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7. Chemical Equilibrium

T= const,P= const:

dGT,P=i

idni 0

equilibrium: G= min

i

idddni= 0

reaction

1J1+2J2+ lJl+mJm+ i

iJi= 0

ipositive for products

inegative for reactants

extent of reaction

ddddddni

i

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independent ofidue to mass balance

dGT,P=iidni=

iiid

dGT,P=

i

ii

d=rGd

= rG= G

T,P

equilibrium:dGT,P= 0=i

iid

dynamic equilibrium

each forward reaction step is exactly balanced by its

back reaction

equilibrium: rG=i

ii= 0

i=i+RTlnai

rG=

i i

i+RTlnai

=i

ii+

i

iRTlnai

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=rG+RT

iln

aii

rG=rG

+RTln

i

aii

reaction quotientQi

aii

rG=rG +RTlnQ

equilibrium rG= 0

equilibrium0=rG+RTlnK

rG=RTlnK

KQeq=iaii,eq equilibrium constant

rG=RTlnK+RTlnQ=RT ln

Q

K

ifQ< K, then rG< 0: forward reaction dominates:

reactants products

ifQ> K, then rG> 0: back reaction dominates:

products reactants

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[Figure:G vs; Atkins 9th ed., Fig. 6.1]

activities

ideal gas ai=

Pi

P

real gas ai= fi

P=

iPi

P

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pure solids and liquids (standard state: pure sub-

stance at1bar): ai= 1

justification (T= const):

RTlnai=i=ii =

PPVidPViP 0

= lnai 0 =ai 1

compared to gases, G for liquids and solids is rather

insensitive to pressure changes, sinceV(s/l)V(g)

[Figure:G vsPfor gases, liquids, and solids; Atkins 9th ed., Fig. 3.21]

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as long asVi andPare not too large, ai(s/l) 1

example: solid with molecular mass200uand density2.00g cm3, i.e.,Vi= 100cm

3mol1, at T= 300Khas

ai= 1.08 1atP= 20bar

= in heterogeneous equilibria involving gases and

solids or liquids: ai(s/l)= 1

solutes:

ai= ici

c ci

c, c = 1M

ideal gas reactions

K=i

aii =

i

Pi

P

i=KP

rG=RTlnKP

Pi= xiP

K=KP=i

xiP

P

i=i

xii

i

P

P

i

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K=KxP

P

, =i

i

ci=ni

V= Pi

RT, Pi=RTci

Pi

P=

RTci

P =

RTcic

Pc =

RTc

P ci

c

K=KP

=i PiP

i

=iRTc

P ci

ci

=i

ci

c

ii

RTc

P

i

K=Kc

RTc

P

real gas reactions

K=iaii =

i

fi

Pi=i

iPi

Pi=iii

i

Pi

Pi

K=KKP

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Kxdepends onP if= 0

>0: Kx

as P

: increasing pressure favors re-actants

< 0: Kx as P: increasing pressure favors

products

illustration of

Le Chateliers principle: When a system at equilibri-um is perturbed, it responds in a way that tends to

minimize the effect of the perturbation.

equilibrium constantK doesdepend onT

KT

P

= T

exprG

RT

P

= exp

rG

RT

1

R

d(rG

/T)

dT

dK

dT=

K

R

d(rG/T)

dT

1

K

dK

dT =

1

R

d(rG/T)

dT

dlnK

dT=

1

R

d(rG/T)

dT

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dlnK

dT=

1

R

(rH)

T2

Gibbs-Helmholtz equation

dlnK

dT=rH

RT2 vant Hoff equation

another illustration of Le Chateliers principle

rH < 0 (exothermic reaction under standard condi-

tions) = K as T: increasing temperature fa-

vors reactants

rH> 0 (endothermic reaction under standard con-

ditions) =K asT: increasing temperature fa-

vors products

integrate vant Hoff equation with the assumption

that rH= const:

lnK2

K1=

rH

R1

T2

1

T1

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[Figure: lnK vs1/T; Atkins 9th ed., Fig. 6.9]

PChem I 7.11