Gherman Group MeetingGherman Group Meeting

Thermodynamics and KineticsThermodynamics and Kinetics and Applicationsand Applications

June 25, 2009June 25, 2009

OutlineOutline

Calculating HCalculating Hff, S, G, S, Gff

Components which contribute to HComponents which contribute to Hff, S, G, S, Gff

Calculating Calculating ΔΔHH°°, , ΔΔSS°°, , ΔΔGG°°Calculating rate constants for chemical reactions Calculating rate constants for chemical reactions

Computing pKComputing pKaa valuesvaluesComputing electron affinitiesComputing electron affinitiesComputing redox potentialsComputing redox potentials

Part 1: ThermodynamicsPart 1: Thermodynamics

Calculating HCalculating Hff

, S, G, S, Gff

HHff°° = E= ESCFSCF + ZPE + E+ ZPE + Esolvationsolvation + H+ Htranstrans + H+ Hrot rot + H+ Hvibvib + H+ Helecelec

Calculating HCalculating Hff

, S, G, S, Gff

HHff°° = E= ESCFSCF + ZPE + E+ ZPE + Esolvationsolvation + H+ Htranstrans + H+ Hrot rot + H+ Hvibvib + H+ Helecelec

HHff

°°

(0 K) = E(0 K) = ESCFSCF

+ ZPE + ZPE

Calculating HCalculating Hff

, S, G, S, Gff

HHff°° = E= ESCFSCF + ZPE + E+ ZPE + Esolvationsolvation + H+ Htranstrans + H+ Hrot rot + H+ Hvibvib + H+ Helecelec

HHff

°°

(0 K) = E(0 K) = ESCFSCF

+ ZPE + ZPE SS°° = S= Stranstrans + S+ Srot rot + S+ Svibvib + S+ Selecelec

Calculating HCalculating Hff

, S, G, S, Gff

HHff°° = E= ESCFSCF + ZPE + E+ ZPE + Esolvationsolvation + H+ Htranstrans + H+ Hrot rot + H+ Hvibvib + H+ Helecelec

HHff

°°

(0 K) = E(0 K) = ESCFSCF

+ ZPE + ZPE SS°° = S= Stranstrans + S+ Srot rot + S+ Svibvib + S+ Selecelec

GGff°° = H= Hff°° -- TSTS°°

Calculating HCalculating Hff

, S, G, S, Gff

HHff°° = E= ESCFSCF + ZPE + H+ ZPE + Htranstrans + H+ Hrot rot + H+ Hvibvib + H+ Helecelec

HHff

°°

(0 K) = E(0 K) = ESCFSCF

+ ZPE + ZPE SS°° = S= Stranstrans + S+ Srot rot + S+ Svibvib + S+ Selecelec

GGff°° = H= Hff°° -- TSTS°°

Components of enthalpy / entropyComponents of enthalpy / entropy……SCF energySCF energyZeroZero--point energypoint energymolecular translationmolecular translationmolecular rotationmolecular rotationmolecular vibrationmolecular vibrationelectronicselectronics

Energy Components (1)Energy Components (1)

ZeroZero--point energypoint energy

sum of energies for lowest vibrational level for each vibrationsum of energies for lowest vibrational level for each vibrationharmonic oscillator approximationharmonic oscillator approximation

modes 12 i

iZPE hω= ∑

Energy Components (1)Energy Components (1)

ZeroZero--point energypoint energy

sum of energies for lowest vibrational level for each vibrationsum of energies for lowest vibrational level for each vibrationharmonic oscillator approximationharmonic oscillator approximation

TranslationTranslationUUtranstrans = 3/2 RT H= 3/2 RT Htranstrans = U= Utranstrans + PV = U+ PV = Utranstrans + RT = 5/2 RT+ RT = 5/2 RT

particle in a 3particle in a 3--dimensional boxdimensional boxV in formula V in formula standard state must be definedstandard state must be defined

standard state Pstandard state P°° = 1 atm (V= 1 atm (V°° = 24.5 L at 298 K)= 24.5 L at 298 K)

modes 12 i

iZPE hω= ∑

3/ 2

2

2ln 5/ 2Btrans

A

Mk T VS Rh N

π °°

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= +⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

Energy Components (2)Energy Components (2)

RotationRotationlinear moleculeslinear molecules

2

2

8ln 1

linear linearrot rot

linear Brot

H U RT

Ik TS Rh

πσ

= =

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

Energy Components (2)Energy Components (2)

RotationRotationlinear moleculeslinear molecules

nonnon--linear moleculeslinear molecules

rigidrigid--rotor approximationrotor approximationI, II, IAA, I, IBB, I, ICC –– moments of inertiamoments of inertiaσσ –– symmetry numbersymmetry number

# of rotations that carry molecule into itself# of rotations that carry molecule into itselfdepends on point group (e.g. Cdepends on point group (e.g. C11=1, C=1, C2v2v=2, T=2, Tdd=12)=12)

2

2

8ln 1

linear linearrot rot

linear Brot

H U RT

Ik TS Rh

πσ

= =

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

3/ 22

2

32

8 3ln2

linear linearrot rot

A B Clinear Brot

H U RT

I I I k TS Rh

π πσ

= =

⎡ ⎤⎡ ⎤⎛ ⎞⎢ ⎥⎢ ⎥= +⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦⎣ ⎦

Energy Components (3)Energy Components (3)

VibrationVibration

harmonic oscillatorsharmonic oscillators

( )

( ) ( )

3 6(5)

/1

3 6(5)/

/1

1

ln 11

i B

i B

i B

Ni

vib vib hw k Ti B

Nhw k Ti

vib hw k Ti B

hH U Rk e

hS R ek T e

ω

ω

−

=

−−

=

= =−

⎡ ⎤⎢ ⎥= − −⎢ ⎥−⎣ ⎦

∑

∑

Energy Components (3)Energy Components (3)

VibrationVibration

harmonic oscillatorsharmonic oscillators

ElectronicElectronicHHelecelec = U= Uelecelec = 0= 0SSelecelec = R ln (2S+1)= R ln (2S+1)

where 2S+1 is the spin multiplicitywhere 2S+1 is the spin multiplicityS = S = ½½ * # of unpaired e* # of unpaired e--

( )

( ) ( )

3 6(5)

/1

3 6(5)/

/1

1

ln 11

i B

i B

i B

Ni

vib vib hw k Ti B

Nhw k Ti

vib hw k Ti B

hH U Rk e

hS R ek T e

ω

ω

−

=

−−

=

= =−

⎡ ⎤⎢ ⎥= − −⎢ ⎥−⎣ ⎦

∑

∑

Energy Components Energy Components --

NotesNotes

Independent of any QM calculationIndependent of any QM calculationtranslational, electronictranslational, electronic

Depend on geometry only:Depend on geometry only:rotational (provided an accurate geometry)rotational (provided an accurate geometry)

Requiring an electronic structure calculationRequiring an electronic structure calculationEESCFSCF

ZPEZPEvibrationalvibrational

Calculating Calculating ΔΔH, H, ΔΔS, S, ΔΔGG

HHff°° = E= ESCFSCF + ZPE + E+ ZPE + Esolvationsolvation + H+ Htranstrans + H+ Hrot rot + H+ Hvibvib + H+ Helecelec

HHff

°°

(0 K) = E(0 K) = ESCFSCF

+ ZPE + ZPE SS°° = S= Stranstrans + S+ Srot rot + S+ Svibvib + S+ Selecelec

GGff°° = H= Hff°° -- TSTS°°

For a chemical reaction: A + B For a chemical reaction: A + B →→ C + DC + DΔΔHH°° = H= Hff°°(C) + H(C) + Hff°°(D) (D) -- HHff°°(A) (A) –– HHff°°(B)(B)ΔΔSS°° = S= S°°(C) + S(C) + S°°(D) (D) -- SS°°(A) (A) –– SS°°(B)(B)ΔΔGG°° = = ΔΔHH°° -- TTΔΔSS°°

Rate ConstantsRate Constants

Transition State TheoryTransition State Theory

compare to Arrhenius expressioncompare to Arrhenius expression

†GB RTk Tk eh

Δ−

=

aERTk Ae

−=

Part 2: Applications of ThermodynamicsPart 2: Applications of Thermodynamics

pKpKaa

Calculations (1)Calculations (1)Calculate pKCalculate pKaa values using a free energy cycle (Bornvalues using a free energy cycle (Born--Haber cycle):Haber cycle):

AH (g) A- (g) + H+ (g)

AH (aq) A- (aq) + H+ (aq)

ΔGg

ΔGa

ΔGsolAH ΔGsol

A- ΔGsolH+

pKpKaa

Calculations (1)Calculations (1)Calculate pKCalculate pKaa values using a free energy cycle (Bornvalues using a free energy cycle (Born--Haber cycle):Haber cycle):

AH (g) A- (g) + H+ (g)

AH (aq) A- (aq) + H+ (aq)

ΔGg

ΔGa

ΔGsolAH ΔGsol

A- ΔGsolH+

( )0.00999 @ 1 & 298

A H AHg g g g

Hg

G G G G

G hartrees P atm T K

− +

+

Δ = Δ + Δ − Δ

Δ = − = =

pKpKaa

Calculations (1)Calculations (1)Calculate pKCalculate pKaa values using a free energy cycle (Bornvalues using a free energy cycle (Born--Haber cycle):Haber cycle):

AH (g) A- (g) + H+ (g)

AH (aq) A- (aq) + H+ (aq)

ΔGg

ΔGa

ΔGsolAH ΔGsol

A- ΔGsolH+

( )

( )

0.00999 @ 1 & 298

264.0 / experimental value

A H AHg g g g

Hg

A H AHa g sol sol sol

Hsol

G G G G

G hartrees P atm T K

G G G G G

G kcal mol

− +

+

− +

+

Δ = Δ + Δ − Δ

Δ = − = =

Δ = Δ + Δ + Δ − Δ

Δ = −

pKpKaa

Calculations (2)Calculations (2)Calculate pKCalculate pKaa values using a free energy cycle (Bornvalues using a free energy cycle (Born--Haber cycle):Haber cycle):

AH (g) A- (g) + H+ (g)

AH (aq) A- (aq) + H+ (aq)

ΔGg

ΔGa

ΔGsolAH ΔGsol

A- ΔGsolH+

( ) ( )/

/log log

a

a

G RTa

G RTa a

K e

pK K e

−Δ

−Δ

=

= − = −

pKpKaa

Calculations (3)Calculations (3)AccuracyAccuracy……

depends mostly upon accuracy of the solvation energy of the ionsdepends mostly upon accuracy of the solvation energy of the ions

and upon the accuracy of and upon the accuracy of ΔΔGGgg

pKpKaa

Calculations (3)Calculations (3)AccuracyAccuracy……

depends mostly upon accuracy of the solvation energy of the ionsdepends mostly upon accuracy of the solvation energy of the ionsapproximately approximately ±±5 kcal/mol5 kcal/mol

and upon the accuracy of and upon the accuracy of ΔΔGGggapproximately approximately ±±5 kcal/mol5 kcal/mol

expected accuracy of pKexpected accuracy of pKaa values approximately values approximately ±±5 units5 units

pKpKaa

Calculations (3)Calculations (3)AccuracyAccuracy……

depends mostly upon accuracy of the solvation energy of the ionsdepends mostly upon accuracy of the solvation energy of the ionsapproximately approximately ±±5 kcal/mol5 kcal/mol

and upon the accuracy of and upon the accuracy of ΔΔGGggapproximately approximately ±±5 kcal/mol5 kcal/mol

expected accuracy of pKexpected accuracy of pKaa values approximately values approximately ±±5 units5 units

prediction of relative pKprediction of relative pKaa values can be more chemically meaningfulvalues can be more chemically meaningful

Electron AffinitiesElectron Affinities

X (g) + e- (g) X- (g)ΔGg

one-electron reduction in the gas phase

X Xg g gElectron Affinity EA G G G

−

= = −Δ = Δ − Δ

Redox Potential Calculations (1)Redox Potential Calculations (1)Calculate redox potentials using free energy cyclesCalculate redox potentials using free energy cycles……

X (g) + e- (g) X- (g)

ΔGsolo ΔGsol

oΔGo = 0

X (sol) + e- (sol) X- (sol)

ΔGg

ΔGredox

reduction potential

Redox Potential Calculations (1)Redox Potential Calculations (1)Calculate redox potentials using free energy cyclesCalculate redox potentials using free energy cycles……

X (g) + e- (g) X- (g)

ΔGsolo ΔGsol

oΔGo = 0

X (sol) + e- (sol) X- (sol)

ΔGg

ΔGredox

X (g) X+ (g) + e- (g)

ΔGsolo ΔGsol

o ΔGo = 0

X (sol) X+ (sol) + e- (sol)

ΔGg

ΔGredox

reduction potential oxidation potential

Redox Potential Calculations (1)Redox Potential Calculations (1)Calculate redox potentials using free energy cyclesCalculate redox potentials using free energy cycles……

X (g) + e- (g) X- (g)

ΔGsolo ΔGsol

oΔGo = 0

X (sol) + e- (sol) X- (sol)

ΔGg

ΔGredox

X (g) X+ (g) + e- (g)

ΔGsolo ΔGsol

o ΔGo = 0

X (sol) X+ (sol) + e- (sol)

ΔGg

ΔGredox

reduction potential oxidation potential

e.g. reduction potential case

n=number of electrons F=Faraday constant=23.060kcal/V*mol

X Xg g g

X Xredox g sol sol

redox

G G G

G G G G

GEnF

−

−

°

Δ = Δ − Δ

Δ = Δ + Δ − Δ

Δ= −

Redox Potential Calculations (2)Redox Potential Calculations (2)

Redox potentials typically reported relative to a standard electRedox potentials typically reported relative to a standard electroderode……

Redox Potential Calculations (2)Redox Potential Calculations (2)

Redox potentials typically reported relative to a standard electRedox potentials typically reported relative to a standard electroderode……

standard hydrogen electrode (SHE)standard hydrogen electrode (SHE)HH++ (aq) + e(aq) + e-- →→ ½½ HH22 (g)(g)reduction potentials: subtract 4.28 Vreduction potentials: subtract 4.28 Voxidation potentials: add 4.28 Voxidation potentials: add 4.28 V

Redox Potential Calculations (2)Redox Potential Calculations (2)

Redox potentials typically reported relative to a standard electRedox potentials typically reported relative to a standard electroderode……

standard hydrogen electrode (SHE)standard hydrogen electrode (SHE)HH++ (aq) + e(aq) + e-- →→ ½½ HH22 (g)(g)reduction potentials: subtract 4.28 Vreduction potentials: subtract 4.28 Voxidation potentials: add 4.28 Voxidation potentials: add 4.28 V

other standard electrodes based upon values relative to SHEother standard electrodes based upon values relative to SHEexample: standard calomel electrode (SCE) example: standard calomel electrode (SCE)

HgHg22

ClCl22

(s) + 2e(s) + 2e--

→→

2Hg(l) + 2Cl2Hg(l) + 2Cl--(aq) E(aq) E°°=+0.24 V (vs. SHE)=+0.24 V (vs. SHE)reduction potentials: subtract (4.28+0.24) Vreduction potentials: subtract (4.28+0.24) Voxidation potentials: add (4.28+0.24) Voxidation potentials: add (4.28+0.24) V