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Project of the IK-CAPE

Model Description

(Translation by DECHEMA e.V.)

Last Update: 28.03.02

Table of contents Page

1 INTRODUCTION 1-1

2 DESCRIPTION OF THE TEMPERATURE DEPENDENCY FOR PURE COMPONENT PROPERTY DATA 2-1

2.1 Equations 2-1

2.2 Extrapolations 2-3

3 CALCULATION OF AVERAGES 3-1

4 ACTIVITY COEFFICIENTS 4-1

4.1 NRTL model 4-1

4.2 UNIQUAC model 4-1

4.3 Wilson model 4-2

4.4 Flory-Huggins model 4-2

4.5 UNIFAC model 4-2

5 POYNTING CORRECTION 5-1

6 HENRY CORRECTION 6-1

7 EQUATIONS OF STATE 7-1

7.1 Redlich-Kwong-Soave equation 7-1

7.2 Peng-Robinson equation 7-2

8 FUGACITY COEFFICIENTS 8-1

8.1 Redlich-Kwong-Soave equation 8-1

8.2 Peng-Robinson equation 8-1

8.3 Virial equation 8-1

8.4 Association in vapor phase 8-2

9 ENTHALPY 9-1

9.1 Enthalpy of liquid with starting phase in liquid 9-1

9.2 Enthalpy of the liquid with starting phase in vapor 9-1

9.3 Enthalpy of a gas with starting phase in liquid 9-2

9.4 Enthalpy of a gas with starting phase in vapor 9-2

9.5 Enthalpy of a solid component 9-2

9.6 The excess enthalpy 9-3 9.6.1 NRTL equation 9-3 9.6.2 UNIQUAC equation 9-4 9.6.3 Wilson equation 9-4 9.6.4 Flory-Huggins equation 9-5 9.6.5 Redlich-Kister equation 9-5

9.7 The isothermal pressure dependency of the enthalpy in vapor phase 9-5 9.7.1 Redlich-Kwong-Soave equation 9-5 9.7.2 Peng-Robinson equation 9-5

9.8 The enthalpy of a gas with simultaneous association 9-6

10 CHEMICAL REACTIONS 10-1

10.1 Equilibrium reactions 10-1

10.2 Kinetically controlled reactions 10-2

10.3 Reactions of general nature 10-3

1-1

1 Introduction Inside of this handbook all models included inside the IK-CAPE Thermodynamics Package are documented. Common abbreviations:

R gas constant in J/kmol/K T temperature in Kelvin P pressure in Pascal Tc critical temperature in Kelvin

Pc critical pressure in Pascal

v volume in m³/kmol xi mol fraction in liquid phase

yi mol fraction in vapor phase

γ i activity coefficients

ϕ i fugacity coefficients

2-1

2 Description of the temperature dependency for pure component property data

2.1 Equations To describe the temperature dependency of any pure component property at current stage the following functions can be used:

• POLY : Polynom

f T a a T a T a T( ) .............= + + + +0 1 22

99

• EPOL : Polynom in exponent

f T a a T a T a T( ) ....= + + + +10 0 1 22

99

• WATS : extended Watson equation

f T a a T aa( ) ( )= − +0 2 31

• ANTO : Antoine equation

ln( ( ))f T aa

T a= −

+01

2

• ANT1 : extended Antoine equation

ln ( ) lnf T aa

T aa T a T a T a= +

++ + +0

1

23 4 5

6

• KIRC : Kirchhoff equation

ln( ( )) lnf T aa

Ta T= − + ∗0

12

2-2

• SUTH : Sutherland equation

f Ta T

a

T

( ) =+

0

11

• WAGN : Wagner equation

ln ( ) ln ( ).f T aT

a a a ar

= + ∗ + + +1 2 31 5

43

561 τ τ τ τ

TT

a

T

a T

a P

r

r

c

c

=

= −==

0

0

1

1τ

• CPL : Equation for the specific heat capacity of liquids

f T a a T a T a Ta

T( ) = + + + +0 1 2

23

3 42

• ICPL : Equation for the specific heat capacity of liquids

f T a a T a T a T a Ta

T( ) = + + + + +0 1 2

23

34

4 5

• VISC : Equation for the dynamic viscosity

f T a e aa

T( ) = +0 2

1

• RACK : Rackett equation

f Ta

aT

aa

( )( )

=+ −

0

11 1

2

3

2-3

• KIR1 : extended Kirchhoff equation

ln( ( )) lnf T aa

Ta T a T a= + + +0

12 3

4

• ALYL : Aly-Lee equation

f T a a

a

Ta

T

a

a

Ta

T

( )sinh cosh

= +

+

0 1

2

2

2

3

4

4

2

• DIP4 : DIPPR function for HVAP and ST

c

r

rrr

hr

Ta

a

TT

TaTaTaah

with

TaTf

=

=

+++=

−=

0

0

35

2432

1 )1()(

• DIP5 : DIPPR function for KVAP and VISV

f Ta Ta

T

a

T

a

( ) =+ +

0

2 32

1

1

2.2 Extrapolations To avoid nonsensical results as well as program crashes, one can use the possibility to extrapolate the above given functions beyond the proofed temperature range using one of the given extrapolation methods. • Linear extrapolation

For linear extrapolation inside the f(T)-diagram applies for the case upperTT > and 0)(’ >upperTf

)(’)(’

)()()(’)(

upper

upperupperupper

TfTf

TfTTTfTf

=

+−∗=

2-4

and for the case lowerTT < and 0)(’ <lowerTf

)(’)(’

)()()(’)(

lower

lowerlowerlower

TfTf

TfTTTfTf

=+−∗=

• Extrapolation to the value B

For upperTT > and 0)(’ <upperTf applies with )(0 upperTfB <≤

[ ] BTTBTf

TfBTfTf upper

upper

upperupper +

−

−−= )(

)(

)(’exp)()(

−

−= )(

)(

)(’exp)(’)(’ upper

upper

upperupper TT

BTf

TfTfTf

For lowerTT < and 0)(’ >lowerTf gilt mit )(0 lowerTfB <≤

[ ] BTTBTf

TfBTfTf lower

lower

lowerlower +

−

−−= )(

)(

)(’exp)()(

−

−= )(

)(

)(’exp)(’)(’ lower

lower

lowerlower TT

BTf

TfTfTf

• linear extrapolation inside a logarithmic diagram

)(ln)()(ln

)(ln upperupperTT

TfTTdT

TfdTf

upper

+−

=

=

For upperTT > and 0)(’ <upperTf applies expressed in f(T)

( )

+−= )(ln

)(

)(’exp)( upperupper

upper

upper TfTTTf

TfTf

)(

)(’)()(’

upper

upper

Tf

TfTfTf =

2-5

• Extrapolation of the Wagner equation for vapor pressure

a T

a Pc

c

0

1

==

Above the critical point the Wagner equation is extrapolated linear inside the ln(p) vs. 1/T diagram

f T a e

f Ta a

Tf T

a aT a( )

’( ) ( )

=

= −

−

1

1 1

2 02

2 00

at T > Tc the vapor pressure represents a virtual value.

3-1

3 Calculation of averages

• MOLA: average based on mole fractions

∑ ⋅=i

ii valuexaverage

• MASS: average based on mass fractions

∑

∑ ∗=

iimass

iiimass

x

valuexaverage

,

,

• MOLG: logarithmic average based on mole fractions

)ln()ln( ii

i valuexaverage ∗= ∑

• MALG: logarithmic average based on mass fractions

∑

∑ ⋅=

iimass

iiimass

x

valuexaverage

,

, )ln()ln(

• LAMB: average for the heat conductivity of gaseous mixtures

λ m = ∗0 5. x xi ii

i

∗ +

∑

∑ λ

λ

1

• VISC: average for the viscosity of gaseous mixtures

ηη

m

i ii

i

i ii

x M

x M=

∗ ∗

∗

∑∑

3-2

• VOLU: average for density based on volume

ρ

ρ

mi

i

x

i

=∑

1

• WILK: average from Wilke for viscosity of gaseous mixtures

+

+

=

= ∑ ∑

j

i

i

j

j

i

ji

ij

jij

ii

Mol

Mol

Mol

Mol

value

value

F

Fy

valueyaverage

18

1

2

4

,

,

• WAMA: average from Wassiljewa,Mason,Saxena für die heat conductivity of gaseous mixtures

ityvisgas

Mol

Mol

Mol

Mol

F

Fy

valueyaverage

j

i

i

j

j

i

ji

ij

jij

ii

cos

18

1

2

4

,

,

=

+

+

=

= ∑ ∑

η

ηη

• DIST: average for the surface tension based on DIPPR

44

=

∑∑

iii

iiii

Volx

valueVolxaverage

3-3

• DIKL: average for the heat conductivity of liquids based on DIPPR

∑

∑∑

=

+=

jjj

iii

i j

ji

ji

Volx

Volxxv

valuevalue

xvxvaverage

11

2

4-1

4 Activity coefficients

4.1 NRTL model

ln, ,

,

,

,,

, ,

,

γτ

ττ

i

j i j i jj

k i kk

j i j

k j kk

ji j

l j l j ll

k j kk

G x

G x

x G

G x

G x

G x= + −

∑∑ ∑∑

∑∑

G ej iS j i j i

,, ,= − τ

τ j i j i

j i

j i j iab

Te T f T, ,

,

, ,ln= + + +

S c d Tj i j i j i, , , ( . )= + − 27315

4.2 UNIQUAC model The volume part is defined by:

Vr x

r xii i

j jj

=∑

The surface part is defined by:

Fq x

q xii i

j jj

=∑

and Fq x

q xii i

j jj

’’

’= ∑

For the activity coefficient applies:

ln ln lnγ γ γi iC

iR= +

with

ln ln lnγ iC i

ii

i

ii

i

ij

jj

V

x

KZq

F

Vl

V

xx l= + + − ∑2

and

ln ’ ln ’’

’,

,

,

γ ττ

τiR

i j j i

j i j

k k jk

jj

q FF

F= − −

∑∑∑1

( ) ( )l r q ri i i i= − − −5 1

4-2

τ j i

ab

Tc T d T

ej i

j ij i j i

,

ln,,

, ,=+ + +

4.3 Wilson model

ln ln ,,

,

γ i j i jj

k k i

j k jj

k

xx

x= −

−∑ ∑∑1 Λ

ΛΛ

Λ i j

ab

Tc T d T

ei j

i ji j i j

,

ln,,

, ,=+ + +

4.4 Flory-Huggins model

( )ln ln ,γ ϕ ϕ χi i iir r

= + − + − ∑11 1

1 1 2 1 22

ln ln , ,γ ϕ χ ϕ χ22 2

1 2 1 2 2 1 22

1kk k

k k i ii

r

r

r

rr= + − + −

∑

with

ϕ

ϕ

11

22 2

=

=

x

rr x

rii i

r x x r

T

i ii

i i i

= +

= +

∑1 2 22

1 2 1 20

1 21 1χ χ χ, , ,

ri segment number of component i

ϕ i volume fraction of component i

χ1 2, i interaction parameter

solvent(1)-solved species(2i)

4.5 UNIFAC model The volume part is defined by:

Vr

r xii

j jj

=∑

r Ri ki

k= ∑υ( )

4-3

The surface part is defined by:

Fq

q xi

i

j jj

=∑

q Qi ki

k= ∑υ( )

For the activity coefficient applies:

ln ln lnγ γ γi iC

iR= +

with the combinatorial part

ln ln lnγ ic

i i ii

i

i

i

V V qV

F

V

F= − + − − +

1 5 1

and the residual part, that describes the interactions between the different functional groups.

( )ln ln ln( ) ( )γ υiR

ki

k ki= −∑ Γ Γ

ln lnΓ Θ ΨΘ Ψ

Θ Ψk k m kmm

m km

n nmn

Q= −

−

∑ ∑∑1

Herein is Θ m the area fraction of the functional group m

Θmm m

n nn

Q X

Q X=

∑

with the mole fraction Xm of the functional group m.

X

x

xm

mj

jj

nj

jnj

=∑∑∑

υ

υ

( )

( )

The parameter Ψji contains the group interaction coefficient between the groups j and i:

Ψ j i

a

Tej i

,

,

=−

5-1

5 Poynting correction

( )ln ,F

v P P

RTp iiL

is

=−

6-1

6 Henry correction

ln

ln ,

H

x H

xi

j i jj

jj

=∑

∑

whereas means: i Index of the Henry components j Index of the non Henry components

ln ln,H ab

Tc T d Ti j = + + +

7-1

7 Equations of state

7.1 Redlich-Kwong-Soave equation The fugacity coefficients can be calculated using the following equation:

( )ln ln ln lnϕ ii

mm m

m

m m

m

m

i

m

i

m

m m

m

B

bz z

v

v b

a

b RT

as

a

B

b

v b

v= − − +

−− −

+1 2

using the following mixture rule:

a x x am ii

jj

i j= ∑ ∑ ,

as x ai j i jj

= ∑ ,

and B bs bi i m= −2

b x x bm ii

j i jj

= ∑ ∑ ,

bs x bi j i jj

= ∑ ,

Herein means:

( )a A A ki j i j i j, ,= −1

( )bb b

ki j

i j

bi j, ,=+

−2

1

The pure component parameters a, b can be received from: A T ai i i( ) = α

( )[ ]α i i rm T= + −1 12

mi i i= + −0 48 1574 0176 2. . .ω ω

aR T

Pic

c

= 0 427482 2

.

7-2

bRT

Pic

c

= 0 0867.

7.2 Peng-Robinson equation Pressure explicit representation:

PRT

v b

a T

v v b b v b=

−−

+ + −( )

( ) ( )

Normal form of the cubic equation for the volume explicit representation:

v bRT

Pv

Pb RTb a

Pv

Pb RTb ab

P3 2

2 3 23 20+ −

−

+ −+

+ −

=

Calculation of compressibility:

zPv

RTmm=

z mm

m m

m m

m m m m m m

v

v b

a v

RT v v b b v b=

−−

+ + −( ) ( )

using the following mixture rules:

a x x am i j i jji

= ∑∑ ,

b x bm i ii

= ∑

Herein means:

a A T A T ki j i j i j, ,( ) ( ) ( )= −1

The pure component parameters a, b can be received from: A T ai i i( ) = α

α i ii

mT

Tc= + −

1 1

2

mi i i= + −0 37464 154226 0 26992 2. . .ω ω

7-3

aR Tc

Pcii

i

= 0 457242 2

.

bRTc

Pcii

i

= 0 07780.

8-1

8 Fugacity coefficients

8.1 Redlich-Kwong-Soave equation

ϕ ϕϕi

i

iv=0

( )ln ln( )

lnϕ i iis

i ii

i

i i

i

zP

RTv b

A T

b RT

v b

v0 1= − − − −

+

and

( )ln ln ln lnϕ iv i

mm m

m

m m

m

m

i

m

i

m

m m

m

B

bz z

v

v b

a

b RT

as

a

B

b

v b

v= − − +

−− −

+1 2

The mixture rules can be find in the previous chapter „Equations of state“.

8.2 Peng-Robinson equation

ϕ ϕϕi

i

iv=0

with

ln ln ( )( )

ϕ i iiS

i ii

i

zP

RTv b

A T

b RT0 1

2 2= − − −

−

( )( )ln

v b

v b

i i

i i

+ +

+ −

1 2

1 2

for a pure component at saturated vapor pressure and

ln ( ) ln ( )ϕ iv i

mm m m

m

m

b

bz

P

RTv b

a

b RT= − − −

−12 2

( )( )

2 1 2

1 2ax a

b

b

v b

v bmj i j

i

mj

m m

m m

, ln−

+ +

+ −

∑

for a component inside a mixture

8.3 Virial equation

ϕ ϕϕi

i

iv=0

8-2

ϕ i

vB

i i

eB

v

i i

0

2

1=

+

,

,

vRT

P

P B

RTi

i i i= + +

2

1 14

0

0,

ϕ iv

vy B

m

eB

v

j i j

j

=+

∑2

1

,

vRT

P

PB

RTm= + +

21 1

4

B y y B

zB

v

m iji

j i j

mm

=

= +

∑∑ ,

1

with Bi i, = Pure component virial coefficient

Bi j, = Cross virial coefficient

8.4 Association in vapor phase Prerequisites:

• The vapor phase behaves ideal, temperature, pressure and concentration are given

• up to 5 components can build Di-, Tetra- and Hexamers as well as mixed dimers among each other

• any desired number of inerts is allowed Notation: "true" concentration zin

i,j ... components n association degree n=1,2,4,6 for i ≤ 5 zi2 = zi4 = zi6 = 0 for i > 5

Mixed dimers zMij i,j ≤ 5

Equilibrium constants: For each association reaction n ⋅ (i) ↔ (i)n

8-3

(i) + (j) ↔ (ij)

equilibrium constants Kz

z pinin

in n

= −1

1

Kz

z z pMijMij

i j

=1 1

are generated. For each applies: ln K AB

Tin inin= +

K K KMij i j= 2 2 2

Set of equations to calculate the true monomer concentrations:

Material balance: K z pin in n

ni1

1 1−∑∑ =

Material rate for each component i:

nK z p K z z p

nK z p K z z p

y

y

in in n

Mij i jj i jn

nn n

M j jj i jn

i

11

1 15

1 111

1 11 15

1

−

≠ ≤−

≠ ≤

+

+=

∑∑∑∑

,

,

→ Set of equations for the variable zin to be solved iterative. Afterwards all zin and zMij can be calculated.

The fugacity coefficients can be calculated afterwards as follows:

ϕ ii

i

z

y= 1

9-1

9 Enthalpy The following abbreviations are used: h i

o Reference enthalpy of the pure component h i

l Enthalpy of the liquid, component i h i

v Enthalpy of the vapor, component i r i Enthalpy of vaporization, component i cp i specific heat capacity of the ideal gas, component i

∆hh

pdp

vp Ts

∂∂

∫

0

( )

9.1 Enthalpy of liquid with starting phase in liquid The enthalpy of the liquid follows from:

h h x hlE

i il

i

= + ∑

with

h h cl dTil

i iT

T

i

= + ∫0

0,

9.2 Enthalpy of the liquid with starting phase in vapor The enthalpy of the liquid follows from:

h h x hlE

i il

i

= + ∑

If the temperature for the phase change is set equal to the systems temperature, the enthalpy of the pure component can be calculated using:

h h cp dTh

pdp r Ti

li i

T

Tiv

p T

i

i is

= + −

−∫ ∫0

0

0, ( )

( )∂∂

If the temperature for the phase change is predetermined, the enthalpy of the pure component has to be calculated using:

h h cp dTh

pdp r T cl dTi

li i

T

T

iv

p T

i ui iT

T

i

ui

is

ui ui

= + −

− +∫ ∫ ∫0

0

0, ( )

( )∂∂

9-2

9.3 Enthalpy of a gas with starting phase in liquid The enthalpy of the gas follows from:

hh

pdp y hv

vP

i iv

i

=

+∫ ∑∂

∂0

If the temperature for the phase change is set equal to the systems temperature, the enthalpy of the pure component can be calculated using:

( )h h cl dT r T

h

pdpi

vi i

T

T

iiv

p Ti is

= + + +

∫ ∫0

0

0,

( )∂∂

If the temperature for the phase change is predetermined, the enthalpy of the pure component has to be calculated using:

h h cl dT r Th

pdp cp dTi

vi i

T

T

i uiiv

p T

iT

T

i

ui

is

ui ui

= + + +

+∫ ∫ ∫0

0

0,

( )( )

∂∂

9.4 Enthalpy of a gas with starting phase in vapor The enthalpy of the gas follows from:

hh

pdp y hv

vP

i iv

i

=

+∫ ∑∂

∂0

with

h h cp dTiv

i iT

T

i

= + ∫0

0,

9.5 Enthalpy of a solid component The enthalpy of a solid component follows from:

h s hs i is

i

= ∑

with

h h cs dTis

i iT

T

i

= + ∫0

0,

9-3

9.6 The excess enthalpy

h T

g

TT

RT xT

RT xT

E

E

ii

ii

i

ii

= − = −

= − ∑∑2 2 2 1∂

∂∂ γ

∂ γ∂γ∂

ln

The excess enthalpy can be calculated using different methods.

9.6.1 NRTL equation

g

RTx

x G

x G

E

i

j j i j ij

j j ij

i

=∑∑∑

, ,

,

τ

h T R xA B B A

BE

ii i i i

ii

= −−∑2

2

’ ’

with

( )

A x G

A

TA x G G

i j j i j ij

ii j j i j i j i j i

j

=

= = +

∑

∑

, ,

’, , , ,’ ’

τ

∂∂

τ τ

B G x

B

TB x G

i j ij

j

ii j

jj i

=

= =

∑

∑

,

’,’

∂∂

G e

S c d T

ab

Te T f T

j iS

j i j i j i

j i j i

j i

j i j i

j i j i

,

, , ,

, ,

,

, ,

, ,

( . )

ln

== + −

= + + +

− τ

τ

27315

∂∂

∂∂

τ∂τ∂

∂τ∂

τ

∂∂

G

TG G

S

T TS

T

b

T

e

Tf

S

Td

j i

j i j i

j i

j i

j i

j i

j i

j i

j i j i

j i

j i

j i

,

, ,

,

,

,

,

,

,

, ,

,

,

,

’

’

= = − +

= = − + +

=

2

9-4

9.6.2 UNIQUAC equation

g

RTx V q x

F

Vq x x F

E

i ii

i ii

iii i j j j i

ji

= + −∑ ∑ ∑∑ln ln ’ ln ’ ,5 τ

h RT q xA

AF T

A A

AE

i ii

i

ij

i j

j i j j

jj

= +−

∑ ∑22’

’’

’,

,

∂τ∂

τ

with

A F

A FT

i j j ij

i j

j i

j

=

=

∑

∑

’

’ ’

,

,

τ

∂τ∂

Fq x

q xii i

j jj

’’

’= ∑

and

τ j i

ab

Tc T d T

ej i

j ij i j i

,

ln,,

, ,=+ + +

9.6.3 Wilson equation

g

TR x x

E

i j i jji

= − ∑∑ ln ,Λ

h RT x xEi

i

ij

j

j i j j i j

ji

= +−

∑∑2

2

ηη

η ηη

’ ’ ’, ,Λ Λ

∂η∂

ηii j i j

jTx= = ∑’ ’ ,Λ

ηi j i jj

x= ∑ Λ ,

∂Λ∂

i j

i j i j

i j i j

i jT

b

T

c

Td

,

, ,

, ,

,’= = − + +

Λ Λ 2

Λ i j

ab

Tc T d T

ei j

i ji j i j

,

ln,,

, ,=+ + +

9-5

9.6.4 Flory-Huggins equation

g

RTx

xx

xx

E

ii

iii i

i

= + +∑ ∑11

12

2

221 2 1 2

2

ln ln ,

ϕ ϕϕ χ

h RTxEi i

i

= ∑1 2 1 22

ϕ χ ,

9.6.5 Redlich-Kister equation

h hEi jE

ji

= ∑∑0 5, ,

( )h

x x

x xA T B T x C T x D T x E T x F T x

x x x

i jE i j

i jd d d d d

d i j

, ( ) ( ) ( ) ( ) ( ) ( )=+

+ + + + +

= −

2 3 4 5

The temperature dependency of the coefficients in most cases can be characterized well by a polynom:

A T a a T a T( ) ...= + +0 1 22

9.7 The isothermal pressure dependency of the enthalpy in vapor phase

9.7.1 Redlich-Kwong-Soave equation The pressure dependency of the pure component is characterized as follows:

( )∆h RT zT

dadT

a

b

b p

zRTi

ii

i

i is

= − +−

+

1 1ln

The pressure dependency of the mixture:

( )∆h RT zT

dadT

a

b

b p

zRTm

mm

m

m= − +−

+

1 1ln

9.7.2 Peng-Robinson equation The pressure dependency of the pure component is characterized as follows:

9-6

∆h RT zb

i i

i

= − −( )11

2 2A T T

A T

T

v b

v bii i i

i i

( )( )

ln( )

( )−

+ ++ −

∂∂

1 2

1 2

with

A T a mT

Tci i ii

( ) = + −

1 1

2

∂

∂A T

Tm

T

Tc

a m

Tc Ti

ii

i i

i

( )= + −

−1 1

The pressure dependency of the mixture:

∆h RT zb

m m

m

= − −( )11

2 2a T

a

T

v b

v bmm m m

m m

−

+ ++ −

∂∂

ln( )

( )

1 2

1 2

with

∂∂

∂∂

∂∂

a

Tx x k

A T A TA T

A T

TA T

A T

Tm

i j i ji jji

i

j

ji= − +

∑∑ ( )

( ) ( )( )

( )( )

( ),1

1

2

9.8 The enthalpy of a gas with simultaneous association

h RE y hv i iv

i

= + ∑

RE

z h z h

nz z

in in Mij Mijj iini

in Mijj iini

=+

+≠

≠

∑∑∑∑∑∑∑∑

∆ ∆

2

( )∆

∆

h RB

hR

B B

in in

Mij i j

= −

= − +2 2 2

10-1

10 Chemical reactions

10.1 Equilibrium reactions

ln ( ) lnf T aa

Ta T a T a T a T= + + + + +0

12 3 4

25

3

• EQLM x f Ti

i

iυ∏ = ( )

• EQVM y f Ti

i

iυ∏ = ( )

• EQLC

x

Vf T

Vx

i

i

i

ii

i

=

=

∏

∑

υ

ρ

( )

• EQVC

y

Vf T

Vz R T

P

i

i

i

=

=⋅ ⋅

∏υ

( )

• EQLA

( )γ υi i

i

x f Ti∏ = ( )

• EQVP

( )Py f Tii

iυ∏ = ( )

• EQLF

10-2

ϕ γ

υi i

si i

ii

P Fp

fx f T

i0

0

=∏ ( )

• EQVF

ϕ

υiv

ii

P

fy f T

i

0

=∏ ( )

10.2 Kinetically controlled reactions

ln ( ) ln

ln ( ) ln

f T aa

Ta T a T

T aa

Ta T a T

= + + +

= + + +

01

2 3

01

2 3ϕ

• KILM

ς ϕ γα=

+

∏∑ ∑∏V f x xk i

ikl i l i

il

i k,

,1

• KIVM

ς ϕ γα=

+

∏∑ ∑∏V f y yk i

ikl i l i

il

i k,

,1

• KILC

ςϕ

γ

ρ

α

α=

+

=

∑ ∏∑ ∑∏

∑

Vf

Vs

xVs

x

Vsx

ki

ik

li l i

il

i

ii

i k

i k

,

,

,1

• KIVC

ς ϕ γα

α=

+

=

∑ ∏∑ ∑∏Vf

Vs

yVs

y

VszRT

P

ki

ik

li l i

ili k

i k

,

,

,1

• KILW

10-3

ς ρ ϕ γα

α=

+

=

∑ ∏∑ ∑∏

∑

V Vsf

Vs

xVs

x

Vs x Mw

mk

iik

li l i

il

i ii

i k

i k

,

,

,1

• KIVW

ς ρ ϕ γ

ρ

α

α=

+

=

=

∑ ∏∑ ∑∏

∑

V Vsf

Vs

yVs

y

Vs y Mw

zRT

P

mk

iik

li l i

il

i ii

m

i k

i k

,

,

,1

10.3 Reactions of general nature • COOR value=ς

• CONV refrrefMratio υς−=*

• STAT ( )1* −= ratioMratio refrref υς