Thermo Chemical Reaction Equilibria

7/26/2019 Thermo Chemical Reaction Equilibria

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CHEMICAL-REACTION

EQUILIBRIA

CHAPTER

13



13.1 THE REACTION COORDINATE

The general chemical reaction :

...... 44332211 ++→++ Av Av Av Av

where | vi | is a stoichiometric coefficient and A

istands for a

chemical formula.

For vi :

Positive ( + for !roduct "egative ( # for reactants



Example :

$%4 + %

2& ' $& + 3%

2

The stoichiometric numers are :

14

−=CH v 12

−=O H v 1=COv 32= H v

The stoichiometric numer for an inert s!ecies is )ero.

*ince

ε d

v

dn

v

dn

v

dn

v

dn===== ...

4

4

3

3

2

2

1

1



The general relation etween a differential change dni in the

numer of moles of a reacting s!ecies and dε is :

dni = vi dε ( i 1 , 2 ,-. N

This variale ε called the reaction coordinate, characteri)es the

etent or degree to which a reaction has ta/en !lace.

∫ ∫ = ε

ε 0

0

d vdn i

n

n i

i

i

or

ε iii

vnn

+=0

( i 1 , 2 ,-. N



*ummation over all s!ecies ields :

∑∑∑ +==i

i

i

i

i

i vnnn ε 0

or n = n0 + vε

where ∑≡i

inn ∑≡i

inn00 ∑≡

i

ivv

Thus the mole fractions yi of the s!ecies !resent are related to ε :

ε

ε

vn

vn

n

n y

iii

i

+

+==

0

0



Example 13.2

$onsider a vessel which initiall contains onl no mol of water

va!or. f decom!osition occurs according to the reaction,

22221 O H O H +→

find e!ressions which relate the numer of moles and the mole

fraction of each chemical s!ecies to the reaction coordinate ε.

Solution 13.2

For the given reaction2

1

2

111 =++−=v

!!lication of 5s. (13.4 and(13.6 ields:



ε −= 02nn O H

ε

ε

2

10

0

2

+

−=n

n y O H

ε =2 H n

ε

ε

2

10

2

+=

n y H

ε 2

12=On

ε

ε

2

12

1

0

2

+=

n

yO

The fractional decom!osition of water va!or is:

( )00

00

0

0 2

nn

nn

n

nn O H ε ε =−−=−

Thus when no 1, ε is identified with the fractional decom!osition of

the water va!or.



∑= j

j jii d vdn ε ,

fter integration :

j j

jiii vnn ε ∑+= ,0

*umming over all s!ecies ields :

j

j i

ji

i

j

j

ji

i

i vnvnn ε ε ∑ ∑∑∑∑

+=+=

,0,0

For the mole fraction:

∑

∑

+

+=

j j j

j

j jii

ivn

vn

yε

ε

0

,0

( i 1 , 2 ,-. N



13.2 APPLICATION OF

EQUILIBRIUM CRITERIA TO

CHEMICAL REACTIONS t the e5uilirium state :

( dGt )T , P

0

The total 7is energ Gt is a minimum

ts differential is )ero.



13.3 THE STANDARD GIBBS-ENERGY

CHANGE

AND THE EQUILIBRIUM CONSTANT The fundamental !ro!ert relation for single#!hase sstems,

!rovides an e!ression for the total differential of the 7is

energ: ( ) ( ) ( ) ∑+−=i

iidndT S dP nV nGd µ

f changes in mole numers ni occur as the result of a single

chemical reaction in a closed sstem , then each dni ma e

re!laced the !roduct vi dε .

( ) ( ) ( )

∑+−=

i

ii d vdT S dP nV nGd ε µ

*ince nG is a state function , the right side of this an eact

differential e!ression :

( ) ( )

P T

t

P T i

ii

GnGv

,,

∂

∂=

∂

∂=∑

ε ε

µ



For chemical#reaction e5uilirium :

0=∑i

iiv µ

8ecall the definition of the fugacit of a s!ecies in solution :

( ) iii f RT T 9ln+Γ= µ

For a !ure s!ecies i in its standard start at the same tem!erature :

( )

iii f RT T G ln+Γ =

The difference etween these two e5uations is :

i

iii

f

f RT G

9ln=− µ



For the e5uilirium state of a chemical reaction :

( )[ ] 09ln =+∑i

iiii f f RT Gv

( ) 09ln =+ ∑∑ i

vii

i

iii f f RT Gv

( ) RT

Gv f f i ii

i

v

ii

i ∑∏ −

=

9ln

( )∏ =i

v

ii K f f i9ln

where

∆−≡

RT

G K

e!



lternative e!ression for :

RT

G K

∆−≡ln

lso definition : ∑∆≡∆i

iGG

K is a function of tem!erature onl .

In spite of its dependence on temperature , K is called the

equilibrium constant for the reaction ;

∑i iiGv

, represented by ΔG˚ , is called the

standard Gibbs-energy change of reaction ,



13.4 EFFECT OF TEMPERATURE

ON THE EQUILIBRIUM CONSTANT

The de!endence of ;G< on T :

( )2 RT

H

dT

RT Gd ∆−=

∆

Then ecome:

2

ln

RT

H

dT

K d ∆=

5uation aove gives the effect of tem!erature on e5uilirium

constant and hence on the e5uilirium conversion.



=hen ; H <

negative eothermic reaction

!ositive endothermic reaction

f ; H < is assumed inde!endent of T, then:

′−

∆

−=′ T T R

H

K

K 11ln

This a!!roimate e5uation im!lies that a !lot of ln K vs the

reci!rocal of asolute tem!erature is a straight line. Figure

13.2 in tetoo/, shows a !lot of ln K vs 1>T for a numer of

common reactions, illustrates this near linearit.



The rigorous develo!ment of the effect of tem!erature on

the e5uilirium constant is ased on the definition of the

7is energ , written for a chemical s!ecies in its standard

state:

G˚ i = H˚

i TS˚

i

?ulti!lication vi and summation over all s!ecies gives:

∑∑∑ −=i

ii

i

ii

i

ii S vT H vGv



s a result of the definition of standard !ro!ert change of

reaction, this reduces to:

;G˚=; H˚ T ;S˚

=here the standard heat of reaction and standard entro!

change is related to tem!erature:

dT RC R H H

T

T P

∫ ∆+∆=∆0

0

∫ ∆+∆=∆

T

T

P

T

dT

R

C RS S

00



%owever

0

000

T

G H S

∆−∆=∆

whence

∫ ∫ ∆

−∆

+∆

+∆−∆

=∆ T

T

P T

T

P

o

T

dT

R

C dT

R

C

T RT

H

RT

H G

RT

G

00

10

0

00

and

RT

G K

∆−=ln



The !receding e5uation ma e reorgani)ed so as to factor

into three terms , each re!resenting a asic contriution

to its value :

K = K 0 K

! K

"

K 0 re!resents the e5uilirium constant at reference

tem!erature T 0:

∆−≡

0

00

RT G K

−∆

=T

T

RT

H K 0

0

01 1e!



∆−

∆−= ∫ ∫

T

T

P T

T

P

T

dT

R

C dT

R

C

T K

00

1e!2

with heat ca!acities given :

( ) ( )( ) ( )

−∆

+

+−

∆+

−

∆+

−

−∆= 2

2

20

22

0

2

02

1

2

121

@

11

2

11

lne! τ

τ

τ

τ τ

τ

τ

τ

τ τ

T

#

CT $T A K



13.5 EVALUATION OF EQUILIBRIUM

CONSTANTS

Example 13.4

$alculate the e5uilirium constant for the va!or#!hase

hdration of ethlene at 146 and at 320A$ from data given in!!. $.

Solution 13.4

First determine values for ;, ;B, ;$, and ;C for the

reaction:

C " H

%(&) + H

"O (&) ' C

" H

OH (&)



The meaning of ; is indicated : (C " H

OH # (C

" H

% #

( H "O. Thus, from the heat#ca!acit data of Tale $. 1:

; 3.61D # 1.424 # 3.4E0 # 1.3E@

;B (20.001 # 14.34 # 1.460 10#3 4.16E l0#3

;$ (#@.002 + 4.32 # 0.000 10#@ #1.@10 10#@

;C (#0.000 # 0.000 # 0.121 106 #0.121 106



Galues of ; H"*

and ;G"*

at 2D.16 for the hdration reaction are

found from the heat#of#formation and 7is#energ#of#formation data

from Tale $.4:

; H"*

#236,100 # 62,610 # (#241,D1D #46,E2 H mol#1

;G"*

#1@D,40 # @D,4@0 # (#22D,6E2 #D,3ED H mol#1

For T 146 + 2E3.16 41D.16 , values of the integrals in 5. (13.1D

are:

C$P%(2D.16,41D.16I#1.3E@,4.16E#3,#1.@10#@,#0.121 +6 #23.121

C$P*(2D.16,41D.16I#1.3E@,4.16E#3,#1.@10#@,#0.121 +6 #0.0@24



*ustitution of values into 5. (13.1D for a reference tem!erature of

2D.16 gives:

( )( ) ( )( ) [email protected]@F24.016.41D

121.23

16.41D314.D

EF2,46

16.2FD314.D

EF2,463ED,D41D

=+

−

+

−

+

+−

=

∆ RT

G

For T 320 + 2E3.16 63.16 ,

C$P%(2D.16,63.16I#1.3E@,4.16E#3,#1.@10#@,#0.121 +6

22.@32

C$P*(2D.16,63.16I#1.3E@,4.16E#3,#1.@10#@,#0.121 +6 0.01E31

=hence,

( ) ( ) ( ) ( )

[email protected]

16.6F3

@32.22

16.6F3314.D

EF2,46

16.2FD314.D

EF2,463ED,D6F3 =−+−+

+−=

∆

RT

G



Finall,

J41D.16: ln K #1.36@ and K 1.443 10#1

J63.16: ln K #6.D2D@ and K 2.42 10#3

!!lication of 5s. (13.21, (13.22, and (13.24 !rovides an

alternative solution to this eam!le. B 5. (13.21,

( ) ( )3@@.2

16.2D314.D

3ED,De!0 == K

( ) ( ) 4E3.1D

16.2FD314.D

EF2,46

0

0 −=−

=∆ RT

H



?oreover,

=ith these values, the following results are readil otained:

T K K K 0

K !

K "

K

2D.16 1 2.3@@ 1 1 2.3@@

41D.16 1.4026 2.3@@ 4.D610#3 0.D@0 1.44310#1

63.16 1.D4 2.3@@ 1.02310#4 0.E4 2.4210#3

$learl, the influence of K !, is far greater than that of K

". This

is a t!ical result, and accounts for the fact that the lines on

Fig. 13.2 are nearl linear.



13.6 REELATION OF EQUILIBRIUM

CONSTANTS TO COMPOSITION

The standard state for gas is the ideal gas#state of the !ure

gas at the standard#state !ressure P < of 1 ar.

*ince for ideal gas : f˚ i = P˚

Thus :

P

f

f

f i

i

i99

=and K

P

f

i

v

i

i

=

∏

9

where the constant K is a function of tem!erature onl .



For a fied tem!erature the com!osition at e5uilirium must

change with !ressure in such a wa that

( ) K P f

i

v

i

i =∏ 9

The fugacit is related to the fugacit coefficient :

P y f iii φ 99 =*ustitution of this e5uation :

( ) K P

P y

v

i

v

ii

i

−

=∏ φ 9

where

∑≡i ivv and P < is the standard Lstate !ressure if 1 ar ,

e!ressed in the same

remains constant .

units used for P .



for a !ure s!ecies can e evaluated from a generali)ed

correlation once the e5uilirium T and P are s!ecified .

For !ressure sufficientl low or tem!erature sufficientl high,

the e5uilirium ehaves essentiall as an ideal gas

For ideal solution :

( ) K P

P

y

v

i

v

ii

i

−

=∏ φ

iφ

( ) K

P

P y

v

i

v

ii

−

=∏



For reaction occurring in li5uid !hase :

K

f

f

i

v

i

i

i

=

∏

9

*ince iiii f - f γ =9

The fugacit ratio e!resses :

==

i

iii

i

iii

i

i

f

f -

f

f -

f

f γ

γ 9

the fugacities of li5uids are wea/ function of !ressure .



For reaction that ta/e !lace in high !ressure

( ) ( )

( )

−= ∑∏

i

ii

i

v

ii V v

RT

P P K - i

e!γ

For low !ressure :

( ) K -i

v

iii =∏ γ

f the e5uilirium miture is an ideal solution , then .i

is unit :

( ) K -i

v

ii =∏

This sim!le relation is /nown as the law of mass action. *ince

li5uids often form nonideal solutions, e5uation aove can e

e!ected in man instances to ield !oor result .



For s!ecies at low concentration in a5ueous solution,

%enrMs law can e use :

iii /0 f =9where

i is a constant which de!endent on tem!erature and

/i is molalit .



13.7 EQUILIBRIUM

CONVERSIONS FOR SINGLE

REACTIONS Example 13.5

The water#gas#shift reaction, $&(g + %2&(g ' $&

2(g + %

2(g

is carried out under the different sets of conditions descried elow.

$alculate the fraction of steam reacted in each case. ssume the

miture ehaves as an ideal gas.

(aThe reactants consist of 1 mol of %2& va!or and 1 mol of $&. The

tem!erature is 1,100 and the !ressure is 1 ar.

(*ame as (a ece!t that the !ressure is 10 ar.

(*ame as (a ece!t that 2 mol of "2 is included in the reactants.



(dThe reactants are 2 mol of %2& and 1 mol of $&. &ther

conditions are the same as in (a.

(eThe reactants are 1 mol of %2& and 2 mol of $&. &ther

conditions are the same as in (a.

(fThe initial miture consists of 1 mol of %2&, 1 mol of $&,and 1 mol of $&

2. &ther conditions are the same as in (a.

(g *ame as (a ece!t that the tem!erature is 1,@60 K .



Solution 13.5

(a For the given reaction at 1,100 , 104>T .06, and

Fig. 13.2 !rovides the value. ln K 0 or K 1. For this reaction

01111 =−−+== ∑i

ivv . *ince the reaction miture is an ideal

1

2

22 == K y y

y y

O H CO

CO H

B 5. (13.6:

2

1 1CO y

ε −=

2

12

1

O H y ε −=

22

1CO y

ε =

22

1 H y

ε =

gas, 5. (13.2D a!!lies, and here ecomes:



*ustitution of these values into 5. ( gives:

( )

11

2

2

=

− 1

1

ε

ε

Therefore the fraction of the steam that reacts is 0.6.

( *ince v 0 , the increase in !ressure has no effect on the

ideal#gas reaction, and ε1 is still 0.6.

or ε1 0.6

(c The "2 does not ta/e !art in the reaction, and serves onl as a

diluent. t does increase the initial numer of moles no from 2 to 4,

and the mole fractions are all reduced a factor of 2. %owever,

5. ( is unchanged and reduces to the same e!ression as

efore. Therefore, ε1 is again 0.6.



(d n this case the mole fractions at e5uilirium are:

3

1 1

CO y

ε −

= 3

22

1

O H y

ε −

= 32

1

CO y

ε

= 32

1

H y

ε

=

and 5. ( ecomes:

( ) ( ) 121

2

=−− 11

1

ε ε

ε

The fraction of steam that reacts is then 0.@@E>2 0.333

or ε1 0.@@E



(e%ere the e!ressions for y$& and y%2& are interchanged , ut this

leaves the e5uilirium e5uation the same as in (d. Therefore ε1

0.@@E, and the fraction of steam that reacts is 0.@@E.

(f n this case 5. ( ecomes:

( )

( ) 1

1

12 =

−

+

1

11

ε

ε ε

The fraction of steam reacted is 0.333.

or ε1 0.333



(g t 1,@60 K , 104>T @.0@, and from Fig. 13.2, in #1.16

or 0.31@.

Therefore 5. ( ecomes:

( )[email protected]

1

2

=− 1

1

ε

ε

The reaction is eothermic, and conversion decreases with increasing

tem!erature.

or ε1 0.3@



13. PHASE RULE AND DUHEM!S

THEOREM FOR REACTING

SYSTEM

This is !hase rule for reacting sstems.

2 = " 3 4 + " 35

where 4 is numer of !hases , " numer of chemical s!ecies

and r is numer of inde!endent chemical reactions at

e5uilirium within the sstem .



13." MULTIREACTION EQUILIBRIA

j

i

v

i

i K f

f ji

=

∏

,9

where j is the reaction inde.

For gas !hase reaction :

j

i

v

i K P

f ji

=

∏

,9

For the e5uilirium miture is an ideal#gas,

( ) j

v

v

i K P

y j

ji

−

=∏

,

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Thermo Chemical Reaction Equilibria

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