TERMODINAMIKA LARUTAN-TEORI - Spada UNS

Termodinamika Larutan: Teori

(Solution Thermodynamics: Theory) (Solution Thermodynamics: Theory)

Lecturer: Bregas S T Sembodo

Pendahuluan

• Sistem nyata biasanya berupa campuran fluida

• Komposisi sering menjadi peubah (variabel) utama

• Dikembangkan teori dasar aplikasi termodinamika untuk campuran gas dan larutan

• Akan diperkenalkan:

• Akan diperkenalkan:

– Potensial kimia (chemical potential) – Sifat parsial (partial properties)

– Fugasitas (fugacity)

– Sifat ekses (excess properties)

– Larutan ideal (ideal solution)

Hubungan sifat-sifat fundamental

• Hubungan antara Energi Gibbs dengan suhu dan tekanan pada sistem tertutup:

dT nS dP

nV T dT

dP nG P

nG nG d

n P n

T

) ( )

) ( (

) ) (

(

, ,

−

 =

 



∂ + ∂





∂

= ∂

– Diterapkan untuk fluida fase tunggal dalam sistem tertutup dimana tidak terjadi reaksi kimia.

• Untuk fase tunggal, sistem terbuka (a single-phase, open system):

∑ 



 





∂ + ∂

 

 

∂ + ∂

 

 

∂

= ∂

i

i n

T i P

n P n

T

n dn dT nG

T dP nG

P nG nG

d

, j

, , ,

) (

) (

) ) (

(

nj

T i P

i n

nG

, ,

)

( 



 





∂

≡ ∂ µ Definisi potensial kimia:

∑

+

−

=

i

i i

dn dT

nS dP

nV nG

d ( ) ( ) ( ) µ

Hubungan sifat fundamental untuk sistem fluida fase tunggal dengan komposisi konstan maupun sebagai variabel:

Jika n = 1,

= − + ∑

i

i i

dx SdT

VdP

dG µ ( , , , ,..., ,...)

2

1

x x

i

x T P G G =

x

T P

S G

,



 





∂

− ∂

=

x

P T

V G

,



 





∂

= ∂

Energi Gibbs dinyatakan sbg fungsi dari variabel-variabel kanonis

Sifat larutan, M Sifat parsial,

Sifat spesies murni, M_i

M

i

Potensial kimia dan kesetimbangan fase

• Untuk sistem tertutup terdiri dari 2 fase dalam kesetimbangan:

∑

+

−

=

i

i i dn dT

nS dP

nV nG

d^{( )α ( )α ( )α} µ^{α α = − +}

∑

i

i i dn dT

nS dP

nV nG

d^{( )β ( )β ( )β} µ^{β β}

i i

β

α

( )

)

( nM nM

nM = +

∑

∑

⁺

+

−

=

i

i i i

i

i dn dn

dT nS dP

nV nG

d^{( ) ( ) ( )} µ^{α α} µ^{β β}

β

α

µ

µ

_i

=

_i

β α

i

i dn

dn = −

Neraca massa:

Multi fase pada T dan P yg sama adalah dalam kesetimbangan jika potensial kimia setiap spesies pada semua fase sama.

Sifat Parsial (Partial properties)

• Definisi sifat molar parsial (partial molar property) spesies i :

arti : perubahan besaran M terhadap perubahan n (jml mol i) jika

nj

T i P

i

n

M nM

, ,

)

( 



 





∂

≡ ∂

arti : perubahan besaran M terhadap perubahan n

_i

(jml mol i) jika P, T dan n

_j

(jml mol spesies lain) tetap

– Potensial kimia adalah Energi Gibbs molar parsial:

– Untuk sifat termodinamika M:

i i

≡ G

µ

,...) ,...,

, , ,

( P T n

₁

n

₂

n

_i

M

nM =

∑

 +

 



∂ + ∂

 

 

∂

= ∂

i

i i n

P n

T

dn M T dT

n M P dP

n M nM

d

, ,

)

(

∑

 +

 



∂ + ∂





∂

= ∂

i

i i n

P n

T

dn M T dT

n M P dP

n M nM

d

, ,

) (

∑

⁺

 +

 



∂ + ∂





∂

= ∂ +

i

i i

i n

P n

T

ndx dn

x M T dT

n M P dP

n M Mdn

ndM ( )

, ,

0

, ,

 =



 



 −

 +









  −



 





∂

− ∂



 





∂

− ∂ ^dT

∑

^{M dx n M}

∑

^{x M dn}

T dP M

P dM M

i

i i i

i i n

P n

T 



0

, ,

=

−



 





∂

− ∂



 





∂

− ∂

∑

i

i i n

P n

T

dx M T dT

dP M P

dM M dan −

∑

= 0

i

i iM x M

Perhitungan sifat-sifat campuran dari sifat- sifat parsialnya

= 0

−

∑

i

i iM n nM

0

, ,

=

−

 

 





∂ + ∂

 

 





∂

∂ ∑

i

i i

n P n

T

M d x T dT

dP M P

M

∑

∑

⁺

=

i

i i i

i

idM M dx

x dM

Persamaan Gibbs/Duhem

Sifat Parsial pada Larutan Biner

• Untuk sistem biner

2 2 1

1M x M

x

M = +

2 2 2

2 1

1 1

1dM M dx x dM M dx

x

dM = x₁dM₁ + M₁dx₁ + x₂dM₂ + M₂dx₂

dM = + + +

P and T tetap, menggunakan pers. Gibbs/Duhem

2 2 1

1dx M dx

M

dM = +

2 1

1 + x =

x

2 1

1

M dx M

dM = −

1 2

1 dx

x dM M

M = +

1 1

2 dx

x dM M

M = −

Constant T, P

Constant T, P

Pada suatu Lab dibutuhkan 2000 cm³ antifreeze solution (larutan antibeku)

mengandung 30 mol-% metanol dalam air. Berapa volume metanol murni dan air murni pada 25°C harus dicampur untuk memperoleh 2000 cm³ antifreeze pada 25°C?

Volume molar parsial dan volume murni masing-masing spesies diketahui.

2 2 1

1V x V

x

V = + V = (0.3)(38.632) +(0.7)(17.765) = 24.025 cm³ / mol V^t 2000

=

=

= n₁ = (0.3)(83.246) = 24.974 mol V mol

n V

t

246 . 025 83

. 24

2000 =

=

= n₁ = (0.3)(83.246) = 24.974 mol mol n₂ = (0.7)(83.246) = 58.272

3 1

1

1 nV (24.974)(40.727) 1017 cm

V ^t = = = V₂^t = n₂V₂ = (58.272)(18.068) =1053 cm³

Entalpi sistem cairan biner dari spesies 1 dan 2 pada T dan P tertentu adalah:

Tentukan dan sbg fungsi x₁, nilai numerik entalpi spesies murni H₁ and H₂, nilai numerik entalpi parsial pada pengenceran tak berhingga dan

H

1

H

₂

∞

H

1 H₂^∞ )

20 40

( 600

400x₁ x₂ x₁x₂ x₁ x₂

H = + + +

) 20 40

( 600

400x₁ x₂ x₁x₂ x₁ x₂

H = + + +

2 1

1 + x =

x

20 3

180

600 x x

H = − − x₁ + x₂ =1 H = 420−60x² + 40x³

40 3

600 x

H = +

3 1

1 20

180

600 x x

H = − −

1 2

1 dx

x dH H

H = +

3 1 2

1

1 420 60x 40x

H = − + H₂ = 600+ 40x₁³

1 = 0 x

mol H₁^∞ = 420 J

1 =1 x

mol H₂^∞ = 640 J

Hubungan antar Sifat Parsial

• Relasi Maxwell :

∑

+

−

=

i

i i

dn G dT

nS dP

nV nG

d ( ) ( ) ( )

S

V 



∂

− ∂

=





∂

∂ Gi (nS)





∂

− ∂

 =

 



∂

∂ G_i (nV)





∂

− ∂

 =

 



∂

∂

n T n

P P

T _,  _,

 

 ∂

−

=



 ∂

nj

T i P

n

P n

T , , ,



 ∂

−

 =

 

 ∂

nj

T i P

n

T n

P _,  _{, ,}

 

 ∂

−

 =

 

 ∂

i x

P

i S

T

G  = −









∂

∂

,

i x

T

i V

P

G  = −









∂

∂

,

PV U

H = +

dT S dP V G

d

_i

=

_i

−

_i

i i

i

U P V

H = +

Campuran Gas Ideal

• Teorema Gibbs’s

– Sifat molar parsial spesies (kecuali volume) dalam camp. gas ideal SAMA DENGAN sifat molarnya sbg gas ideal murni pada suhu campuran tetapi pada

gas ideal murni pada suhu campuran tetapi pada tekanan yg sama dengan tekanan parsialnya dalam campuran.

ig i ig

i

V

M ≠ M

_i^ig

( T , P ) = M

_i^ig

( T , p

_i

)

Untuk yg tergantung tekanan, mis. dS

_i^ig

= − Rd ln P const . T

i i

i i

ig i ig

i R y

P y R P

p R P

p T S P

T

S ( , ) − ( , ) = − ln = − ln = ln

Untuk yg tidak tergantung tekanan, mis.

) , ( )

, ( )

,

( T P H T p H T P

H

_i^ig

=

_i^ig _i

=

_i^{ig =}

∑

i

ig i i

ig y H

H ⁼

∑

i

ig i i

ig yU

U

) , ( )

,

( _i^ig _i

ig

i T P M T p

M =

i ig

i ig

i T P S T P R y

S ( , )− ( , ) = ln

) , ( )

,

( _{i i}

i T P M T p

M =

∑

∑

⁻

=

i

i i

i

ig i i

ig y S R y y

S ln

ig i ig

i ig

i H TS

G = − ^{H (T,P) H (T,P)}

G

_i^ig

= H

_i^ig

− TS

_i^ig

+ RT ln y

_i

ig i ig

i =

i ig

i ig

i T P S T P R y

S ( , ) − ( , ) = ln

i ig

i ig

i ig

i

≡ G = G + RT ln y

µ P y RT

T

_i

i ig

i

= Γ ( ) + ln

µ

or P

y y

RT T

y G

i

i i

i

i i

ig ⁼

∑

^{Γ ( ) +}

∑

^ln

G T RT P

i ig

i

= Γ ( ) + ln

∑

∑ −

=

i

i i

i

ig i i

ig

y S R y y

S ln

y S y

R y

y R

S y S

i i

i i

i i

i

ig i i

ig

− ∑ = − ∑ ln = ∑ ln 1 = ∆ >0

Perubahan entropi pencampuran gas ideal selalu positif ATAU entropi campuran gas ideal lebih besar dari pada entropi spesies murni konstituennya

Fugasitas dan Koefisien Fugasitas

• Potensial kimia:

– Menyediakan kriteria dasar kesetimbangan fase

nj

T i P

i n

nG

, ,

)

( 



 





∂

≡ ∂ µ

– Namun, Energi Gibbs juga µ

_i

didefinisikan

berhubungan dengan energi dalam dan entropi (nilai absolut tidak diketahui, sulit ditentukan nilainya dari T dan P).

• Fugasitas:

– Suatu kunatitas yg menggantikan µ

_i

i i

i

T RT f

G ≡ Γ ( ) + ln

Dengan satuan tekanan

i i

i

T RT f

G ≡ Γ ( ) + ln P RT

T

G

_i^ig

= Γ

_i

( ) + ln

P RT f

G

G

_i

−

_i^ig

= ln

ⁱ

G

_i^R

= RT ln φ

_i

P f

_i

i

=

φ

Residual Gibbs energy Fugacity coefficient

) . (

) 1 (

ln dP const T

P

Z

∫ −

φ = ( 1 ) ( . )

ln

0

const T

P Z dP

P i

i

= ∫ −

φ

Kesetimbangan Uap-Cair (VLE) untuk Spesies Murni

• Uap Jenuh:

• Cairan Jenuh:

v i i

v

i

T RT f

G = Γ ( ) + ln

l i i

l

i

T RT f

G = Γ ( ) + ln

f

v l i

v l i

i v

i

f

RT f G

G − = ln

VLE

0

ln =

=

− _l

i v l i

i v

i f

RT f G

G

sat i l

i v

i f f

f = =

sat i l

i v

i

φ φ

φ = =

For a pure species coexisting liquid and vapor phases are in equilibrium when they have the same temperature, pressure, fugacity and fugacity coefficient.

Fugasitas Cairan Murni

• Fugasitas spesies murni i sebagai cairan tertekan (compressed liquid):

sat i i i

RT f G

G − = ln

^sat

G − G

^sat

= ∫

^P

V dP ( isothermal process )

i i

i

G RT f

G − = ln G G V dP ( isothermal process )

P i

sat i

i sat

∫

i

=

−

∫

=

^P

P i

sat i

i

sat i

dP RT V

f

f 1

ln

Karena V_i sedikit dipengaruhi P (weak function)

RT P P

V f

f

_i^l _i^sat

sat i

i

( )

ln = −

sat i sat i sat

i

P

f = φ RT

P P

P V f

sat i l

sat i i sat i i

)

exp ( −

= φ

For H₂O at a temperature of 300°C and for pressures up to 10,000 kPa (100 bar) calculate values of f_i and φ_i from data in the steam tables and plot them vs. P.

For a state at P:

G

_i

= Γ

_i

( T ) + RT ln f

_i

For a low pressure reference state:

G

_i^*

= Γ

_i

( T ) + RT ln f

_i^*

) 1 (

ln

_{* i i}^*

i

i

G G

RT f

f = −

i i

i

H TS

G = −



 





− − −

= 1 ( )

ln

^*

*

* i i

i i

i

i

S S

T H H

R f

f

The low pressure (say 1 kPa) at 300°C:

J gK S_i^* =10.3450 kPa

P

f_i^* = ^* =1 J g H_i^* = 3076.8

S_i =10.3450 gK kPa

P

f_i = =1

For different values of P up to the saturated pressure at 300°C, one obtains the values of f_i ,and hence φ_i . Note, values of f_i and φ_i at 8592.7 kPa are obtained

Fig 11.3 Values of f_i andφ_i at higher pressure:

RT P P

P V f

sat i l

sat i i sat i i

)

exp ( −

= φ

Fugasitas dan Koefisien Fugasitas:

spesies dalam larutan

• Untuk spesies i dalam campuran gas nyata atau dalam larutan cair:

i i

i

≡ Γ ( T ) + RT ln f ˆ

µ

• Multi fase pada T dan P yg sama adalah dalam

kesetimbangan jika fugasitas masing-masing spesies konstituen sama pada semua fase:

Fugasitas spesies i dalam larutan (mengganti tekanan parsial y_iP)

π β

α

i i

i

f f

f ˆ = ˆ = ... = ˆ

ig

R

M M

M = −

Sifat Residual:

ig i i

R

i

M M

M = −

Sifat residual parsial

ig i i

R

i

G G

G = −

i i

i

≡ Γ ( T ) + RT ln f ˆ

µ

P y RT

T

_i

i ig

i

= Γ ( ) + ln

µ

P y RT f

i ig i

i i

ˆ

= ln

− µ µ

i R

i

RT

G = ln φ ˆ

i n

T i P

i G

n nG

j

 =



 





∂

≡ ∂

, ,

) µ (

P y

f

i i i

ˆ ≡ ˆ φ

Koefisen fugasitas spesies i dalam larutan

Untuk gas ideal,

= 0

R

G

i

1 ˆ = ˆ =

P y

f

i i

φi fˆ_i = y_iP

Fundamental residual-property relation

RT dT nG nG

RT d RT

d nG 1 ( ) ₂

−

=



 





∑

+

−

=

i

i idn dT

nS dP

nV nG

d^{( ) ( ) ( )} µ

TS H

G = H −TS G = −

∑

+

−

=



 





i

i i dn RT dT G

RT dP nH

RT nV RT

d nG ₂ f (P,T,n_i)

RT nG =

G/RT as a function of its canonical variables allows evaluation of all other

thermodynamic properties, and implicitly contains complete property information.

The residual properties:

∑

+

−

 =









i

i R i R

R R

RT dn dT G

RT dP nH

RT nV RT

d nG _{2 } ^{= − +}

∑









i

i i R

R R

dn RT dT

dP nH RT

nV RT

d nG ₂ ^lnφ^ˆ

or

∑

+

−

 =









i

i R i R

R R

RT dn dT G

RT dP nH

RT nV RT

d nG _{2 } ^{= − +}

∑









i

i i R

R R

dn RT dT

dP nH RT

nV RT

d nG ₂ ^lnφ^ˆ

Fix T and composition:

x T R

R

P RT G

RT V

,

) /

( 









∂

= ∂

Fix P and composition:

x P R

R

T RT T G

RT H

,

) /

( 









∂

− ∂

=

Fix T and P:

nj

T i P

R

i n

RT nG

, ,

) /

ˆ (

ln 









∂

= ∂ φ

Develop a general equation for calculation of values form compressibility- factor data.

φ^ˆi

ln

nj

T i P

R

i n

RT nG

, ,

) /

ˆ (

ln 









∂

= ∂ φ

∫

⁻

= ^P

R

P n dP RT nZ

nG

0 ( )

∫

^∂ _∂ ^{− }

= ^P

i P

dP n

n nZ ) ˆ (

lnφ

∫

^

 

 ∂

=

n T i P

i n P

j

0

, ,

lnφ

i i

n Z nZ =

∂

∂( )

=1

∂

∂ ni

n

∫ −

=

^P _i

i

P

Z dP

0

( 1 )

ln φ ˆ

Integration at constant temperature and composition

Fugacity coefficient from the virial E.O.S

• The virial equation:

– the mixture second virial coefficient B:

– for a binary mixture:

RT Z = 1 + BP

∑∑

=

i j

ij j

i

y B

y B

– for a binary mixture:

• n mol of gas mixture:

22 2 2 21

1 2 12

2 1 11

1

1

y B y y B y y B y y B

y

B = + + +

RT n nBP nZ = +

2

2 1 ,

, 1 ,

1

) 1 (

) (

n T n

T

P n

nB RT

P n

Z nZ 



 





∂ + ∂

 =



 





∂

= ∂

2

2 1 ,

0

1 , 1

) ( )

( ˆ 1

ln

n T P

n

T n

nB RT

dP P n

nB

RT 



 





∂

= ∂



 





∂

=

∫

∂ φ

2

2 1 ,

0

1 , 1

) ( )

( ˆ 1

ln

n T P

n

T n

nB RT

dP P n

nB

RT 



 





∂

= ∂



 





∂

=

∫

∂ φ

22 2 2 21

1 2 12

2 1 11

1

1

y B y y B y y B y y B

y

B = + + +

22 11

12

12

≡ 2 B − B − B

δ y

_i

= n

_i

/ n

(

^{11 22 12}

)

ˆ1

lnφ ^{B y} δ

RT

P +

= Similarly: ^lnφˆ2

(

^{B22 y12δ12}

)

RT

P +

=

For multicomponent gas mixture, the general form:











 + −

=

∑∑

^{(2 )}

2 ˆ 1

ln _{ik ij}

i j

j i kk

k B y y

RT

P δ δ

φ where

kk ii

ik

ik

≡ 2 B − B − B

δ

Determine the fugacity coefficients for nitrogen and methane in N₂(1)/CH₄(2) mixture at 200K and 30 bar if the mixture contains 40 mol-% N₂.

cm mol B

B

B

_{12 11 22} ³

12

≡ 2 − − = 2 ( − 59 . 8 ) + 35 . 2 + 105 . 0 = 20 . 6 δ

( ) [

^{35.2 (0.6) (20.6)}

]

^0.0501

) 200 )(

14 . 83 ( ˆ 30

lnφ₁ = ^B₁₁ + ^y₂²δ₁₂ = − + ² = −

RT P

9511 .

ˆ 0

1 = φ

( ) [

^{105.0 (0.4) (20.6)}

]

^0.1835

) 200 )(

14 . 83 ( ˆ 30

lnφ₂ = ^B₂₂ + ^y₁²δ₁₂ = − + ² = −

RT P

8324 .

ˆ 0

2 = φ

Generalized correlations for the fugacity coefficient

∫ −

=

^Pr _r

r r i

i

const T

P Z dP

0

( 1 ) ( . )

ln φ

1

0

Z

Z

Z = + ω

) (

1 B⁰ B¹

T Z P

r

r +ω

=

−

) (

ln B

⁰

B

¹

T P

r

r

ω

φ = +

1

0

Z

Z

Z = + ω

∫

^{− +}

∫

= ^{Pr Pr} _r

r r r

r const T

P Z dP P

Z dP

0 0

1

0 1) ( . )

(

lnφ ω

or

1

0

ln

ln

ln φ = φ + ω φ

∫

⁻

≡ ^Pr

r r

P Z dP

0

0

0 ( 1)

lnφ ^≡

∫

^Pr

r r

P Z dP

0 1

lnφ1

with

Table E1:E4 or Table E13:E16

) ,

,

( TR PR OMEGA

= PHIB φ

For pure gas

For pure gas

Estimate a value for the fugacity of 1-butene vapor at 200°C and 70 bar.

127 .

=1 Tr

731 .

=1 Pr

191 .

= 0 ω

627 .

0 = 0

φ and φ^{1 =1.096}

1

0

ln

ln

ln φ = φ + ω φ

Table E15 and E16

638 .

= 0

φ f = φ P = ( 0 . 638 )( 70 ) = 44 . 7 bar

For gas mixture:

For gas mixture:











 + −

=

∑∑

^{(2 )}

2 ˆ 1

ln _{ik ij}

i j

j i kk

k B y y

RT

P

δ δ

φ

) ( B

⁰

B

¹

P

B RT

_ij

cij cij

ij

= + ω

2

j i

ij

ω

ω = ω + T

_cij

= ( T

_ci

T

_cj

) ( 1 − k

_ij

)

Prausnitz et al. 1986

cij cij cij

cij

V

RT P = Z

2

cj ci

cij

Z Z Z +

=

^{1/3 1/3 3}

2  





 



 +

=

^{ci cj}

cij

V V V

Empirical interaction parameter

Estimate and for an equimolar mixture of methyl ethyl ketone (1) / toluene (2) at 50°C and 25 kPa. Set all k_ij = 0.

ˆ

1

φ φ ˆ

₂

2

j i

ij

ω ω = ω +

) 1

( )

(

_{ci cj ij}

cij

T T k

T = −

cij cij cij

cij

V

RT P = Z

2

cj ci

cij

Z Z Z +

=

^{1/3 1/3 3}

2  





 



 +

=

^{ci cj}

cij

V V V

) ( B

⁰

B

¹

P

B RT

_ij

cij cij

ij

= + ω δ

¹²

≡ 2 B

¹²

− B

¹¹

− B

²²

( )

^0.0128

lnφˆ^ˆ = ^P

(

^B + ^y2δ

)

= −^0.0128 φˆ = 0.987 lnφ₁ = ^B₁₁ + ^y₂²δ₁₂ = −

RT P

( )

^0.0172

lnφˆ₂ = ^B₂₂ + ^y₁²δ₁₂ = − RT

P

987 . ˆ 0

1 = φ

983 . ˆ 0

2 = φ

The ideal solution

• Serves as a standard to be compared:

i i

id

i

G RT x

G = + ln = ∑

i i

+ ∑

i i

id

x G RT x x

G ln

∑

=

i

id i i

id

x M

M

i i

i

G RT x

G = + ln

cf.

G

_i^ig

= G

_i^ig

+ RT ln y

_i

i P

i x

P id id i

i R x

T G T

S G ln

,

 −









∂

− ∂

 =









∂

− ∂

=

S

_i^id

= S

_i

− R ln x

_i

T i x

T id id i

i

P

G P

V G











∂

− ∂



=









∂

− ∂

=

,

i id

i

V

V =

i i

i i

id i id

i id

i

G T S G RT x TS RT x

H = + = + ln + − ln H

_i^id

= H

_i

∑

∑

i

i i

i

i i

i i

i i

i i

id x S R x x

S ⁼

∑

⁻

∑

ln

i i

i

id

x V

V = ∑

i i

i

id

x H

H = ∑

i i

id

i

G RT x

G = + ln

i i

i

≡ Γ ( T ) + RT ln f ˆ

µ

The Lewis/Randall Rule

• For a special case of species i in an ideal solution:

id i i

id i id

i

= G = Γ ( T ) + RT ln f ˆ

µ

i i

i

G RT x

G = + ln

i i

i

T RT f

G ≡ Γ ( ) + ln

i i id

i

x f

f ˆ =

The Lewis/Randall rule

i id

i

φ

φ ˆ =

The fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P.

Excess properties

• The mathematical formalism of excess properties is analogous to that of the residual properties:

id

E

M M

M ≡ −

– where M represents the molar (or unit-mass) value of any extensive thermodynamic property (e.g., V, U, H, S, G, etc.)

– Similarly, we have:

∑

+

−

 =









i

i E i E

E E

RT dn dT G

RT dP nH

RT nV RT

d nG ₂

The fundamental excess-property relation

(1) If C^E_P is a constant, independent of T, find expression for G^E, S^E, and H^E as

functions of T. (2) From the equations developed in part (1), fine values for G^E , S^E, and H^E for an equilmolar solution of benzene(1) / n-hexane(2) at 323.15K, given the following excess-property values for equilmolar solution at 298.15K:

C^E_P=-2.86 J/mol-K, H^E = 897.9 J/mol, and G^E = 384.5 J/mol From Table 11.1:

x P E E

P T

T G C

, 2 2









∂

− ∂

=

. const a

C_P^E = = T

a T

G

x P

E  = −









∂

∂

, 2 2

b T T a

G^E

+

−

 =









∂

∂ ln

integration From Table 11.1:

E E

T

S G 









∂

− ∂

= a T b

T P x

+

−

 =

 

 ∂ ln

,

c bT T

T T a

G^E = − ( ln − )+ +

integration From Table 11.1:

x

T P

S

,



 ∂

−

=

b T a

S^E = ln −

c aT TS

G

H ^E = ^E + ^E = +

integration

86 .

−2

=

= a C_P^E

c

a +

= (298.15) 9

. 897

c b

a − + +

−

= ((298.15)ln(298.15) (298.15) (198.15) 5

. 384

We have values of a, b, c and hence the excess- properties at 323.15K

The excess Gibbs energy and the activity coefficient

• The excess Gibbs energy is of particular interest:

id

E

G G

G ≡ −

i i

i

T RT f

G = Γ ( ) + ln ˆ

i i i

id

i

T RT x f

G = Γ ( ) + ln

i i E i

i

x f

RT f G

ˆ

= ln

i i

i

i

x f

≡ fˆ γ

i E

i

RT

G = ln γ<