Show that pKw = 14.00 at 298 K and use it to relate pH and pOH.

Kw = 1.00 × 10⁻¹⁴ mol² dm⁻⁶, so pKw = −log₁₀(1.00 × 10⁻¹⁴) = 14.00. Taking negative logarithms of Kw = [H⁺][OH⁻] turns the product into a sum, giving pH + pOH = 14.00 at this temperature.

Explain why the Arrhenius equation is used in its logarithmic form when analysing rate data.

Taking natural logarithms of k = Ae^(−Ea/RT) gives ln k = ln A − Ea/(RT). This has the form of a straight line, y = mx + c, with ln k plotted against 1/T. Rate constants vary over many orders of magnitude, so the logarithm both linearises the relationship and makes the values manageable.

How are ln x and log₁₀ x related, and why does the distinction matter in Arrhenius work?

ln x = 2.303 log₁₀ x. It matters because the Arrhenius equation in logarithmic form uses natural logarithms; if base-ten logarithms are used by mistake, the gradient of the plot, and hence the activation energy, comes out wrong by a factor of 2.303.

A rate constant increases from 1.0 × 10⁻⁵ to 1.0 × 10⁻² s⁻¹ on heating. Express this change on a logarithmic scale and comment on why logarithms are appropriate here.

ln(1.0 × 10⁻²) − ln(1.0 × 10⁻⁵) = ln(10³) = 6.91, and on a base-ten scale the change is exactly three units. Logarithms are appropriate because the rate constant has risen by a factor of one thousand, a change that a linear axis would render almost unreadable.

Why can taking a logarithm of a quantity with units be misleading, and how is this handled for pH and for ln k?

Strictly, only a pure number can be the argument of a logarithm. In practice the quantity is understood to be divided by its standard unit, so pH uses the numerical value of [H⁺] in mol dm⁻³ and an Arrhenius plot uses the numerical value of k in its own units. This is why the units of k must be stated separately and are not carried into ln k.

A student takes logarithms of both sides of rate = k[A]ⁿ. What linear relationship results, and what is the gradient?

Taking base-ten logarithms gives log₁₀(rate) = log₁₀ k + n log₁₀[A]. Plotting log₁₀(rate) against log₁₀[A] gives a straight line whose gradient is the order n and whose intercept is log₁₀ k.

WJEC A-Level Chemistry 1410QS

Formulae and Equations

Topic overview Charges on common ions Building formulae for ionic and molecular compounds Assigning oxidation numbers Using oxidation numbers to follow redox changes Balancing equations and using state symbols Ionic and half equations

Basic Ideas About Atoms

Topic overview Radioactive decay and nuclear equations Alpha, beta and gamma radiation Half-life, dating and the uses of radioisotopes: half-life of radioactive decay Half-life, dating and the uses of radioisotopes: adverse consequences for living cells Ionisation energies and electronic structure Orbitals and the filling of sub-shells Atomic spectra and the energy of light: atomic emission spectrum of the hydrogen atom Atomic spectra and the energy of light: relationship between energy and frequency Atomic spectra and the energy of light: significance of the frequency

Chemical Calculations

Topic overview Relative masses and the mass spectrometer: relative atomic Relative masses and the mass spectrometer: how a mass spectrometer works Relative masses and the mass spectrometer: molecular ion The mole, the Avogadro constant and molar mass: relative mass terms and simple mass spectra The mole, the Avogadro constant and molar mass: moles and percentage composition by mass Empirical and molecular formulae Concentration, solubility and titration calculations: concentration in mol dm-3 and in g dm-3 Concentration, solubility and titration calculations: stoichiometry and the limiting reagent Concentration, solubility and titration calculations: titration calculations and estimating a titre Gas volumes and the ideal gas equation Yield, atom economy and experimental uncertainty: concepts of atom economy and percentage yield Yield, atom economy and experimental uncertainty: how to estimate the percentage error

Bonding

Topic overview Ionic bonds and covalent bonds: ionic bonding in terms of ion formation Ionic bonds and covalent bonds: covalent bonding and forces Electronegativity, polarity and in-between bonds: concepts of electronegativity and bond polarity Electronegativity, polarity and in-between bonds: forces Forces between molecules Hydrogen bonding Shapes of molecules and ions: VSEPR principle and its use Shapes of molecules and ions: bond angles associated with linear

Solid Structures

Topic overview Ionic crystals: sodium chloride and caesium chloride Giant covalent structures: diamond and graphite Simple molecular solids: iodine and ice Metals and the electron sea Using structure to explain physical properties

The Periodic Table

Topic overview How the Periodic Table is organised Oxidation and reduction as electron transfer Trends across periods and down groups Group 2: reactions, trends and the reasons behind them — reactions with oxygen, water and dilute acid Group 2: reactions, trends and the reasons behind them — trends down the group Group 2: reactions, trends and the reasons behind them — thermal stability of the carbonates Group 2: reactions, trends and the reasons behind them — oxides and hydroxides Identifying ions, making salts and gravimetric analysis: identifying cations with sodium hydroxide Identifying ions, making salts and gravimetric analysis: preparing a soluble salt from an acid Group 7: the halogens — physical properties and the trend in volatility Group 7: the halogens — reactions of the halogens with metals Group 7: the halogens — displacement reactions and the trend Group 7: the halogens — halide ions as reducing agents, uses of chlorine and the chlorate equilibrium

Simple Equilibria and Acid-Base Reactions

Topic overview Reversible reactions and dynamic equilibrium Le Chatelier's principle and shifting the position of equilibrium The equilibrium constant Kc Acids, bases, salts and neutralisation pH, strong acids and weak acids Acid-base titrations and their calculations

Thermochemistry

Topic overview Enthalpy changes and their definitions Calculating enthalpy changes from experimental measurements Determining enthalpy changes in the laboratory Hess's law and energy cycles Average bond enthalpies

Rates of Reaction

Topic overview Collision theory and the conditions that change rate Measuring rate and relating it to concentration Energy profiles and activation energy Temperature and the Boltzmann distribution Catalysts and how they work Following reactions in the laboratory

The Wider Impact of Chemistry

Topic overview Social and economic consequences of chemical manufacture and energy supply Environmental consequences of chemical processes and energy production Green chemistry and sustainable practice

Organic Compounds

Topic overview Drawing and reading organic structures Naming organic compounds Chain length, functional groups and physical properties Structural isomerism Electrophiles, nucleophiles, radicals and bond fission

Hydrocarbons

Topic overview Alkane bonding, combustion and fossil fuels: combustion reaction of alkanes and benefits Alkane bonding, combustion and fossil fuels: combustion calculations and acid rain Radical substitution of alkanes The C=C bond and why alkenes are reactive E–Z isomerism in alkenes Electrophilic addition, Markovnikov's rule and testing for alkenes: mechanism of electrophilic addition Electrophilic addition, Markovnikov's rule and testing for alkenes: Markovnikov's rule and the bromine water test Electrophilic addition, Markovnikov's rule and testing for alkenes: testing for a C=C bond Hydrogenation and addition polymers: conditions required Hydrogenation and addition polymers: nature of addition polymerisation

Halogenoalkanes

Topic overview Nucleophilic substitution by hydroxide ions Elimination to form alkenes Reactivity of the carbon–halogen bond and the silver nitrate test Uses of halogenoalkanes, CFCs and the ozone layer: uses of halogenoalkanes and CFCs Uses of halogenoalkanes, CFCs and the ozone layer: radical mechanism Uses of halogenoalkanes, CFCs and the ozone layer: replacements for CFCs Heating under reflux in the laboratory

Alcohols and Carboxylic Acids

Topic overview Two routes to ethanol: hydration and fermentation — hydration of ethene Two routes to ethanol: hydration and fermentation — fermentation Two routes to ethanol: hydration and fermentation — comparing the two routes on rate Classifying alcohols and dehydrating them Oxidising alcohols to aldehydes, ketones and acids Chemical tests for alcohols and carboxylic acids Carboxylic acids, bases and salts Making an ester and separating it by distillation: esterification Making an ester and separating it by distillation: carrying out the preparation Making an ester and separating it by distillation: separating and purifying Making an ester and separating it by distillation: yield

Instrumental Analysis

Topic overview Reading a mass spectrum Infrared spectra and functional groups NMR spectra and atomic environments Putting the three techniques together

Redox and Standard Electrode Potential

Topic overview Electron transfer and oxidation numbers Half-equations, half-cells and cell diagrams The hydrogen electrode and standard electrode potentials Cell EMF and the feasibility of reactions The hydrogen fuel cell

Redox Reactions

Topic overview Building ion–electron half-equations Combining half-equations into full equations Copper, iodide and thiosulfate Carrying out and calculating a redox titration: procedure for a manganate titration Carrying out and calculating a redox titration: redox titration calculations

Chemistry of the P-Block

Topic overview Amphoteric behaviour of aluminium and lead Stability of oxidation states down Groups 3, 4 and 5: increasing stability of the inert pair cations Stability of oxidation states down Groups 3, 4 and 5: change in relative stability Valence shells that break the octet rule: how some Group 3 elements can form compounds Valence shells that break the octet rule: structure and bonding in Al2Cl6 and formation Boron nitride: two structures, two sets of properties Group 4 oxides and chlorides, and lead in solution: structure Group 4 oxides and chlorides, and lead in solution: hydrolysis of the Group 4 chlorides Group 4 oxides and chlorides, and lead in solution: chloride hydrolysis and lead(II) precipitates Group 4 oxides and chlorides, and lead in solution: distinguishing lead from silver ions Chlorine, chlorate and the sodium halides: chlorine with water and with cold and hot alkali Chlorine, chlorate and the sodium halides: bleaching and bactericidal action of Cl2 Chlorine, chlorate and the sodium halides: differences in behaviour of NaCl

Chemistry of the D-Block Transition Metals

Topic overview Variable oxidation states and the colours of their solutions: how the d-block elements attain various Variable oxidation states and the colours of their solutions: most important oxidation states of Cr Complex ions: ligands, coordination number and shape Why transition metal complexes are coloured Ligand exchange and changes in coordination number Transition metals as catalysts: why transition metals Transition metals as catalysts: variable oxidation state Transition metals as catalysts: heterogeneous catalysis Transition metals as catalysts: surface area Transition metal ions with aqueous hydroxide

Chemical Kinetics

Topic overview Following a reaction by sampling and quenching Finding the order of a reaction from experimental results The rate equation and the rate constant Rate-determining steps and reaction mechanisms Temperature, catalysts and the Arrhenius equation

Enthalpy Changes for Solids and Solutions

Topic overview Defining the enthalpy changes Born-Haber cycles for ionic compounds Dissolving: lattice breaking against hydration What the enthalpy of formation says about stability

Entropy and Feasibility of Reactions

Topic overview Entropy as the freedom of particles Comparing entropy across states and reactions Calculating entropy changes Gibbs free energy change Feasibility, temperature and spontaneous endothermic change

Equilibrium Constants

Topic overview Writing equilibrium constant expressions and deriving their units Calculating Kc and equilibrium amounts Partial pressures and Kp calculations Temperature, magnitude and the position of equilibrium

Acid-Base Equilibria

Topic overview Proton transfer and conjugate pairs Strong and weak acids: Ka and pKa Water, Kw and the pH scale pH calculations for acids and alkalis: how to use pH pH calculations for acids and alkalis: strong bases, dilution and mixtures Buffers: how they hold pH steady and why they matter Titrations: curve shapes, salt pH and choosing an indicator — pH curve shapes for strong and weak acids and bases Titrations: curve shapes, salt pH and choosing an indicator Titrations: curve shapes, salt pH and choosing an indicator — how an indicator works and choosing one Titrations: curve shapes, salt pH and choosing an indicator — using a pH probe

Stereoisomerism

Topic overview Structural isomerism and stereoisomerism compared E‒Z isomerism and the priority rules Chiral centres, enantiomers and racemic mixtures Optical activity and plane-polarised light

Aromaticity

Topic overview Benzene: shape, bonding and delocalisation Aromatic stability and resistance to addition Electrophilic substitution: nitration, halogenation and alkylation How ring and substituent affect each other

Alcohols and Phenols

Topic overview Making primary and secondary alcohols Reactions of primary and secondary alcohols Phenol: acidity and characteristic reactions Recognising a phenol

Aldehydes and Ketones

Topic overview Making aldehydes and ketones from alcohols Distinguishing aldehydes from ketones Reduction of carbonyl compounds with NaBH₄ Nucleophilic addition of hydrogen cyanide 2,4-dinitrophenylhydrazine as a test for a carbonyl group The triiodomethane test

Carboxylic Acids and Their Derivatives

Topic overview Comparing the acidity of carboxylic acids, phenols, alcohols and water Making carboxylic acids by oxidation Reduction and decarboxylation of carboxylic acids: reduction of carboxylic acids using LiAlH₄ Reduction and decarboxylation of carboxylic acids: decarboxylation of carboxylic acids Esters and acid chlorides Turning acids and halogenoalkanes into amides and nitriles Hydrolysis and reduction of nitriles and amides

Amines

Topic overview Preparing aliphatic and aromatic amines The basicity of amines Ethanoylation of primary amines Primary amines and cold nitric(III) acid Coupling reactions and azo dyes: coupling of benzenediazonium salts with phenols Coupling reactions and azo dyes: role of the —N=N— chromophore in azo dyes The origin of colour

Amino Acids, Peptides and Proteins

Topic overview Structure and classification of α-amino acids Zwitterions, amphoteric character and physical properties Peptide links: dipeptides, polypeptides and proteins Levels of protein structure and the bonds that hold them Proteins at work: enzymes and biological function

Organic Synthesis and Analysis

Topic overview Planning a synthesis as a sequence of reactions Working up a preparation: separation and purification Melting temperature as a test of purity Addition and condensation polymerisation: polyesters and polyamides Polymers: hydrolysis, properties and identifying repeat units High-resolution ¹H NMR and the elucidation of structure: low- and high-resolution proton NMR High-resolution ¹H NMR and the elucidation of structure: chemical shift, TMS, solvents and D₂O exchange High-resolution ¹H NMR and the elucidation of structure: combining mass spectra and NMR to identify compounds High-resolution ¹H NMR and the elucidation of structure: combining NMR with mass spectrometry Chromatography and the composition of mixtures

Practical Skills and Techniques

Topic overview Measuring mass, volume, time and temperature Standard solutions and titration technique Indicators and acid–base titration curves Heating, refluxing, distilling and purifying: Heating Heating, refluxing, distilling and purifying: purifying Melting temperature, chromatography and pH measurement Electrochemical cells, rate methods and working safely: Electrochemical cells Electrochemical cells, rate methods and working safely: rate methods and working safely

Planning, Analysis and Evaluation

Topic overview Planning an investigation and controlling variables Recording observations and presenting data Graphs, gradients and processing data Errors, uncertainty, accuracy and precision Evaluating a method and suggesting improvements Research and referencing

Core Experiments and Their Chemistry

Topic overview Gravimetric and acid–base analysis Enthalpy and rate experiments: Enthalpy Enthalpy and rate experiments: rate experiments Redox titrations and electrochemical cells Equilibrium constants and following a titration with a pH probe Organic synthesis, separation and identification: Organic synthesis Organic synthesis, separation and identification: identification tests and chromatography

Mathematical Skills in Chemistry

Topic overview Units, standard form and proportion in chemical calculation: volume, equilibrium, rate and energy units Units, standard form and proportion in chemical calculation: standard form Units, standard form and proportion in chemical calculation: proportion Units, standard form and proportion in chemical calculation: checking answers, buffers, rate constants and units Significant figures, means and uncertainty in experimental data: significant figures in multiplication Significant figures, means and uncertainty in experimental data: relative atomic mass as a weighted mean Significant figures, means and uncertainty in experimental data: absolute and percentage uncertainty Symbols, rearranging equations and logarithms: chemical symbols and what each one asserts Symbols, rearranging equations and logarithms: rearranging an equation Symbols, rearranging equations and logarithms: substituting into equations for Kc, enthalpy and pH Symbols, rearranging equations and logarithms: logarithms, the Arrhenius equation and rearranging formulae Graphs: plotting, gradients, intercepts and tangents — reading a spectrum Graphs: plotting, gradients, intercepts and tangents — reaction order from data and plotting good graphs Graphs: plotting, gradients, intercepts and tangents — Arrhenius plots, zero-order graphs and tangents Graphs: plotting, gradients, intercepts and tangents — initial rates from tangents and interpreting graphs Shape, symmetry and three-dimensional structure: Shape Shape, symmetry and three-dimensional structure: symmetry

Mathematical Skills in Chemistry

Symbols, rearranging equations and logarithms: logarithms, the Arrhenius equation and rearranging formulae

2 things to know
All units

Overview

pKw at 298 K, the relationship between ln and log₁₀, solving the Arrhenius equation for temperature and activation energy, half-life calculations, and rearranging mole and percentage yield formulae.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

Rearrange the expression for percentage yield to find the theoretical mass of product, given a yield of 75.0% and an actual mass of 3.00 g.

Key term

A gaseous equilibrium has Kc = 4.0 × 10⁻³ mol dm⁻³. Comment on the position of equilibrium and express the comparison using an appropriate symbol.

Questions from this subtopic

Each one with the model answer underneath.

1
Show that pKw = 14.00 at 298 K and use it to relate pH and pOH.
Model answer
Kw = 1.00 × 10⁻¹⁴ mol² dm⁻⁶, so pKw = −log₁₀(1.00 × 10⁻¹⁴) = 14.00. Taking negative logarithms of Kw = [H⁺][OH⁻] turns the product into a sum, giving pH + pOH = 14.00 at this temperature.
2
Explain why the Arrhenius equation is used in its logarithmic form when analysing rate data.
Model answer
Taking natural logarithms of k = Ae^(−Ea/RT) gives ln k = ln A − Ea/(RT). This has the form of a straight line, y = mx + c, with ln k plotted against 1/T. Rate constants vary over many orders of magnitude, so the logarithm both linearises the relationship and makes the values manageable.
3
How are ln x and log₁₀ x related, and why does the distinction matter in Arrhenius work?
Model answer
ln x = 2.303 log₁₀ x. It matters because the Arrhenius equation in logarithmic form uses natural logarithms; if base-ten logarithms are used by mistake, the gradient of the plot, and hence the activation energy, comes out wrong by a factor of 2.303.
4
A rate constant increases from 1.0 × 10⁻⁵ to 1.0 × 10⁻² s⁻¹ on heating. Express this change on a logarithmic scale and comment on why logarithms are appropriate here.
Model answer
ln(1.0 × 10⁻²) − ln(1.0 × 10⁻⁵) = ln(10³) = 6.91, and on a base-ten scale the change is exactly three units. Logarithms are appropriate because the rate constant has risen by a factor of one thousand, a change that a linear axis would render almost unreadable.
5
Why can taking a logarithm of a quantity with units be misleading, and how is this handled for pH and for ln k?
Model answer
Strictly, only a pure number can be the argument of a logarithm. In practice the quantity is understood to be divided by its standard unit, so pH uses the numerical value of [H⁺] in mol dm⁻³ and an Arrhenius plot uses the numerical value of k in its own units. This is why the units of k must be stated separately and are not carried into ln k.
6
A student takes logarithms of both sides of rate = k[A]ⁿ. What linear relationship results, and what is the gradient?
Model answer
Taking base-ten logarithms gives log₁₀(rate) = log₁₀ k + n log₁₀[A]. Plotting log₁₀(rate) against log₁₀[A] gives a straight line whose gradient is the order n and whose intercept is log₁₀ k.

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