Lecture 4: Equilibrium28-09-2010

• Lecture plan:– equilibrium

• equilibrium and Gibbs free energy• description of equilibrium• response of equilibrium to conditions (P, T, pH)

– equilibrium electrochemistry• representing redox reactions in terms of half-reactions• electrochemical cells• the Nernst equation• standard potentials and electrode calibration

– problems

EQUILIBRIUM

Chemical Equilibrium

• Chemical reaction tend to move towards a dynamic equilibrium in which both reactants and products are present but have no tendency to undergo net change

A B C D⎯⎯→+ +←⎯⎯

The question: How to predict the composition of mixture at various condition

The Gibbs energy minimum• Spontaneous change at const P and T happens towards lower values of

the Gibbs energy• Let’s consider reaction A B⎯⎯→←⎯⎯

A

B

dn ddn d

ξξ

= −= +

If some amount of A changed into B: dξextent of the reaction

Reaction Gibbs energy (definition):,

rP T

GGξ

⎛ ⎞∂Δ = ⎜ ⎟∂⎝ ⎠

( )A A B B A B B AdG dn dn d d dμ μ μ ξ μ ξ μ μ ξ= + = − + = −

,r B A

P T

GG μ μξ

⎛ ⎞∂Δ = = −⎜ ⎟∂⎝ ⎠

At equilibrium 0rGΔ =

Difference between chemical potentials of the products and the reactants at the composition fo the reaction mixture

The Gibbs energy minimum

• Spontaneity reaction at const P, T

0rGΔ <

0rGΔ =

0rGΔ >

Forward reaction is spontaneous, reaction exergonic (work-producing)

Reverse reaction is spontaneous, reaction endergonic i.e.required work to go in forward reaction

Reaction at equilibrium

Exergonic reaction,e.g. glucose oxidation Endergonic reaction,

e.g. protein synthesis

The description of equilibrium• Perfect gas equilibrium

0 0

0

( ln ) ( ln )

ln

r B A B B A A

Br

A

G RT p RT ppG RTp

μ μ μ μΔ = − = + − + =

= Δ +

Q – reaction quotient

At equilibrium: 0

0

0 ln

ln

Br r

A

r

pG G RTp

RT K G

Δ = = Δ +

= −Δ

K- equilibrium constant

Why reaction doesn’t go till the end:

( ln ln )mix A A B BG nRT x x x xΔ = +

dimensionless: 0Ap

p

The description of equilibrium

• General case of a reaction 2 3A B C D+ ⎯⎯→ +

0 3 2C D A B= + − −

0 lnr rG G RT QΔ = Δ +0 0 0

products reactantsr f fG G Gν νΔ = Δ − Δ∑ ∑

activitiesof productsactivitiesof reactants

Q =jv

jj

Q a=∏

3

2C D

A B

a aQa a

=For example, for the reaction above:

standard Gibbs free energies of formation

The description of equilibrium

0ln rRT K G= −Δ

jvj

j equilibrium

K a⎛ ⎞

= ⎜ ⎟⎝ ⎠∏At equilibrium:

Example: Find degree of dissociation of water vapour at 2300K and 1 bar if standard Gibbs energy for decomposition is 118 kJ/mol

2 2 21( ) ( ) ( )2

H O g H g O g⎯⎯→ +←⎯⎯

0 33118*10ln 2.08*10

8.3*2300GK KRTΔ

= − = =

2 2

2

12 3 2 1 2

1 2(1 )(2 )H O

H O

p p pKp

αα α

= =− + 0.0205α =

The description of equilibrium0ln rRT K G= −Δ

0 0 0/ /r r rG RT H RT S RK e e e−Δ −Δ Δ= =Increase with reaction entropy

decrease with reaction enthalpy

Boltzmann distribution interpretation:

/

/

i

i

E kT

i E kT

i

ep Ne

−

−=∑

Due to higher density of energy levels (i.e. higher S), B is dominant at equilibrium

The description of equilibrium

• Relation between equilibrium constants

• Using biological standard state

C D C D C Db

A B A B A B

a a b bK K Ka a b b γ

γ γγ γ

= = × =

At low concentration: bK K≈

2( ) ( ) ( ) ( )NADH aq H aq NAD aq H g+ ++ ⎯⎯→ +

If a biological reaction involves H+ ions, we have to take into account that standard biological condition is at log 7

HpH a += − =

0

3

7 ln10

21.8 / 16.1 8.3 10 / 310 19.7 /r rG G RT

kJ mol kJ K mol K kJ mol

⊕

−

Δ = Δ + × =

= − + × × × =

The response of equilibria to the conditions• Equilibria will respond to temperature, pressure and

concentration changes

Pressure dependence:

0T

KP

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠

Depends on standard (standard pressure)

0rGΔ

• Pressure increase by injecting inert gas: no change as partial pressures of reactants and products stay the same .• Pressure increase by compression: system will adjust partial pressures so the constant stays the same.

2

0( ) 2 ( ) B

A

pA g B g Kp p

⎯⎯→ =←⎯⎯

0ln rRT K G= −Δ

The response of equilibria to the conditions• Le Chatelier principle:

A system at equilibrium, when subjected to disturbance responds in a way that tends to minimize the effect of disturbance

2

0( ) 2 ( ) B

A

pA g B g Kp p

⎯⎯→ =←⎯⎯(1 )nα−

Extent of dissociation, α:

2 nα

Mole fractions at equilibrium:

2 2 2 2

2

½

0

(1 ) 1 2(1 ) 2 1 1

41

11 4

A B

B B

A A

nn n

p p pKp p

p Kp

α α ακ κα α α ακ ακ α

α

− −= = =

− + + +

= = =−

⎛ ⎞= ⎜ ⎟+⎝ ⎠

The response of equilibria to the conditions• Temperature response

Equilibrium will shift in endothermic direction if temperature is increased and in exothermic direction if temperature is lowered.

Van’t Hoff equation:

0ln rRT K G= −Δ0 0 0

2

( / ) ( / )ln 1 r r rd G T d G T Hd KdT R dT dT T

Δ Δ Δ= − = −

Gibbs-Helmholtz equation

0 0

2

ln ln(1/ )

r rH Hd K d KdT RT d T R

Δ Δ= = −

i.e. for exothermic reaction:

0 ln0 0rd KH

dTΔ < <

0

2 12 1

1 1ln ln r HK KR T T

⎛ ⎞Δ− = − −⎜ ⎟

⎝ ⎠So, we can predict the equilibrium constant at another temperature:

The response of equilibria to the conditions

• Boltzmann distribution interpretation

The response of equilibria to the conditions

• Noncalorimetric measuring reaction enthalpy

0

slope: r HR

Δ

0ln(1/ )

r Hd Kd T R

Δ=

The response of equilibria to the conditions

• Value of K at different temperatures

0ln(1/ )

r Hd Kd T R

Δ=

2

1

1/ 00

2 12 11/

1 1 1ln ln (1/ )T

rr

T

HK K H d TR R T T

⎛ ⎞Δ− = Δ = −⎜ ⎟

⎝ ⎠∫

Equilibria and pH

3log

H OpH a += −

at low concentration equal to molarity

3

32

2 3

142

2 ( ) ( ) ( )

10 298H O OHw H O OH

H O

H O l H O aq OH aqa a

K a a at Ka

+ −

+ −

+ −

−

⎯⎯→ +←⎯⎯

= = =

• Dissociation of water (autoprotolysis)

Ionic dissociation constant of water

For pure water:3

710H O OH

a a+ −−= =

The response of equilibria to pH

Acidity constant, Ka:

32 3( ) ( ) ( ) ( ) H O A

aHA

a aHA aq H O l H O aq A aq K

a+ −+ −⎯⎯→+ + =←⎯⎯ loga apK K= −

Basicity constant, Kb:

2( ) ( ) ( ) ( ) HB OHb

B

a aB aq H O l HB aq OH aq K

a+ −+ −⎯⎯→+ + =←⎯⎯ logb bpK K= −

• Arrhenius acid: increases concentration of H3O+ in solutionArrhenius base: increases concentration of OH- in solution

3 2 4

( ) ( ) ( )

( ) ( ) ( ) ( )

NaOH aq Na aq OH aq

NH aq H O l NH aq OH aq

+ −

+ −

⎯⎯→ +←⎯⎯⎯⎯→+ +←⎯⎯

Can be done via donation of OH- or removing of H+.

Conjugate base

Conjugate acid

Example: dissociation of formic acid

Answer: 33[ ] 1.22 10

2.91H OpH

+ −= ×

=

• Example: pK of formic acid is 3.77 at 298K. What is pH of 0.01M solution?What would happen if it were strong acid?

2 3( )HCOOH H O l HCOO H O− +⎯⎯→+ +←⎯⎯2 4

2b b acx

a±− ± −

=

[ ]3 41.695 10a

H O HCOOK

HCOOH

+ −−

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= = × 2 4 61.695 10 1.695 10 0x x− −+ × − × = 31.22 10x −= ×

• What would be dissociation degree at pH=4 and pH=10?

Equilibrium electrochemistry

• Any redox reaction can be expressed as difference of two half-reactions, which are conceptual reactions showing gain of electrons

Ox Redeν −+ →

2 2( ) ( ) ( ) ( )Zn s Cu aq Zn aq Cu s+ ++ ⎯⎯→ +

2( ) ( ) 2Zn s Zn aq e+⎯⎯→ +2 ( ) 2 ( )Cu aq e Cu s+ + ⎯⎯→

+

Equilibrium electrochemistry

1 1Ox Redeν −+ →

The electrode where oxidation occurs is called anode, the electrode where reduction occurs is called cathode.

Electrolyte

• Two half-reactions will run in the opposite directions in two half cells

Cathode

2 2Red Ox eν −→ +Anode

Electrochemical cells

4 4( ) | ( ) ( ) | ( )Zn s ZnSO aq CuSO aq Cu sM 4 4( ) | ( ) || ( ) | ( )Zn s ZnSO aq CuSO aq Cu s

Notation:

Liquid junction Liquid junction potential assumed eliminated

Phase boundary

Types of half-cells

• Metal in a solution of it’s ions

• Gas in contact with a solution of it’s ions

• Two different oxidation states of the same species

• Metal in contact with its insoluble salt

( ) ( )AgCl s e Ag s Cl− −+ → +

3 2Fe e Fe+ − ++ →22 2H e H+ −+ →

2 2Zn e Zn+ −+ →

The Nernst equation• A cell where overall cell reaction hasn’t reached chemical

equilibrium can do electrical work as the reaction drives electrons through an external circuit

,maxe r

r

w GFE Gν

= Δ

− = ΔFaraday constant AF eN=

rGEFν

Δ= −

Cell emf:

00ln lnrG RT RTE Q E Q

F F Fν ν νΔ

= − − = −

• As there is no potential difference at equilibrium:0

ln FEKRT

ν=

Standard potentials and SHE• Although it’s not possible to measure potentials of electrodes separately,

we can define a particular electrode as having zero potential at all temperatures.

Standard Hydrogen Electrode (SHE) at aH+=1 (pH=0) and p=1bar

02( ) | ( ) | ( ) 0Pt s H g H aq E+ =

22 2H e H+ −+

Standard potentials and SHE

2

01/ 2

0 0 2 2

( / , ) ln

ln ln ln

H Cl

H

H Cl

a aRTE E AgCl Ag ClF a

RT RT RTE E a a E bF F F

γ

+ −

+ −

−

±

= −

= − = − −

• All other electrodes can be calibrated using SHE:(“Harned cell” – calibration of Ag/AgCl electrode)

2

2

( ) | ( ) | ( ) | ( ) | ( ) | ( )1 ( ) ( ) ( ) ( )2

Pt s H g H aq HCl aq AgCl s Ag s

H g AgCl s HCl aq Ag s

+

+ → +

1202 lnRTE b E Cb

F+ = +For experimental calibration:

Standard efm can be found from the offset

Electrochemical series• Cell emfs are convenient source for data on equilibrium constants, Gibbs

energies etc.

0 0 01 1 2 2 2 1Red ,Ox ||Red ,Ox E E E= −

Red1 has thermodynamic tendency to reduce Ox2 if: 0 02 1E E>

low reduces high

Species selective electrodes

• Ion-selective electrode is an electrode that generates a potential in response to the presence of a solution of specific ions

Determination of thermodynamic functions by emf

• By measuring emf Gibbs energy can be determined:0 0

rG FEνΔ = −

00rSdE

dT FνΔ

=

• The temperature coefficient of standard emf gives standard entropy of the reaction:

doesn’t depend on pressure

00 0 0 0

r r rdEH G T S F E TdT

ν⎛ ⎞

Δ = Δ + Δ = − −⎜ ⎟⎝ ⎠

• and therefore provides non-calorimetric way to measure enthalpy

Application: Batteries• Lead-acid rechargeable battery (inv. 1859)

2 02 4 4 2

2 04 4

4 ( ) ( ) 2 ( ) 2 1.685

( ) ( ) ( ) 2 0.356

PbO H aq SO aq e PbSO s H O E

Pb s SO aq PbSO s e E

+ − −

− −

+ + + → + =

+ → + = −

• During the charging, the reactions are reversed

• Life time is limited due to mechanical stress due to formation and dissolution of solid material

02

02 2 2 3

( ) 2 ( ) ( ) 2 1.1

2 ( ) 2 ( ) 2 ( ) 0.76

Zn s OH aq ZnO s H O e E

MnO s H O e Mn O s OH aq E

− −

− −

+ → + + =

+ + → + = −

• Alkaline cell:

Structure of metal-electrolyte interface• Formation of electrical double layer due to

specifically and non-specifically adsorbed ions

Measuring absolute half-cell potential• The energy diagram of the cell:

Fermi levels of two metals should be offset by the Gibbs energy

work function in solution

double layer

Measuring absolute half-cell potential• Gomer and Tryson experiment (J.Chem.Phys 66(1977), 4413):

variable DC voltage is applied to the gold-electrode capacitor with a vibrating plate, AC voltage is measured

Vdc=0

Vdc=Vm2Au

2Au M Au AuE V φ= −• Absolute half cell potential for gold air electrode can be measured

• Absolute SHE potential 0 4.73SHEE V= −

Class problems:• Atkins 6.8b: In the gas phase reaction A+B=C+2D it was found

that when 2mol A, 1mol B and 3 mol D were mixed and allowed to come to equilibrium at 25C, the mixture contained 0.79mol of C at 1 bar. Calculate mol fraction of every species at equilibrium, Kx, K and ∆rG0.

• Atkins 6.21(a) Devise cells in which the following are the reactions and calculate the standard emf in each case:(a) Zn(s) + CuSO4(aq) → ZnSO4(aq) + Cu(s)(b) 2 AgCl(s) + H2(g) → 2 HCl(aq) + 2 Ag(s)(c) 2 H2(g) + O2(g) → 2 H2O(l)

• Atkins 6.22(a) Use the Debye-Huckel limiting law and the Nernst equation to estimate the potential of the cell

Ag|AgBr(s)|KBr(aq,0.050mol/kg)||Cd(NO3)2(aq, 0.010mol/kg) |Cd

at 25°C.