Gherman Group Meeting Gherman Group Meeting Thermodynamics and Kinetics Thermodynamics and Kinetics and Applications and Applications June 25, 2009 June 25, 2009
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
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
−=
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