Chapter 6 Reaction Equilibrium in Ideal Gas Mixtures Physical Chemistry Chapter 6.

Chapter 6

Reaction Equilibrium in

Ideal Gas Mixtures

Physical Chemistry

Chapter 6

Chemical Potentials in an Ideal Gas Mixture

Chemical Potential of a Pure Ideal Gas

VdPSdTdG (4.36)

dPP

RTdPVd m

isothermal, pure ideal gas

2

1

12

1

P

PdP

PRTd

1

212 ln),(),(

P

PRTPTPT isothermal, pure ideal gas (6.1)

oo

P

PRTT ln)( pure ideal gas, Po1 bar (6.2)

dPVdTSddG mmm

(4.86)*im

PT

Gn

G,

,ii

one-phase pure substance

Physical Chemistry

Chapter 6

Chemical Potentials in an Ideal Gas Mixture

Fig. 6.1 T is fixed

P/Po

- o

01 2 3

RT

-RT

-2RT

mmm TSHG

)(TfHm

)(PfSm

0 Pas

For a pure ideal gas

Physical Chemistry

Chapter 6

Chemical Potentials in an Ideal Gas Mixture

Fig. 6.2

),(),(),,,,( **21 iiiii PTPxTxxPT

An ideal gas mixture

Ideal gas mixture at T and P

PxP ii

*iP

Pure gas mixture at

and T

membrane

(1) RTnPV tot

(2) The mixture is separated by thermally conducting rigid membrane permeable to gas i only, and PxP ii

barPo 1

ideal gas mixture (6.3)

Physical Chemistry

Chapter 6

Chemical Potentials in an Ideal Gas Mixture

Fig. 6.2

),(),(),,,,( **21 iiiii PTPxTxxPT

Ideal gas mixture at T and P

PxP ii

*iP

Pure gas mixture at

and T

membrane

ideal gas mixture(6.3)

oio

ii P

PRTT ln)(

ideal gas mixture (6.4)*barPo 1

The fundamental thermodynamic equation

No intermolecular interaction, no effect on from other gases in the mixture.

i

Physical Chemistry

Chapter 6

Chemical Potentials in an Ideal Gas Mixture

Fig. 6.1 (modified)

Pi/Po

i - i o

01 2 3

RT

-RT

-2RT

)()( , TGT oim

oi

)(Tfoi

barPo 1

For an ideal gas mixture

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

For the ideal-gas reaction

dDcCbBaA (4.36)

the equilibrium condition i ii 0

DCBA dcba 0 BADC badc

oio

ii P

PRTT ln)( (6.4)*

0lnln

lnln

oBo

BoAo

A

oDo

DoCo

C

P

PbRTb

P

PaRTa

P

PdRTd

P

PcRTc

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

(6.5))lnlnlnln(oB

oA

oD

oC

oB

oA

oD

oC

P

Pb

P

Pa

P

Pd

P

PcRT

badc

0lnln

lnln

oBo

BoAo

A

oDo

DoCo

C

P

PbRTb

P

PaRTa

P

PdRTd

P

PcRTc

oB

oA

oD

oCi

i

oii

oiTmi

oT badcTGG )(,,

b

oB

a

oA

d

oD

c

oCo

P

P

P

P

P

P

P

PRTG lnlnlnln

Physical Chemistry

Chapter 6

the equilibrium condition

Ideal Gas Reaction Equilibrium

(6.6)

(6.7)

)lnlnln(lnb

oB

a

oA

d

oD

c

oCo

P

P

P

P

P

P

P

PRTG

b

oB

a

oA

d

oD

c

oC

o

PP

PP

PP

PP

RTG ln

b

oB

a

oA

d

oD

c

oC

oP

PP

PP

PP

PP

K

barPo 1

Physical Chemistry

Chapter 6

oP

o KRTG ln

Ideal Gas Reaction Equilibrium

oP

o KRTG ln

(6.7)b

oB

a

oA

d

oD

c

oC

oP

PP

PP

PP

PP

K

barPo 1

0]ln[ ,

o

eqioi

ii

iii P

PRT

0ln)( ,

io

eqioi

ii P

PRTT (6.8)

i i iii ii i iii accaandbaba )(

Physical Chemistry

Chapter 6

0i

iithe equilibrium condition

Ideal Gas Reaction Equilibrium

(6.9))(,, TGG oi

ii

oiTm

ii

oT

(6.10)

n

iinn

n

ii aaaaaaaa

12121

1

ln)ln(lnlnlnln

oiTm

oi GT ,,)(

io

eqi

o

eqi

ii

oT

i

P

PRT

P

PRTG

,, lnln

Physical Chemistry

Chapter 6

n

ini aaaa

121 (6.11)*

Ideal Gas Reaction Equilibrium

(6.12)

in

io

eqioT P

PRTG

1

,ln

n

ini aaaa

121 (6.11)*

i

io

eqioP P

PK

,

oP

oT KRTG ln

(6.13)*

(6.14)*

RT

GoP

o

eK

(6.15)

Physical Chemistry

Chapter 6

n

ii

n

ii aa

11

lnln

i

i

o

eqioT P

PRTG

,ln (6.10)

Ideal Gas Reaction Equilibrium

RT

GoP

o

eK

(6.15)

(6.9))(,, TGG oi

ii

oiTm

ii

oT

Since depends only on T, for a given ideal-gas reaction is a function of T only.

oTG o

PK

)()( TKKTGG oP

oP

oo

At a given temperature, is constant for a given reaction.oPK

the standard equilibrium constant

the standard pressure equilibrium constant

oPK

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

i

eqiPiPK

, (6.19)

dDcCbBaA

Since is dimensionless, the standard equilibrium constant is dimensionless.

oi PP

oPK

has dimensions of pressure raised to the change in mole numbers for the reaction as written.

PK

the equilibrium constant

the pressure equilibrium constantPK

i

io

eqioP P

PK

,

(6.13)*

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

badf

o

o

boeqB

aoeqA

doeqD

foeqF

boeqB

aoeqA

doeqD

foeqFo

P

P

RTc

cccc

cccc

PRTcPRTc

PRTcPRTcK

)/()/(

)/()/(

)/()/(

)/()/(

,,

,,

,,

,,

(6.23)

dDfFbBaA i

io

eqioP P

PK

,

(6.13)*

the molar concentrationV

nc i

i (6.21)*

ideal gas mixtureRTcV

RTnP i

ii (6.22)

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

badf

o

o

boeqB

aoeqA

doeqD

foeqF

boeqB

aoeqA

doeqD

foeqFo

P

P

RTc

cccc

cccc

PRTcPRTc

PRTcPRTcK

)/()/(

)/()/(

)/()/(

)/()/(

,,

,,

,,

,,

(6.23)

dDfFbBaA

n/mol

same dimension as Po3/1/1 dmmollitermolco

i

io

eqioP c

cK

,

(6.24)

i

io

eqioP P

PK

,

(6.13)*

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

i

io

eqioP c

cK

,

(6.24)

i

io

eqioP P

PK

,

(6.13)*

the standard equilibrium constant

the concentration equilibrium constant

ocK

the standard equilibrium constant

the standard pressure equilibrium constant

oPK

3/1/1 dmmollitermolco

barPo 1

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

badf

o

o

boeqB

aoeqA

doeqD

foeqF

boeqB

aoeqA

doeqD

foeqFo

P

P

RTc

cccc

cccc

PRTcPRTc

PRTcPRTcK

)/()/(

)/()/(

)/()/(

)/()/(

,,

,,

,,

,,

(6.23)

moln

o

ooc

oP P

RTcKK

/

(6.25)

i

ieqix xK

, (6.26)

moln

oxoP P

PKK

/

(6.27)

Physical Chemistry

Chapter 6

van’t Hoff Equation

RT

GK

ooP

lnFrom (6.14)

dT

Gd

RTRT

G

dT

Kd oooP )(1ln

2

(6.31)

(6.9))(,, TGG oi

ii

oiTm

ii

oT

(6.32) i

oim

ii

oimi

o

dT

dGG

dT

dG

dT

d ,,)(

oim

ii

o GG ,

dT

Gd

RTdT

Td

R

G

dT

Kd oooP )(1/1ln

2

11

TTdT

d

Physical Chemistry

Chapter 6

van’t Hoff Equation

dPVdTSdG mmm From

mP

m ST

G

(6.31)

(6.33)oim

oim S

dT

dG,

,

(6.34)o

i

oimi

o SSGdT

d ,)(

(6.32) i

oim

ii

oimi

o

dT

dGG

dT

dG

dT

d ,,)(

Physical Chemistry

Chapter 6

van’t Hoff Equation

22

ln

RT

STG

RT

S

RT

G

dT

Kd oooooP

ooo STHG

(6.36)*2

ln

RT

H

dT

Kd ooP

(6.34)o

i

oimi

o SSGdT

d ,)(

(6.32) i

oim

ii

oimi

o

dT

dGG

dT

dG

dT

d ,,)(

Physical Chemistry

Chapter 6

Gibbs-Helmholtz Equation

TSHG

T

H

T

G

T

HG

T

G

P

T

GHS

T

H

T

G

T

G

P

PP T

G

TT

T

G

T

G

PPPTT

GT

G

TT

G

T

11

T

G

T

G

TT

G

T

G

TT

G

T PPP

112

Physical Chemistry

Chapter 6

Gibbs-Helmholtz Equation

T

G

T

G

TT

G

T

G

TT

G

T PPP

112

T

H

T

G

TT

P

2T

H

T

G

TP

2T

H

T

G

TP

When it is applied to changes

Physical Chemistry

Chapter 6

Differentiation of lnKPo with respect to temperature gives

van’t Hoff Equation

RT

GK

ooP

lnFrom (6.14)

(6.36)*

T

G

dT

d

RdT

Kd ooP 1ln

The differentials are complete because K and G depend only on temperature, not on pressure. Using Gibbs-Helmholtz equation

2T

H

T

G

TP

2

ln

RT

H

dT

Kd ooP

Physical Chemistry

Chapter 6

van’t Hoff Equation

2

1

2

12

)(ln

T

T

oT

T

oP dT

RT

THK

(6.36)*2

ln

RT

H

dT

Kd ooP

dTRT

HKd

ooP 2

ln

2

12

1

2 )(

)(

)(ln

T

T

o

oP

oP dT

RT

TH

TK

TK(6.37)

42 ETCTBTAH oT (6.38)

211

2 11

)(

)(ln

TTR

H

TK

TK o

oP

oP (6.39)

2112

11)(ln)(ln

TTR

HTKTK

ooP

oP

Physical Chemistry

Chapter 6

van’t Hoff Equation

(6.36)*2

ln

RT

H

dT

Kd ooP

R

H

Td

Kd ooP

1ln (6.40)

2

11

TTdT

d

dTTT

d2

11

The van’t Hoff equation is an expression for the slope of a graph of the equilibrium constant (specially, lnK) plotted against the temperature. It may be expressed in either of two ways:

Physical Chemistry

Chapter 6

Ideal Gas Reaction Equilibrium

)(2)(3)( 322 gNHgHgN

The equilibriumextent of reaction

the reaction quotient3

2

22

3

HN

NHP PP

PQ (6.41)

PP KQ The reaction goes to right

PP KQ The reaction reaches equilibrium

PP KQ The reaction goes to left

0eq

0eq0eq

i

eqiPiPK

, (6.19) the equilibrium constant

The extent of reaction

Physical Chemistry

Chapter 6

Simultaneous EquilibriaA system with several simultaneous reactions that have species in common.

(6.47)

molnmoln

molnmolnmoln

COH

COOHCH

2,1

,1,1,1

,0,0

,0,0,0

2

224

23

1,

24

2)(

toto

OHCH

HCOoP nP

P

nn

nnK (6.48)

224 3HCOOHCH

2224 42 HCOOHCH

2

2

4

2, )(

)(

24

22

toto

OHCH

HCOoP nP

P

nn

nnK

Physical Chemistry

Chapter 6

Simultaneous EquilibriaAt 600 K, CH3Cl(g) and H2O(g) react to form CH3OH, and then form (CH3)2O with a simultaneous equilibrium shown:

(1)

,6.10,00154.0 2,1, oP

oP KK

Starting with equal amount of CH3Cl and H2O, find the equilibrium amounts of all species.

)()()()( 323 gHClgOHCHgOHgClCH

)()()()(2 2233 gOHgOCHgOHCH (2)

Physical Chemistry

Chapter 6

Simultaneous EquilibriaSuppose that a system initially contains 1 mole of CH3Cl(g) and H2O(g), x moles of HCl and y moles of (CH3)2O are formed at equilibrium.

(1))()()()( 323 gHClgOHCHgOHgClCH

)()()()(2 2233 gOHgOCHgOHCH (2)

1-x 1-x+y x-2y x

x-2y y 1-x+y

6.10)2(

)1(

00154.0)1)(1(

)2(

22,

1,

yx

yxyK

yxx

xyxK

oP

oP

Physical Chemistry

Chapter 6

Simultaneous EquilibriaSuppose that a system initially contains 1 mole of CH3Cl(g) and H2O(g), x moles of HCl and y moles of (CH3)2O are formed at equilibrium.

6.10)2(

)1(

00154.0)1)(1(

)2(

22,

1,

yx

yxyK

yxx

xyxK

oP

oP

009.0

048.0

y

x

Physical Chemistry

Chapter 6

Simultaneous Equilibria

The equilibrium amounts of all species are

(1))()()()( 323 gHClgOHCHgOHgClCH

)()()()(2 2233 gOHgOCHgOHCH (2)

0.952 0.961 0.03 0.048

0.03 0.009 0.961

Physical Chemistry

Chapter 6