : 1, e.g. H2SO4 + 2NaOH → Na2SO4 + 2H2O Reciprocals These are most obvious in gas laws, using 1 / V, and concentration versus rate graphs, using 1 / T For more information about gas laws, see our revision notes in the Ideal Gases topic For more information about concentration and rates, see our revision notes in the How Fast? The Rate of Chemical Change topic Logarithmic functions These are most obvious in pH and Arrhenius calculations For more information about pH, see our revision note on The pH Scale For more information about Arrhenius, see our revision note on The Arrhenius Equation Exponential functions (Additional Higher level) © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 2 This is most obvious in Arrhenius calculations Section 1 of the Data Booklet contains a list of the mathematical equations that you are expected to be able to manipulate and work with Your notes Careful: This is not an exhaustive list - there are other equations that you will be expected to know that are not given in Section 1 of the Data Booklet, e.g. percentage yield Examiner Tips and Tricks Make sure your final answers are written as proper numbers, not left as fractions Leaving them as fractions will typically lose you a mark What is the mean average? The mean average is often just called the “average” It is the total of all the values divided by the number of values, i.e. add all the numbers together and divide by how many there are For example, two common isotopes of chlorine are chlorine-35 and chlorine-37, which exist in a 3 : 1 ratio The information shows that there are three chlorine-35 atoms for every one chlorine-37 atom Overall, there is a total of four atoms to be considered in the calculation So, the mean average = 35 + 35 + 35 + 37 = 35.5 4 Problems with the mean average occur when there are anomalous results These should not be included in a mean average calculation What mean average calculations are in chemistry? Two main calculations in chemistry that involve mean average calculations are: Calculating relative atomic mass from isotopic abundance data For more information, see our revision note on Isotopes Calculating average titres as part of a titration calculation These calculations will typically have values that should not be considered because they are not concordant Worked Example Calculate the average titre for the following experimental results. © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 3 Rough titre / cm3 Titre 1 / cm3 Titre 2 / cm3 Titre 3 / cm3 Titre 4 / cm3 15.50 14.90 15.15 14.95 14.95 Your notes Answer: The three values that are used for the mean average calculation are: 14.90 14.95 14.95 The rough titre should never be used in the mean average calculation Any results that are not concordant are considered to be outlying / anomalous results and should not be included in the mean average calculation So, the average titre = . + 14 . 95 + 14 . 95 = 14.93 cm3 14 90 3 Range The range can only be applied to numerical data It is a measure of how spread out the data is, which means that it is the difference between the highest and lowest values You can remember this as "Hi - Lo" It can be expressed as: A range of values e.g. 9.2 - 8.4 A single value e.g. 9.2 - 8.4 = 0.8 The range can be affected when the highest and / or lowest data are anomalous results themselves What is scientific notation? Scientific notation is also known as standard form It is a system of writing and working with very large or very small numbers For example, Avogadro's number 602 000 000 000 000 000 000 000 is 6.02 x 1023 in scientific notation Numbers in scientific notation are written as: a × 10n They follow these rules: a is a number above 1 and below 10 © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 4 For large numbers, n is an integer that is greater than 0 i.e It shows how many times a is multiplied by 10 Your notes For small numbers, n is an integer that is less than 0 i.e It shows how many times a is divided by 10 n < 0 for small numbers i.e how many times a is divided by 10 Applying scientific notation to numbers The scientific notation for numbers greater than 1 has the x 10 raised to a positive power while the scientific notation for numbers less than 1 has the x 10 raised to a negative power When rounding a number in standard form to a certain number of significant figures, only the value of a is rounded (the × 10n value will not be significant) For example, 4.37 × 106 to 2 significant figures is 4.4 × 106 Orders of magnitude When a number is expressed to an order of 10, this is an order of magnitude Example: If a number is described as 3 × 108 then that number is actually 3 × 100 000 000 The order of magnitude of 3 × 108 is just 108 Orders of magnitude follow rules for rounding The order of magnitude of 6 × 108 is 109 as the magnitude is rounded up A quantity is one order of magnitude larger than another quantity if it is about ten times larger Similarly, two orders of magnitude would be 100 times larger, or 102 In chemistry, orders of magnitude can be very large or very small Approximation and estimation Approximation and estimation are both methods used to obtain values that are close to the true or accurate values While they share some similarities, they have distinct characteristics and are used in different contexts © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 5 An approximation is a value that is close to the actual value but not exact. Used when: Your notes A precise calculation is unnecessary A quick answer is sufficient The exact value is hard or time-consuming to obtain Approximations are typically easy and fast to work out For example, the pH of a strong acid is generally between pH 1.0 and 3.0 A useful approximation is to say pH = 1.0. This is not exact, but it is widely accepted and practical for many uses Estimationis making a reasoned guess based on available data. Used when the true value is unknown or can't be directly measured. For example, estimating percentage yield Start with a known amount of reactant. Calculate the theoretical mass of product. Perform the reaction and measure actual product mass Percentage yield = actual yield × 100 theoretical yield Appreciate when some effects can be ignored and why this is useful During calculations using acid and base dissociation constants the assumption made is that the value of [H+] = [HA] Or the concentration of hydrogen ions is the same as the concentration of the acid This is because The weak acid has a low degree of dissociation so you can assume the concentration of the acid is the same value at equilibrium and use it in the expression for Ka CH3COOH ⇌ CH3COO– + H+ A point to remember is that Ka values are different depending on the acid, so the degree of dissociation also varies. Therefore the error using this assumption for some acids will be larger than others All the H+ ions are assumed to come from the acid as water will contribute a very small number of H+ ions So the Ka expression: © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 6 Ka = ⎡ ⎢ ⎣ H + A− HA ⎤ ⎡ ⎥ ⎢ ⎦ ⎣ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ Your notes can be simplified to ⎡ ⎢ ⎣ Ka = H+ 2 HA ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ Remember to give the full expression in an exam when asked Percentage change and percentage difference Percentage change and percentage difference are commonly used to express the relative change between two values They are useful for comparing experimental results, determining reaction yields and analysing other chemical data Percentage change Percentage change is used to express the relative change between an initial value and a final value It is calculated using the following formula: Percentage Change = ( Final Value − Initial Value × 100 Initial value ) Worked Example During the course of a chemical reaction, the initial of chemical species A increases from 0.05 mol dm–3 to 0.08 mol dm–3. Calculate the percentage change in concentration. Answer: − Final Value Initial Value Initial value 0 08 0 05 100 = 60% Percentage change = 0 05 Percentage change = ( ( . − . . ) ) × 100 × So, the concentration of the chemical species increased by 60% during the reaction Percentage difference Percentage difference is used to compare two values to determine how much they differ from each other as a percentage It is calculated using the following formula: © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 7 Percentage Difference = value 1 − value 2 × 100 average of value 1 and value 2 ( ( ) ) Your notes Worked Example The melting points of different samples B and C are measured: B = 75°C C = 81°C Calculate the percentage difference in melting points. Answer: Percentage difference = ( Percentage difference = ( − + 75 81 75 81 2 ( ( Percentage difference = − value 1 value 2 average of value 1 and value 2 ( ) × 100 ) ) × 100 ) ) −6 × 100 = –7.69% 78 When calculating percentage difference, you can ignore the minus sign in front of the calculation So, the melting points of B and C differ by 7.69% Percentage Error Percentage error is used to express the difference between a final calculated answer and an accepted or literature value It is calculated using the following formula Percentage error = accepted value − experimental value × 100 accepted value You should be able to comment on any differences between the experimental and literature values Worked Example Experimental results showed the enthalpy of combustion of propan-1-ol to be –1.5 x 103 kJ mol–1. The literature value for this enthalpy change is -2021 kJ mol-1. Calculate the percentage error. © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 8 Answer: − accepted value experimental value accepted value 2021 1500 100 = 26 % Percentage error = 2021 Percentage error = − × 100 Your notes × Percentage uncertainty Percentage uncertainties are a way to compare the significance of an absolute uncertainty on a measurement This is not to be confused with percentage error, which is a comparison of a result to a literature value It is calculated using the following formula Percentage uncertainty = absolute uncertainty × 100 measured value Diagram with examples of percentage uncertainty calculations for common laboratory apparatus The absolute uncertainty for analogue measurements is ± half a division and for digital measurements is ± the last significant division For more information about uncertainties, see our revision note on Processing Uncertainties in Chemistry © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 9 Your notes Examiner Tips and Tricks Percentage uncertainty can be reduced by: Using equipment with a smaller uncertainty Increasing the measured value, e.g. using a sample with greater mass or volume Mathematical skills linked to graphs and tables There are several specification points linked to graphs and tables For more information about working specifically with graphs, see our revision note on Graphing in Chemistry These include, but are not limited to, being able to: Distinguish between qualitative and quantitative data, incorporating continuous and discrete variables Understand direct and inverse proportionality, as well as positive and negative correlations between variables Determine rates of change from tabulated data Qualitative and quantitative data Qualitative data usually describes something in words, not numbers For example: Copper sulfate solution is blue A more dilute copper sulfate solution is pale blue, while a more concentrated copper sulfate solution is a darker blue Quantitative data uses numbers to count / measure something For example: The neutralisation of 25.0 cm3 sodium hydroxide by 25.0 cm3 hydrochloric acid increases the temperature of the system by 2.5 oC Discrete data is quantitative It consists of separate, distinct and countable values For example: The stoichiometric coefficients representing the relative number of molecules or atoms involved in the reaction are discrete values and must be integers Electrons can only occupy certain discrete energy levels, e.g. 1s, 2s, etc Continuous data is also quantitative © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 10 It is based on measurements and can include decimal numbers or fractions This allows for an infinite number of values Your notes For example: The temperature of an exothermic reaction as time progresses The volume of gas produced during the thermal decomposition of calcium carbonate Direct and inverse proportionality There are a number of terms that are commonly applied to trends, particularly in graphs Directly proportional This applies to a trend that has a clearly linear relationship Mathematically, this can be described as y = kx, where k can be positive or negative In most situations, it is clear that k is positive This means that the relationship can be described as "when one variable increases, the other increases" or "if x doubles, then y doubles" A directly proportional relationship is always a straight line through the origin with a fixed gradient Inversely proportional This can be described as y = k , where k can be positive or negative x This means that the relationship can be described as "when one variable increases, the other decreases" or "if x doubles, then y halves" When plotted, inverse proportionality is not a straight line and does not pass through the origin Positive correlation This term is best applied to the gradient of a graph The gradient of the graph is positive / slopes or curves upwards It describes a relationship where as x increases, y also increases Negative correlation This term is, also, best applied to the gradient of a graph The gradient of the graph is negative / slopes or curves downwards It describes a relationship where as x increases, y decreases Examiner Tips and Tricks © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 11 Careful: A common mistake made by students is to describe any graph with a straight line going diagonally upwards as directly proportional This is not correct because direct proportionality must go through the origin A graph that does not go through the origin can correctly be described as proportional, but it is not directly proportional Your notes Determine rates of change from tabulated data To determine rates of change from tabulated data, you can use the average rate of change or gradient, if the data has been plotted as a graph The average rate of change between two points on a graph or in a table is: Rate of change = the change in the dependent variable y − axis the change in the independent variable x − axis ( ) ( ) Worked Example An experiment is run to measure the amount of chemical Y produced as time progresses: Time / seconds 10 20 30 40 50 Amount of Y / cm3 3.0 7.0 10.0 14.0 18.0 Calculate the rate of change for this reaction between 10 and 30 seconds. Answer: Rate of change = ( ( Rate of change = − − the change in the dependent variable y axis the change in the independent variable x axis ( . − . − ) ) 10 0 3 0 = 0.35 cm3 s–1 30 10 ( ) ) So, on average, the amount of Y increases by 3.5 cm3 every 10 seconds over the interval from t = 10 to t = 30 seconds © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 12 Using Units, Symbols & Numerical Values in Chemistry Your notes Using units, symbols & numerical values in Chemistry International system of units (SI) In science, there are 7 base SI units which are used to measure various physical quantities SI base units table Quantity SI base unit Symbol length metre m mass kilogram kg time second s temperature Kelvin K amount of substance mole mol current Ampere A luminous intensity candela cd These base SI units form the foundation for measuring various properties and quantities in chemistry and other sciences Other common units can be derived from these base units for specific applications, but they are all based on the International System of Units (SI) Concentration, c [mol dm–3] Joules, J [N m] Molar mass, Mr [ g mol–1] Pascals, Pa [kg / m s2] Table of common units in chemistry Quantity Unit Abbreviation energy joule J © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 13 pressure pascal Pa electrical charge coulomb C enthalpy kilojoules per mole kJ mol–1 entropy joules per Kelvin J K–1 potential difference volts V concentration moles per cubic decimetre mol dm–3 Your notes Prefixes Measurements of physical quantities can require very large and very small values, for example: The diameter of an atom is about 10–10 m or 0.0000000001 m One mole of a substance contains 6.02 × 1023 or 602 000 000 000 000 000 000 000 particles Powers of ten are numbers that can be achieved by multiplying 10 times itself These come under two categories of units: Multiples e.g. 102, 103 Sub-multiples e.g. 10-1, 10-2 Each power of ten is defined by a prefix, the most common ones used in chemistry are listed in the table below The complete list of prefixes can be found in Section 3 of the data booklet Table of common prefixes in chemistry Prefix Abbreviation Power of ten kilo- k 103 centi- c 10–2 milli- m 10–3 micro- μ 10–6 nano- n 10–9 © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 14 pico- 10–12 p Your notes Example conversions A mass of 5.2 kg 5.2 kg = 5.2 kilograms = 5.2 x 103 = 5200 grams The diameter of an aluminium atom is 184 pm 184 pm = 184 picometres = 184 x 10–12 m Correctly given in standard form, this is a value of 1.84 x 10–10 m The energy required to heat 10 dm3 of liquid water at constant pressure from 0 °C to 100 °C is approximately 4.2 MJ 4.2 MJ = 4.2 megajoules = 4.2 x 106 = 4 200 000 J Symbols in chemistry There is a large number of symbols used in chemistry: State symbols e.g. solid (s), liquid (l), gas (g) Chemical symbols - from the Periodic Table, Section 7 of the data booklet e.g. lithium = Li, carbon = C, Copper = Cu Physical constants - given in Section 2 of the data booklet e.g. Planck's constant = h, the speed of light in a vacuum = c Terms in equations - relevant equations are given in Section 1 of the data booklet e.g. n = cv, where n is the number of moles, c is the concentration and v is the volume Units for quantities e.g. specific heat capacity measured in J g–1 K–1 Other abbreviations used in chemistry e.g. STP for standard temperature and pressure While the data booklet can assist with using the correct symbols, it is essential to know the symbols specifically linked to physical constants, terms in equations and units for quantities For example, the letter c, depending on capitalisation (c or C), could represent: The speed of light in a vacuum in the c = f λ equation The specific heat capacity in the Q = mcΔT equation © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 15 Concentration in the n = cv equation The units of electrical charge Your notes The prefix centi-, e.g. cm3 The centigrade / Celsius units of temperature The chemical symbol for carbon The symbol for combustion in the enthalpy of combustion term ΔH c term (if subscripts are included) What are significant figures? Significant figures must be used when dealing with quantitative data Significant figures are the digits in a number that are reliable and absolutely necessary to indicate the quantity of that number There are some important rules to remember for significant figures All non-zero digits are significant Zeros between non-zero digits are significant 4107 (4.s.f.) 29.009 (5.s.f) Zeros that come before all non-zero digits are not significant 0.00079 (2.s.f.) 0.48 (2.s.f.) Zeros after non-zero digits within a number without decimals are not significant 57,000 (2.s.f) 640 (2.s.f) Zeros after non-zero digits within a number with decimals are significant 689.0023 (7.s.f) When rounding to a certain number of significant figures: Identify the significant figures within the number using the rules above Count from the first significant figure to the specified number Use the next number as the ‘rounder decider’ If the decider is 5 or greater, increase the previous value by 1 The same approach can be applied to decimal places, although significant figures are more common © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 16 Worked Example Your notes Write 1.0478 to 3 significant figures. Answer: 1. Identify the significant figures They are all significant figures 2. Count to the specified number The question says to 3 significant figures, so the fourth digit is the 'rounder decider' 1.0478 3. Round up or down 1.05 Examiner Tips and Tricks Exam questions sometimes state: To give an answer to a certain number of significant figures, commonly 3 To give an answer to an appropriate number of significant figures Make sure you keep an eye out for this as it can be an easy and frustrating mark to lose after all your hard work in the calculation An appropriate number of significant figures The appropriate number of significant figures depends on: The precision of the measurement The limitations of the equipment used to make the measurement When performing calculations involving measured values, it's essential to maintain the proper number of significant figures throughout the calculation to avoid rounding errors An easy way to avoid rounding errors is to continue using the calculator value until the final answer Tip: Avoid rounding any calculation to 1 significant figure during a calculation as this typically introduces rounding errors and can sometimes, automatically, lose you a mark In the final result, the number of significant figures should not exceed the value with the least number of significant figures used in the calculation Worked Example © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 17 Calculate the number of moles in 35.75 cm3 of a 0.015 mol dm–3 solution of HCl. Give your answer to an appropriate number of significant figures. Answer: Convert 35.75 cm3 to dm3 Your notes . 35 75 = 0.03575 dm3 1000 Moles = concentration x volume Moles = 0.015 x 0.03575 = 5.3625 x 10–4 The volume is given to 4 significant figures The concentration is given to 2 significant figures Therefore, the appropriate number of significant figures is 2 So, the final answer is 5.4 x 10–4 moles Examiner Tips and Tricks For numbers such as the Avogadro constant and Gas constant, the number of significant figures is not limited by measurement precision but rather by the definition of the constant itself In these cases, use the defined number of significant figures provided for that constant e.g. Avogadro = 6.02 × 1023 mol−1 has 3 significant figures © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 18 Processing Uncertainties in Chemistry Processing uncertainties in chemistry Your notes What is uncertainty? Uncertainty is quantitative indication of the quality of the result It is the difference between the actual reading taken (caused by the equipment or techniques used) and the true value It is a range of values around a measurement within which the true value is expected to lie and is an estimate Uncertainties are not the same as errors Errors arise from equipment or practical techniques that cause a reading to be different from the true value Uncertainties in measurements are recorded as a range (±) to an appropriate level of precision Table showing different uncertainties Uncertainty in a reading ± half the smallest division in a measurement at least ±1 smallest division in repeated data half the range i.e. ± ½ (largest - smallest value) in digital readings ± the last significant digit (unless otherwise quoted) Types of uncertainty Uncertainty is grouped into three main types: Absolute uncertainty The actual amount by which the quantity is uncertain e.g.if v = 5.0 ± 0.1 cm, the absolute uncertainty in v is 0.1 cm Fractional uncertainty The absolute uncertainty divided by the quantity itself © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 19 e.g.if v = 5.0 ± 0.1 cm, the fractional uncertainty in v is Percentage uncertainty . . 0 1 1 = 5 0 50 Your notes The ratio of the expanded uncertainty to the measured quantity on a scale relative to 100% This is calculated using the following formula: Percentage uncertainty = uncertainty measured value × 100 How to calculate absolute, fractional and percentage uncertainty The key pieces of information from this burette reading are the smallest division and the reading The uncertainties in this reading are: Absolute Uncertainty = . 0 1 = 0.05 cm3 2 Reading = 19.6 ± 0.05 cm3 Fractional Uncertainty = . uncertainty 0 1 1 = = cm3 value 19 6 196 . Percentage © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 20 Uncertainty = uncertainty value × 100 = 190 . .16 × 100 = 0.5% Reading = 19.6 ± 0.5% cm3 Your notes Propagating uncertainties in processed data Uncertainty propagates in different ways depending on the type of calculation involved Adding or subtracting measurements When you are adding or subtracting two measurements then you add together the absolute measurement uncertainties For example: Using a balance to measure the initial and final mass of a container Using a thermometer to measure the temperature at the start and end Using a burette to measure the initial and final readings In all of these cases, the instrument is read twice to obtain the final value (e.g. change in mass, temperature, or volume) If the measurement is uncertain by a fixed amount each time, then the total uncertainty is potentially doubled For example: If each reading has an uncertainty of ±0.1 g, the total uncertainty becomes ±0.2 g Multiplying or dividing measurements When you multiply or divide experimental measurements then you add together the percentage uncertainties You can then calculate the absolute uncertainty from the sum of the percentage uncertainties Exponential measurements (HL only) When experimental measurements are raised to a power, you multiply the fractional or percentage uncertainty by the power The coefficient of determination, R2 The coefficient of determination is a measure of fit that can be applied to lines and curves on graphs The coefficient of determination is written as R2 It is used to evaluate the fit of a trend line / curve: R2 = 0 The dependent variable cannot be predicted from the independent variable. © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 21 R² is usually greater than or equal to zero R2 between 0 and 1 The dependent variable can be predicted from the independent variable, although the degree of success depends on the value of R2 Your notes The closer to 1, the better the fit of the trend line / curve R2 = 1 The dependent variable can be predicted from the independent variable The trend line / curve is perfect Note: This does not guarantee that the trend line / curve is a good model for the relationship between the dependent and independent variables © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 22 Graphing in Chemistry Your notes Graphing in Chemistry Sketch graphs Sketch graphs are a way to represent qualitative trends where the variables shown are often proportional or inversely proportional Sketch graphs do not have scales or data points, but they must have labels as these examples from the Gas Laws show: Sketch graphs of Boyle's Law Sketched graphs show relationships between variables Graphical relationships In the first sketch graph above you can see that the relationship is a straight line going through the origin This means as you double one variable the other variable also doubles so we say the independent variable is directly proportional to the dependent variable The second sketched graph shows a shallow curve which is the characteristic shape when two variables have an inversely proportional relationship The third sketched graph shows a straight horizontal line, meaning as the independent variable (x-axis) increases the dependent variable does not change or is constant Worked Example Which graph shows the correct relationship between the number of moles of a gas, n, and the temperature, T, at constant pressure and volume? © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 23 Your notes Answer: The correct option is D The ideal gas equation is PV = nRT If P, V and R are constant then PV / R = nT, which is a constant n must be inversely proportional to T, which gives graph D General guidance on drawing graphs The types of graphs that students are expected to be able to draw include: Bar charts Histograms Scatter graphs Line / curve graphs Graphs need to have: Clear title Labelled axes Units on the axes Appropriate linear scales without any jumps This means the plotted graph must occupy at least half or more of the sheet or grid © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 24 A rough rule of thumb is that if you can double the scale and still fit all the points on, then your scale is not appropriate Clearly shown data points Your notes The most common convention is to use small crosses to show the data points Graph of concentration versus time Graphs must show appropriate scales, labelling and units. The independent variable usually goes on the x-axis and the dependent variable on the y-axis Remember: The independent variable is the one you control or manipulate and the dependent variable is the one that changes as a result of your manipulation Always draw data points in pencil as it makes it easier to make corrections and adjustments Best fit lines Students often confuse the term lines of best fit with straight lines Lines of best fit can be straight lines or curves (just like the example above) and: They show the trend of the data It does not have to go through all the points, but shows the general trend They must go through the majority of the points Where the data is scattered the points should be evenly distributed on either side of the best fit line Extrapolation and interpolation © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 25 Extrapolation is the term used to describe the process of extending a line of best fit Extrapolation on a temperature correction graph Your notes This temperature correction graph from a calorimetry investigation shows how the two best fit lines are extrapolated to find the maximum temperature rise Interpolation is the term used to describe the process of assuming a trend line applies between two points Extrapolation and interpolation on a graph © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 26 Interpolation uses the line of best fit within the plotted points and extrapolation extends the best fit line beyond the plotted points Your notes Interpolation uses the line of best fit within the plotted points and extrapolation extends the best fit line beyond the plotted points Examiner Tips and Tricks You will have to decide if the origin, point (0,0) should be included as a data point If it does, it will be a good place to anchor the graph as it will be the most accurate data point Other features of graphs Gradient The gradient of a graph can be found by: For straight-line graphs Draw a triangle Then, use the equation for a straight line For curves Draw a tangent to the graph, using a ruler to line up against the curve at the point where the gradient is to be measured Then, use the equation for a straight line to calculate the gradient How to draw a tangent to a curve © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 27 Your notes Lining up a ruler against the curve is essential to drawing a tangent accurately The triangle should be as large as possible to minimise precision errors The equation for a straight line is y = mx + c, where: y = dependent variable x = independent variable m = slope c = y-intercept Therefore, the gradient or slope, m = ∆y / ∆x Changes in gradient Graphs with curves of best fit have changing gradients This means that multiple gradients can be calculated to show: The progressing rate of a reaction The effects of factors, such as concentration, on the rate of reaction A rate kinetics graph illustrating the calculation of rates from a curve © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 28 Your notes The gradient can be found at different points on a curve. Each rate has been divided by 60 to convert it from minutes-1 to seconds-1 Intercepts Intercepts are the points where a line / curve of best fit crosses an axis on a graph The most common use of intercepts is in graphs of free energy versus temperature Graph of free energy versus temperature for the synthesis of ammonia © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 29 The x-intercept shows the temperature when the reaction ceases to be feasible, in this case at 460 K (187 oC) and the y-intercept shows that the reaction is exothermic as the extrapolated value is approximately -46 kJ mol -1 Your notes Maxima and minima The maxima and minima are the highest and lowest points respectively Maxima - the gradient goes from positive to 0 to negative Minima - the gradient goes from negative to 0 to positive A literal application of maxima and minima is in energy profile diagrams The energy profile diagram for H2 + Cl2 → 2HCl Maxima represent the energy of the transition state and minima represent the energy of the reactants and products Areas A common use of areas of graphs in chemistry is Maxwell-Boltzmann curves A Maxwell-Boltzmann distribution curve © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 30 Your notes The total area under the curve represents the total number of molecules / particles, while the Ea line further divides this area into particles with and without the minimum activation energy to react Uncertainty bars The uncertainty in a measurement can be shown on a graph as an uncertainty bar This bar is drawn above and below the point and shows the uncertainty in that measurement Uncertainty bars are plotted on graphs to show the absolute uncertainty of values plotted Usually, these bars will be in the vertical direction, for y-values but they can be plotted horizontally, for x-values The size of the uncertainty bar can be used as an indication of the amount of error / uncertainty in the measurement Graph to show uncertainty bars © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 31 Your notes Uncertainty bars show the error / uncertainty in a specific measurement © 2025 Save My Exams, Ltd. Get more and ace your exams at savemyexams.com 32