Why does the exponential term make the rate of a reaction so sensitive to temperature?

The temperature appears in the denominator of an exponent, so a small change in T produces a proportionally much larger change in the value of the exponential, and it is that factor which multiplies the rate constant.

Calculate the value of the exponential factor for a reaction of activation energy 50.0 kJ mol⁻¹ at 298 K.

The exponent is −50 000 ÷ (8.314 × 298) = −20.18, and exp(−20.18) is 1.7 × 10⁻⁹, a pure number with no units.

Show by calculation what happens to the rate constant of a reaction of activation energy 50.0 kJ mol⁻¹ when the temperature rises from 298 K to 308 K.

The ratio of the two rate constants is exp[(Ea/R)(1/298 − 1/308)] = exp(6014 × 1.09 × 10⁻⁴) = exp(0.655) = 1.9, so the rate constant very nearly doubles for a rise of ten kelvin.

What does the exponential term in the Arrhenius equation represent physically?

It is the fraction of the colliding molecules that possess at least the activation energy, so it measures how many collisions have enough energy to lead to reaction.

Eduqas A-Level Chemistry A410QS

Formulae and Equations

Topic overview Formulae of common ions Writing formulae for ionic compounds Oxidation numbers and the rules for assigning them Oxidation numbers in ions and less obvious cases Constructing and balancing chemical equations Ionic equations and state symbols

Basic Ideas About Atoms

Topic overview Radioactive decay and nuclear equations Behaviour of α, β and γ radiation Half-life and radioactive decay calculations Radiation hazards and the uses of radioisotopes Ionisation energy and its variation between elements Successive ionisation energies and electronic structure Orbitals and electron configurations of elements 1 to 36 Emission and absorption spectra The hydrogen spectrum, E = hf and the convergence limit

Chemical Calculations

Topic overview Relative atomic, isotopic, formula and molecular mass Principles of the mass spectrometer Mass spectra, isotopic abundance and relative atomic mass Empirical and molecular formulae The mole, the Avogadro constant and molar mass Moles, grams and numbers of particles Concentration and solubility Molar volume and gas volume corrections The ideal gas equation Stoichiometry and reacting quantities Acid-base titration calculations Atom economy, percentage yield and percentage error

Bonding

Topic overview Ionic bonding and the formation of ions Covalent bonding and the balance of forces Coordinate bonding Electronegativity and bond polarity The ionic–covalent continuum Intermolecular forces, permanent and temporary dipoles Hydrogen bonding VSEPR and the shapes of simple molecules and ions

Solid Structures

Topic overview Sodium chloride and caesium chloride structures Diamond and graphite Iodine and ice Metallic bonding and the electron sea model Structure, bonding and physical properties

The Periodic Table

Topic overview Arrangement of the elements and the s-, p- and d-blocks Oxidation and reduction as electron transfer Trends across periods and down groups Reactions of Group 2 elements with oxygen, water and steam Reactions of the aqueous cations Mg²⁺, Ca²⁺ and Ba²⁺Flame colours Reactivity of Group 1 and Group 2 metals Thermal stability of the Group 2 carbonates and hydroxides Solubility of the Group 2 hydroxides and sulfates Basic character of the Group 1 and Group 2 oxides and hydroxides Volatility of the halogens and their reactions with metals Oxidising power, displacement reactions and the silver nitrate test Water treatment, salt preparation and gravimetric analysis

Simple Equilibria and Acid-Base Reactions

Topic overview Reversible reactions and dynamic equilibrium Le Chatelier's principle The equilibrium constant Kc Acids and bases as donors and acceptors of H⁺Strong and weak acids, pH and hydrogen ion concentration Acid-base titrations

Thermochemistry

Topic overview Enthalpy changes of reaction, combustion and formation Hess's law and energy cycles Average bond enthalpies Calculating enthalpy changes from experimental data Laboratory procedures for determining enthalpy changes

Rates of Reaction

Topic overview Collision theory and the conditions that change rate Energy profiles and activation energy The Boltzmann distribution and the effect of temperature Catalysts and how they work Calculating rates from experimental data Methods of following the course of a reaction

The Wider Impact of Chemistry

Topic overview Social, economic and environmental impact of chemical synthesis The production of energy and the choices it forces Green chemistry and sustainability

Organic Compounds

Topic overview Representing organic compounds Naming organic compounds Boiling temperature and solubility of organic compounds Structural isomerism Bond fission, electrophiles, nucleophiles and radicals

Hydrocarbons

Topic overview Alkanes and the sigma-bonded framework Combustion of alkanes and the use of fossil fuels Free radical substitution and the photochlorination of alkanes Alkenes and the reactivity of the C=C bond E–Z isomerism Electrophilic addition of bromine to alkenes Tests for alkenes Markovnikov addition of hydrogen bromide and carbocation stability Catalytic hydrogenation of alkenes Addition polymerisation

Halogenoalkanes

Topic overview Nucleophilic substitution of primary halogenoalkanes Elimination reactions forming alkenes Bond polarity, bond enthalpy and the ease of substitution The silver nitrate test for halogenoalkanes Uses of halogenoalkanes and the regulation of their use CFCs and their effect in the upper atmosphere Heating under reflux

Alcohols and Carboxylic Acids

Topic overview Industrial preparation of ethanol from ethene Fermentation and the distillation of ethanol Biofuels and the issues they raise Classifying alcohols as primary, secondary and tertiary Dehydration of alcohols to alkenes Oxidation of primary alcohols to aldehydes and carboxylic acids Oxidation of secondary alcohols to ketones The dichromate(VI) test for alcohols Carboxylic acids and the sodium hydrogencarbonate test Reactions of carboxylic acids with bases, carbonates and hydrogencarbonates Esterification and separation by distillation

Instrumental Analysis

Topic overview Mass spectra and molecular structure Infrared spectroscopy Carbon-13 NMR spectroscopy Low resolution proton NMR spectroscopy

Redox and Standard Electrode Potential

Topic overview Redox systems and half-equations in cell diagrams Standard electrode potential and the standard hydrogen electrode Combining electrodes to form electrochemical cells Cell EMF and the feasibility of reactions The hydrogen fuel cell

Redox Reactions

Topic overview Constructing ion/electron half-equations Combining half-equations into a stoichiometric redox equation The reaction of Cu²⁺ with iodide ions Carrying out a redox titration Redox titration calculations

Chemistry of the P-Block

Topic overview Amphoteric behaviour of aluminium and lead The inert pair effect in Groups 3, 4 and 5 Electron-deficient compounds and expanded octets Al₂Cl₆ and donor-acceptor compounds Hexagonal and cubic boron nitride Stability of oxidation states II and IV in Group 4 Carbon dioxide and lead(II) oxide Bonding in the Group 4 chlorides Hydrolysis of the Group 4 chlorides Reactions of aqueous lead(II) ions Chlorine with cold and warm aqueous sodium hydroxide Bleaching and bactericidal action of chlorine and chlorate(I)Sodium halides with concentrated sulfuric acid

Chemistry of the D-Block Transition Metals

Topic overview Oxidation states of the d-block elements Important oxidation states and the colours of aqueous ions Bonding in tetrahedral and octahedral complexes The origin of colour and the splitting of d-orbitals Ligand exchange and change of coordination number Colours and formulae of copper and cobalt complex ions Heterogeneous and homogeneous catalysis Nickel in hydrogenation and iron in the Haber process Vanadium(V) oxide and manganese(IV) oxide as catalysts Reactions of Cr³⁺, Fe²⁺, Fe³⁺ and Cu²⁺ with excess aqueous hydroxide

Chemical Kinetics

Topic overview Measuring reaction rate by sampling and quenching The rate equation, order and the rate constant Finding reaction order from experimental results The rate-determining step and reaction mechanism The Arrhenius equation and activation energy

Enthalpy Changes for Solids and Solutions

Topic overview Enthalpy changes of atomisation, lattice and hydration Enthalpy change of solution and the solubility of ionic compounds The Born–Haber cycle Standard enthalpy of formation as an indicator of stability

Entropy and Feasibility of Reactions

Topic overview Entropy as a measure of freedom Entropy of solids, liquids and gases Calculating entropy changes from absolute entropies Gibbs free energy change Feasibility, spontaneity and endothermic processes

Equilibrium Constants

Topic overview Writing Kc and Kp expressions and their units Calculating Kc and the amounts present at equilibrium Kp, partial pressures and mole fractions Temperature and the magnitude of the equilibrium constant

Acid-Base Equilibria

Topic overview The Lowry–Brønsted theory and conjugate pairs Strong and weak acids and the dissociation constant Ka The ionic product of water, Kw pH of strong acids and strong bases pH of weak acids from Ka and pKa Titration curves How buffer solutions work Buffer calculations Buffers in living systems and industry Salt hydrolysis and the choice of indicator

Stereoisomerism

Topic overview Stereoisomerism and structural isomerism Chiral centres and enantiomers Optical activity and plane-polarised light Racemic mixtures

Aromaticity

Topic overview Structure and bonding in benzene Delocalisation and resistance to addition Electrophilic substitution: nitration and halogenation Friedel-Crafts alkylation and the ring's interaction with substituents

Alcohols and Phenols

Topic overview Forming alcohols from halogenoalkanes and carbonyl compounds Alcohols with hydrogen halides, ethanoyl chloride and carboxylic acids Acidity of phenol Reactions of phenol and the iron(III) chloride test

Aldehydes and Ketones

Topic overview Preparing aldehydes and ketones by oxidation of alcohols Distinguishing aldehydes and ketones with Tollens’ reagent and Fehling’s solution Reduction of aldehydes and ketones using NaBH₄ Nucleophilic addition of HCN to aldehydes and ketones Reaction with 2,4-dinitrophenylhydrazine The triiodomethane (iodoform) test

Carboxylic Acids and Their Derivatives

Topic overview Relative acidity of carboxylic acids, phenols, alcohols and water Preparing carboxylic acids by oxidation Reduction and decarboxylation of carboxylic acids Acid chlorides and their reactions Esters and their hydrolysis Amides and their hydrolysis Nitriles from carboxylic acids and halogenoalkanes Hydroxynitriles, hydrolysis and reduction of nitriles

Amines

Topic overview Amines from halogenoalkanes and ammonia Amines from the reduction of nitriles Aromatic amines from nitrobenzene Basicity of amines Ethanoylation of primary amines Reaction of primary amines with cold nitric(III) acid Azo coupling, azo dyes and the origin of colour

Amino Acids, Peptides and Proteins

Topic overview α-amino acids and their general formula Zwitterions and amphoteric behaviour Melting temperature, solubility and the isoelectric point Dipeptides, polypeptides and the peptide link Protein structure and the role of proteins

Organic Synthesis and Analysis

Topic overview Planning a multi-step synthesis Choosing between alternative routes Manipulation and separation techniques Purification and melting temperature Addition and condensation polymerisation Polyesters and polyamides High resolution proton NMR Combining spectra to elucidate a structure Interpreting chromatographic data Gas chromatography and HPLC

Specified Practical Work

Topic overview Gravimetric analysis and qualitative analysis of unknown solutions Standard solutions, standardising an acid and preparing a soluble salt Back titration and double titration Determining enthalpy changes by calorimetry Following a rate by gas collection, the iodine clock and reaction order Electrochemical cells, redox titration and the estimation of copper Determining an equilibrium constant and following a titration with a pH probe Hydrolysis of 1-bromobutane and the preparation of an ester Organic synthesis, purification and identification

Practical Skills and Techniques

Topic overview Choosing apparatus and recording measurements Volumetric technique with pipette, burette and volumetric flask Titration procedure and the choice of indicator Heating, heating under reflux and distillation Filtration, purification and qualitative testing Melting temperature, chromatography and electrochemical cells Variables, errors and risk in evaluating a method

Mathematical Skills in Chemistry

Topic overview Standard form, units and significant figures Ratios, fractions and percentages Changing the subject and substituting values Logarithms, pH and pKa Exponentials, the Arrhenius equation and rate Graphs, gradients and intercepts Tangents and rates of change Area under a curve Uncertainty and combining errors The gas equation and equilibrium expressions

Mathematical Skills in Chemistry

Exponentials, the Arrhenius equation and rate

3 things to know
Component 3

Overview

Recall questions on the exponential form of the Arrhenius equation and its logarithmic form, what is plotted to obtain a straight line, what the gradient and intercept give, calculating an activation energy from two rate constants, and why a small temperature rise changes the rate so much.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

State the Arrhenius equation and explain what each symbol stands for.

Numbers

A first-order reaction has a rate constant of 2.50 × 10⁻³ s⁻¹ and starts at 0.800 mol dm⁻³. Calculate the concentration remaining after 300 s.

Key term

Explain why the half-life of a first-order reaction does not depend on the starting concentration.

Questions from this subtopic

Each one with the model answer underneath.

1
State the Arrhenius equation and explain what each symbol stands for.
Model answer
The equation is k = A exp(−Ea/RT), in which k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant and T is the absolute temperature.
2
Why does the exponential term make the rate of a reaction so sensitive to temperature?
Model answer
The temperature appears in the denominator of an exponent, so a small change in T produces a proportionally much larger change in the value of the exponential, and it is that factor which multiplies the rate constant.
3
Calculate the value of the exponential factor for a reaction of activation energy 50.0 kJ mol⁻¹ at 298 K.
Model answer
The exponent is −50 000 ÷ (8.314 × 298) = −20.18, and exp(−20.18) is 1.7 × 10⁻⁹, a pure number with no units.
4
Show by calculation what happens to the rate constant of a reaction of activation energy 50.0 kJ mol⁻¹ when the temperature rises from 298 K to 308 K.
Model answer
The ratio of the two rate constants is exp[(Ea/R)(1/298 − 1/308)] = exp(6014 × 1.09 × 10⁻⁴) = exp(0.655) = 1.9, so the rate constant very nearly doubles for a rise of ten kelvin.
5
What does the exponential term in the Arrhenius equation represent physically?
Model answer
It is the fraction of the colliding molecules that possess at least the activation energy, so it measures how many collisions have enough energy to lead to reaction.
6
A first-order reaction has a rate constant of 2.50 × 10⁻³ s⁻¹ and starts at 0.800 mol dm⁻³. Calculate the concentration remaining after 300 s.
Model answer
The concentration follows [A] = [A]₀ exp(−kt), and kt is 0.750, so [A] = 0.800 × exp(−0.750) = 0.800 × 0.472 = 0.378 mol dm⁻³.

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AThe equation is k = A exp(−Ea/RT).

BHere k is the rate constant.

CA is the pre-exponential factor.

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Kc is a fixed number at a given temperature.

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