Entropy, free energy and Entropy, free energy and equilibriumequilibrium

SpontaneitySpontaneityEntropyEntropyFree energy and equilibriumFree energy and equilibrium

Learning objectivesLearning objectives

Discuss what is meant by spontaneityDiscuss what is meant by spontaneity Discuss energy dispersal and its relevance Discuss energy dispersal and its relevance

to spontaneityto spontaneity Describe the concept of a reversible Describe the concept of a reversible

processprocess Define entropyDefine entropy

EquilibriumEquilibrium

At an equilibrium point, the system resists small At an equilibrium point, the system resists small disturbances (not necessarily large ones)disturbances (not necessarily large ones)

At equilibrium, the rates of the forward and At equilibrium, the rates of the forward and backward processes are equalbackward processes are equal

Locally stable more

stable

unstable

Spontaneity Spontaneity

The tendency for a process to advance to The tendency for a process to advance to equilibrium without external influenceequilibrium without external influence

Something that happens naturally is spontaneousSomething that happens naturally is spontaneous Any process will be spontaneous in one directionAny process will be spontaneous in one direction The reverse is non-spontaneousThe reverse is non-spontaneous If work needs to be done, it is not spontaneousIf work needs to be done, it is not spontaneous

A rock naturally rolls down a hill - spontaneousA rock naturally rolls down a hill - spontaneous It must be pushed back up - nonspontaneousIt must be pushed back up - nonspontaneous A hot object naturally cools - spontaneousA hot object naturally cools - spontaneous

Indicators of spontaneityIndicators of spontaneity

What is the indicator of spontaneity?What is the indicator of spontaneity? Heat evolved?Heat evolved?

But…endothermic reactions occur spontaneously But…endothermic reactions occur spontaneously as well (ice melting, salt dissolving)as well (ice melting, salt dissolving)

Enthalpy is not an indicator of spontaneity, Enthalpy is not an indicator of spontaneity, although most spontaneous processes are although most spontaneous processes are exothermic – energy is conserved not createdexothermic – energy is conserved not created

The amount of energy does not change in any The amount of energy does not change in any process – but it is redistributed...process – but it is redistributed...

Various spontaneities: dispersalVarious spontaneities: dispersal

Matter disperses – gas fills a container, two Matter disperses – gas fills a container, two liquids mixliquids mix

Heat disperses – hot object cools on cold Heat disperses – hot object cools on cold surfacesurface

Motion disperses – a ball stops bouncingMotion disperses – a ball stops bouncing

The reverses of these three well known The reverses of these three well known processes processes nevernever occur spontaneously occur spontaneously

ReversibilityReversibility

A A reversiblereversible process is one where the process is one where the system and surroundings are changed to system and surroundings are changed to original values without any changeoriginal values without any change

An An irreversibleirreversible process is one where the process is one where the system and surroundings cannot be system and surroundings cannot be restoredrestored

A reversible process produces the A reversible process produces the maximum possible workmaximum possible work

Reversibility and realityReversibility and reality

Reversibility only occurs when the system is Reversibility only occurs when the system is in or almost at equilibrium – at an in or almost at equilibrium – at an infinitesimal rateinfinitesimal rate

In reality this does not obtainIn reality this does not obtain Real processes produce less work than Real processes produce less work than

ideal processesideal processes Spontaneous processes are irreversibleSpontaneous processes are irreversible The reverse of a spontaneous process is The reverse of a spontaneous process is

nonspontaneousnonspontaneous

Spontaneity and speedSpontaneity and speed

The speed of a reaction is The speed of a reaction is notnot an indicator of an indicator of its spontaneity.its spontaneity.

Spontaneity is determined by the relative Spontaneity is determined by the relative positions of the initial and final states positions of the initial and final states (thermodynamic state functions)(thermodynamic state functions)

Speed is determined by the pathway Speed is determined by the pathway (kinetics)(kinetics)

Two independent regimesTwo independent regimes

Entropy – the mixing (distributing) linkEntropy – the mixing (distributing) link

entropyentropy measures distribution of energy over measures distribution of energy over statesstates

The more states available, the more entropyThe more states available, the more entropy It is a “state function” – depends only on initial and It is a “state function” – depends only on initial and

final states, not the pathway.final states, not the pathway. The entropy change for a process isThe entropy change for a process is

Systems move spontaneously to a state of greater Systems move spontaneously to a state of greater entropy – greater distribution of energyentropy – greater distribution of energy

Disorder provides more states for energy Disorder provides more states for energy distribution than ordered systemsdistribution than ordered systems

initialfinal SSS

Why do crystals form at all?Why do crystals form at all?

Entropy is distribution of energy over Entropy is distribution of energy over microstatesmicrostates

Crystals are highly ordered arrangementsCrystals are highly ordered arrangements Crystals should spontaneously fly apart to Crystals should spontaneously fly apart to

maximize disordermaximize disorder

But...This view ignores energy of the latticeBut...This view ignores energy of the lattice Energy input to break bonds corresponds to Energy input to break bonds corresponds to

entropy entropy decrease decrease (localization of energy) (localization of energy)

Don’t let them fool youDon’t let them fool you

A popular argument against evolution is that the A popular argument against evolution is that the formation of organized DNA molecules from a formation of organized DNA molecules from a random soup of atoms and molecules contravenes random soup of atoms and molecules contravenes Second LawSecond Law

Just as crystals appear in a dish spontaneously so Just as crystals appear in a dish spontaneously so can DNA form from smaller unitscan DNA form from smaller units

N.B. Order (energy concentration) can appear N.B. Order (energy concentration) can appear spontaneously locally provided greater disorder spontaneously locally provided greater disorder (energy dispersal) is occurring elsewhere(energy dispersal) is occurring elsewhere

Entropy and solubilityEntropy and solubility

Hydration increases entropy of the ions in Hydration increases entropy of the ions in the latticethe lattice Ions in solution have greater disorderIons in solution have greater disorder

but can decrease entropy of the solventbut can decrease entropy of the solvent Solvent molecules now have greater orderSolvent molecules now have greater order

Excessive hydration by highly polarizing Excessive hydration by highly polarizing ions can reduce entropy of solventions can reduce entropy of solvent CaSOCaSO4 4 is only slightly soluble is only slightly soluble

What will these socks ne’er be What will these socks ne’er be matched?matched?

Would you be stunned if Would you be stunned if the tumble dryer the tumble dryer matched the socks?matched the socks?

Okay, you never match Okay, you never match the socks anywaythe socks anyway

Chaos in the sock Chaos in the sock drawer is naturaldrawer is natural

The same principles are The same principles are applied to chemical applied to chemical change (sort of)change (sort of)

Chance meeting: entropy and Chance meeting: entropy and probabilityprobability

Ordered states are less likely because there are fewer ways to Ordered states are less likely because there are fewer ways to obtain themobtain them Do our socks become matched spontaneously?Do our socks become matched spontaneously? No, only one of many possible arrangementsNo, only one of many possible arrangements

With only a few molecules the ordered state becomes massively With only a few molecules the ordered state becomes massively less probable than a disordered stateless probable than a disordered state

Boltzmann and disorderBoltzmann and disorder

W is the number of possible arrangements W is the number of possible arrangements of the stateof the state

kk is Boltzmann’s constant = is Boltzmann’s constant = R/NR/NAA = 1.38x10 = 1.38x10-23 -23 J/KJ/K

The entropy is proportional to the natural log The entropy is proportional to the natural log of the number of arrangements of the stateof the number of arrangements of the state

WkS ln

Entropy of a disordered systemEntropy of a disordered system

An ordered arrangement has W = 1, S = 0An ordered arrangement has W = 1, S = 0 Entropy of one mole of disordered Entropy of one mole of disordered

molecules molecules

S = 5.76 J/KS = 5.76 J/K

2ln2ln2lnln RkNkWkS AN A

Entropy and gas expansionEntropy and gas expansion

There is only one possible way for all the gas There is only one possible way for all the gas molecules to fill A and leave B empty. There are molecules to fill A and leave B empty. There are ways for N ways for NAA molecules to occupy A molecules to occupy A and Band B

Entropy associated with gas mixingEntropy associated with gas mixing Entropy associated with gas expansion:Entropy associated with gas expansion:Doubling the volume doubles the number of positions Doubling the volume doubles the number of positions

(microstates) for distribution of energy(microstates) for distribution of energy

initial

final

V

VRS ln

2 AN

Making sense of units and definition Making sense of units and definition of entropyof entropy

Units of entropy are J/KUnits of entropy are J/K How do these units connect to disorder and How do these units connect to disorder and

probability?probability? Disorder is Disorder is notnot entropy entropy Disorder increases the number of microstates Disorder increases the number of microstates

availableavailable Clausius definition of entropy is:Clausius definition of entropy is:

Change in entropy Change in entropy = (heat supplied)/temperature= (heat supplied)/temperature

sys

qS

T

Work and gas expansionWork and gas expansion

Work associated with isothermal Work associated with isothermal reversiblereversible expansion of gasexpansion of gas

Isothermal means that Isothermal means that ΔΔE = 0 = qE = 0 = qrevrev + w + wrevrev

But...But...

Equivalent to result obtained from consideration of Equivalent to result obtained from consideration of arrangementsarrangements

2

1

lnrev

Vw nRT

V

2

1

lnrev rev

Vq w nRT

V

2

1 2

1

ln

lnrevsys

VnRT

Vq VS nR

T T V

Les Regles du JeuLes Regles du Jeu(Rules of the game)(Rules of the game)

Thermodynamics is the LawThermodynamics is the Law First Law: The total energy of a system and its First Law: The total energy of a system and its

surroundings is constant in any processsurroundings is constant in any process

Second Law: In any Second Law: In any spontaneousspontaneous process, the total process, the total entropy of a system and its surroundings increasesentropy of a system and its surroundings increases

sysE q w

0totS

Third Law of ThermodynamicsThird Law of Thermodynamics

The entropy of a perfectly ordered The entropy of a perfectly ordered crystalline substance at 0K is zerocrystalline substance at 0K is zero Increasing T causes increase in entropy through Increasing T causes increase in entropy through

molecular motion (rotational, vibrational and molecular motion (rotational, vibrational and translational), and changes of statetranslational), and changes of state

Entropy and temperatureEntropy and temperature

Disorder and motionDisorder and motion Greater motion corresponds to greater number of Greater motion corresponds to greater number of

microstates – entropy increases with Tmicrostates – entropy increases with T

Entropy of a system increases with TEntropy of a system increases with T

Increasing T increases entropy through Increasing T increases entropy through greater molecular motiongreater molecular motion In a solid an increased number of vibrational In a solid an increased number of vibrational

energy states – more ways to distribute energyenergy states – more ways to distribute energy

Phase changes cause step change because Phase changes cause step change because of increased number of microstates in less of increased number of microstates in less condensed phasecondensed phase

Standard molar entropyStandard molar entropy

S° The entropy of one mole of the pure S° The entropy of one mole of the pure substance at 1 atm pressure and a specified substance at 1 atm pressure and a specified temperature, usually 25°C temperature, usually 25°C Determined experimentally from heat capacity Determined experimentally from heat capacity

measurementsmeasurements

Comparison of different substancesComparison of different substances

Gases have highest valuesGases have highest values Solids have the lowest valuesSolids have the lowest values

Standard entropy of reactionStandard entropy of reaction

In the reaction NIn the reaction N22OO44 = 2NO = 2NO22

Products have more particles than reactantsProducts have more particles than reactants Predicting entropy change from chemical Predicting entropy change from chemical

equation by counting particlesequation by counting particles

oreactantsSSS o

productso

oN 422

2 OoNO

o SSS

Entropy: connecting the microscopic Entropy: connecting the microscopic to the macroscopicto the macroscopic

Microscopic:Microscopic: Measure of microstates and disorderMeasure of microstates and disorder

Macroscopic:Macroscopic: Indicator of spontaneous processIndicator of spontaneous process

Three resultsThree results

SStotaltotal > 0 the process is spontaneous > 0 the process is spontaneous

SStotal total < 0 the process is nonspontaneous < 0 the process is nonspontaneous

SStotaltotal = 0 the process is at equilibrium = 0 the process is at equilibrium

surrsystot SSS

SurroundingsSurroundings

Entropy change for the system is obtained Entropy change for the system is obtained from the entropies of the initial and final from the entropies of the initial and final statesstates

What about the surroundings?What about the surroundings? At constant pressure, the entropy change in At constant pressure, the entropy change in

the surroundings is related to the enthalpy the surroundings is related to the enthalpy change for the systemchange for the system

THSsurr

Enthalpy change of system determines Enthalpy change of system determines entropy change of surroundingsentropy change of surroundings

Heat released by the Heat released by the systemsystem increases the increases the disorder of the disorder of the surroundingssurroundings. .

The effect of this is modulated by the The effect of this is modulated by the temperature: temperature: At low temperature the effect is much more At low temperature the effect is much more

significantsignificant At high temperature, where there is already At high temperature, where there is already

considerable disorder, the effect is muted – the considerable disorder, the effect is muted – the difference between tossing a rock into a calm difference between tossing a rock into a calm pool (low T) and a storm-tossed ocean (high T)pool (low T) and a storm-tossed ocean (high T)

Land of the Free... EnergyLand of the Free... Energy

Since Since SStotaltotal is obtained from the is obtained from the SSsystemsystem and and

the the ΔΔHHsystemsystem, everything can be written in , everything can be written in

terms of the system:terms of the system: Gibbs free energyGibbs free energy

State function, depends only on initial and final State function, depends only on initial and final statesstates

STHG

TSHG

Significance of Significance of ΔΔGG

But from before,But from before,

So…So…

surrsystot SSS

T

HSS syssystot

syssystot HSTST

GST tot

ΔΔG is indicator of spontaneityG is indicator of spontaneity

1.1. ΔΔG < 0 reaction always spontaneousG < 0 reaction always spontaneous

2.2. ΔΔG > 0 reaction always nonspontaneousG > 0 reaction always nonspontaneous

3.3. ΔΔG = 0 reaction at equilibriumG = 0 reaction at equilibrium

The Gibbs Free Energy is a measure of the The Gibbs Free Energy is a measure of the total entropy changetotal entropy change

totSTG

Four possible conditionsFour possible conditions

ΔΔHH ΔΔSS ΔΔGG SpontaneitySpontaneity ExampleExample

-- ++ --Spontaneous at all TSpontaneous at all T 2NO2NO22 = N = N22 + 2O + 2O22

-- --- Or - Or

++

Spontaneous at low T, Spontaneous at low T, nonspontaneous at high Tnonspontaneous at high T

NN22 + 3H + 3H22 = 2NH = 2NH33

++ -- ++Nonspontaneous at all TNonspontaneous at all T 3 O3 O22 = 2 O = 2 O33

++ ++- Or - Or

++

Spontaneous at high T, Spontaneous at high T, nonspontaneous at low Tnonspontaneous at low T

2HgO = Hg + O2HgO = Hg + O22

Standard free energiesStandard free energies

Solids liquids and gases in pure form at 1atm Solids liquids and gases in pure form at 1atm pressurepressure

Solution at 1 M concentrationSolution at 1 M concentration Standard temperature usually 25°CStandard temperature usually 25°C Standard free energy change: Standard free energy change:

ΔΔG°G°

Change in free energy that occurs when reactants in Change in free energy that occurs when reactants in standard states are converted to products in their standard states are converted to products in their standard statesstandard states

ΔΔG° as a predictor of reactionsG° as a predictor of reactions

Consider the reaction Consider the reaction

We want to calculate We want to calculate ΔΔG°G° Need Need ΔΔH° and H° and ΔΔS°S° ΔΔH° is equal to H° is equal to ΔΔH°(formation) for NHH°(formation) for NH33

ΔΔS° comes from the S° values for the reactants and S° comes from the S° values for the reactants and productproduct

ΔΔH° = 2 x -46.1 kJ/mol = -92.2 kJ/molH° = 2 x -46.1 kJ/mol = -92.2 kJ/molΔΔS° = -198.7J/mol KS° = -198.7J/mol K

ΔΔG°(25°C) = -92.2 – 298x-198.7x.001 kJ/molG°(25°C) = -92.2 – 298x-198.7x.001 kJ/mol

= -33.0 kJ/mol= -33.0 kJ/mol

)(2)(3)( 322 gNHgHgN

Standard free energy of formationStandard free energy of formation

The free energy of formation of one mole of The free energy of formation of one mole of the substance in its standard state from the the substance in its standard state from the most stable forms of the elements in their most stable forms of the elements in their standard statesstandard states

Thus the Thus the ΔΔG°G°ff for NH for NH33 is given by is given by

-33.0/2 kJ/mol = -16.5 kJ/mol-33.0/2 kJ/mol = -16.5 kJ/mol Elements in the standard state have Elements in the standard state have ΔΔG°G°ff = 0 = 0

Standard free energy of formation of Standard free energy of formation of some common compoundssome common compounds

SubstanceSubstance FormulaFormulaΔΔGGffº/º/

kJ/molkJ/molSubstanceSubstance FormulaFormula

ΔΔGGffºº//

kJ/molkJ/mol

AcetyleneAcetylene CC22HH22 209.2209.2 Nitrogen Nitrogen dioxidedioxide NONO22 51.351.3

AmmoniaAmmonia NHNH33 -16.5-16.5 WaterWater HH22OO -237.2-237.2

Carbon Carbon dioxidedioxide COCO22 -394.4-394.4 DiamondDiamond CC 2.92.9

EthyleneEthylene CC22HH44 68.168.1 GraphiteGraphite CC 00

The importance of the stateThe importance of the state

Using reactants in different states will Using reactants in different states will require modification to calculation for require modification to calculation for ΔΔG°G°

Consider graphite and diamond – two forms Consider graphite and diamond – two forms of carbon. Is it perhaps surprising that of carbon. Is it perhaps surprising that diamond is less stable than graphite?diamond is less stable than graphite?

ΔΔG°G°ff (diamond) = 2.9 kJ/mol (diamond) = 2.9 kJ/mol Free energy change for conversion of Free energy change for conversion of

diamond into COdiamond into CO22 is larger than for is larger than for conversion of graphite into COconversion of graphite into CO22

ΔΔG°G°ff and stability and stability

Many common compounds are unstable Many common compounds are unstable with respect to their elements – NOwith respect to their elements – NO22, C, C22HH44

and Cand C22HH22

They will spontaneously decompose to the They will spontaneously decompose to the elements according to thermodynamicselements according to thermodynamics

However, they are generally regarded as stableHowever, they are generally regarded as stable Kinetic barriers prevent rapid decompositionKinetic barriers prevent rapid decomposition

Thermodynamic functions and Thermodynamic functions and chemical processes chemical processes

The foolish man builds his process on a foundation without The foolish man builds his process on a foundation without ΔΔGGffºº

ΔΔG°G°ff calculations can save a lot of unnecessary work calculations can save a lot of unnecessary work Indicates favourability of a reaction under standard Indicates favourability of a reaction under standard

conditionsconditions If unfavourable, need to either modify the conditions or If unfavourable, need to either modify the conditions or

explore alternative synthesis pathwaysexplore alternative synthesis pathways Example:Example: The formation of NO is not favoured from NThe formation of NO is not favoured from N22 and O and O22; ; but it is favoured by reaction of Obut it is favoured by reaction of O22 with NH with NH33

3

2

22

3

HN

NH

pp

pQ

Accounting for actual conditionsAccounting for actual conditions In most reactions, the reactants and products are In most reactions, the reactants and products are

not in standard statesnot in standard states

Q is the reaction quotient – similar in form to KQ is the reaction quotient – similar in form to K Pressures for gases Pressures for gases Concentrations for liquidsConcentrations for liquids

)(2)(3)( 322 gNHgHgN

QRTGG o ln

Free energy and equilibriumFree energy and equilibrium

Q Q «1, «1, ΔΔG < 0 Drives towards productsG < 0 Drives towards products Q » 1, Q » 1, ΔΔG > 0 Drives back towards reactantsG > 0 Drives back towards reactants At equilibrium, At equilibrium, ΔΔG = 0G = 0

ΔΔG° = -RT ln KG° = -RT ln K

Relationship between Relationship between ΔΔGGffº and Kº and K

ΔΔGGºº Ln KLn K KK CommentComment

< 0< 0 > 0> 0 > 1> 1Mainly productsMainly products

>0>0 < 0< 0 < 1< 1Mainly reactantsMainly reactants

= 0= 0 = 0= 0 11Even ratio of reactants and Even ratio of reactants and productsproducts