Solution Thermo(3)

CPE553 CHEMICAL ENGINEERING THERMODYNAMICS 4/14/2013

1

SOLUTION THERMODYNAMICS:

THEORY

• FUGACITY AND FUGACITY

COEFFICIENT: SPECIES IN SOLUTION

• GENERALIZED CORRELATIONS FOR THE

FUGACITY COEFFICIENT

• THE IDEAL SOLUTION MODEL

2

• The definition of a species in solution is parallel to the definition of the pure-species fugacity.

• For species i in a mixture of real gases or in a solution of liquids,

where is the fugacity of species i in solution, replacing the partial pressure yiP in eq. (11.29) for ideal gas.

• For species in solution, the phase equilibrium criteria is:

Multiple phases at the same T and P are in equilibrium when the fugacity of each constituent species is the same in all phases.

• For the specific case of multicomponent vapor/liquid equilibrium, eq. (11.47) becomes:

3

ˆlni i iT RT f (11.46) ^

if

ˆ ˆ ˆ... 1, 2, ...,i i if f f i N (11.47)

ˆ ˆ 1, 2, ...,l

i if f i N (11.48)

lnig ig

i i i iG T RT y P (11.29)

• The definition of a residual property is

where M is the molar (or unit mass) value of a thermodynamic property and Mig is the value that the property would have for an ideal gas of the same composition at the same T and P.

• The defining equation for a partial residual property follows from this equation. Multiplied by n mol of mixture, it becomes

• Differentiation with respect to ni at constant T, P and nj gives

• Reference to eq. (11.7) shows that each term has the form of a partial molar property. Thus,

R ignM nM nM

4

R igM M M (6.41)

R

iM

4

, ,, , , ,jj j

R ig

i i iP T nP T n P T n

nM nMnM

n n n

R ig

i i iM M M (11.49)


2

• Written for the residual Gibbs energy, eq. (11.49) becomes

an equation which defines the partial residual Gibbs energy.

• Subtracting eq. (11.29) from eq. (11.46), both written for the same T and P,

yields

• This result combined with eq. (11.50) and the identity gives

where by definition,

• The dimensionless ratio is called the fugacity coefficient of species i in

solution.

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R ig

i i iG G G (11.50)

ˆlnig i

i i

i

fRT

y P

i iG

ˆlnR

i iG RT (11.51)

ˆˆ ii

i

f

y P (11.52)

ˆi

lnig ig

i i i iG T RT y P

(11.29)

ˆlni i iT RT f (11.46)

Although most commonly applied to gases, the fugacity coefficient may also

be used for liquids, in this case mole fraction yi is replaced by xi.

Because eq. (11.29) for an ideal gas is a special case of eq. (11.46), then

Thus the fugacity of species i in an ideal gas mixture is equal to its partial

pressure. Moreover, , and for an ideal gas

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ˆ ig

i if y P (11.53)

ˆ 1ig

i 0R

iG

ˆˆ 1

igig i i

i

i i

f y P

y P y P ˆln ln 1 0R ig

i iG RT RT

Values of for species i in solution are readily found from equations of state.

The simplest form of the virial equation provides a useful example.

Written for a gas mixture it is exactly the same as for a pure species:

The mixture second virial coefficient B is a function of temperature and composition.

where y represents mole fractions in a gas mixture, the indices i and j identify species, and both run over all species present in the mixture.

The virial coefficient Bij characterizes a bimolecular interaction between molecule i and molecule j, and therefore Bij = Bji.

The summations account for all possible bimolecular interactions.

ˆi

7

1BP

ZRT

(3.38)

i j ij

i j

B y y B (11.61)

7

For a binary mixture i = 1, 2 and j = 1, 2; the expansion of eq. (11.61) then

gives

or

B11 and B22 are pure species virial coefficient and B12 is a mixture property,

known as a cross coefficient. Both are function of temperature only.

Expressions such as eqs. (11.61) and (11.62) relate mixture coefficients to

pure species and cross coefficients. They are called mixing rules.

2 2

1 11 1 2 12 2 222B y B y y B y B

8

1 1 11 1 2 12 2 1 21 2 2 22B y y B y y B y y B y y B

(11.62)

8

i j ij

i j

B y y B (11.61)


3

Eq. (11.62) allows derivation of expressions for for a binary gas mixture that obeys eq. (3.38). For n mol of gas mixture,

Differentiation with respect to n1 gives

Because and , this reduces to

From eq. (11.60) yields

where the integration is elementary, because B is not function of pressure. All that remains is evaluation of the derivative.

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1 2ˆ ˆln and ln

nBPnZ n

RT

2

1

1 ,

1T n

nBPZ

RT n

2 2

10

1 1, ,

1ˆlnP

T n T n

nB nBPdP

RT n RT n

0

ˆln 1P

i i

dPZ

P (11.60)

2 2 21 1 1, , , , ,P T n P T n T n

nZ nBn P

n n RT n

1in n i inZ n Z

Eq. (11.62) for the second virial coefficient may be written:

or

Multiplying by n and substituting yi = ni/n gives,

By differentiation,

Therefore from previous eqn.,

Similarly,

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1 2 11 1 2 12 2 1 22

1 11 1 2 11 1 2 12 2 22 1 2 22

1 2 1

2

B y y B y y B y y B

y B y y B y y B y B y y B

1 11 2 22 1 2 12 12 12 11 22 with 2B y B y B y y B B B

1 21 11 2 22 12

nnnB nB n B

n

2

1 111 2 12 11 2 122

1 ,

2

11 1 2 12 11 2 12

1 11

1

T n

nB n nB n B n

n n n n n

B y y B y

2

1 11 2 12ˆln

PB y

RT

2

2 22 1 12ˆln

PB y

RT

(11.63a)

(11.63b)

1 11 2 22 1 2 12nB ny B ny B ny y or

2 2

1 11 1 2 12 2 222B y B y y B y B (11.62)

2 2

10

1 1, ,

1ˆlnP

T n T n

nB nBPdP

RT n RT n

Eqs. (11.63) are readily extended for application to multicomponent gas

mixtures; the general equation is

where i and j run over all species, and

with

11

1ˆln 22

k kk i j ik ij

i j

PB y y

RT

(11.64)

2 2ik ik ii kk ij ij ii ijB B B B B B

0, 0, etc., and , etc.,ii kk ki ik

11

Determine the fugacity coefficients as given by eqs. (11.63) for nitrogen and

methane in a N2(1)/CH4(2) mixture at 200K and 30 bar if the mixture

contains 40 mole % N2. Experimental virial coefficient data are as follows:

Solution:

By definition,

Substitution of numerical values in eqs.(11.63) yields

Thus,

3 -1

11 22 1235.2 105.0 59.8 cm molB B B

3 -1

12 12 11 222 2 59.8 35.2 105.0 20.6 cm molB B B

2

1

2

2

30ˆln 35.2 0.6 20.6 0.050183.14 200

30ˆln 105.0 0.4 20.6 0.183583.14 200

12

1 2ˆ ˆ0.9511 and 0.8324

R=8.314 m3 Pa/mol.K = 83.14 cm3 bar/mol.K 12


4

Note that the second virial coefficient of the mixture as given by eq. (11.62)

is

Substitution in eq. (3.38) yields a mixture compressibility factor,

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2 2

1 11 1 2 12 2 22

2 2

3 -1

2

0.4 35.2 2 0.4 0.6 59.8 0.6 105.0

72.14 cm mol

B y B y y B y B

1

72.14 301 0.870

83.14 200

BPZ

RT

Eq. (11.35) is put into generalized form by substitution of the relations,

Hence,

where the integration is at constant Tr. Substitution for Zi by Eq. (3.57)

yields

where for simplicity i is omitted.

0

ln 1rP r

i i

r

dPZ

P

14

c r c rP P P dP P dP

0

ln 1 const TP

i i

dPZ

P (11.35)

(11.65)

0 1

0 0ln 1

r rP Pr r

r r

dP dPZ Z

P P

0 1

iZ Z Z (3.57)

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This equation may be written in alternative form:

where

The integrals in these equations may be evaluated numerically or graphically for various values of Tr and Pr from data for Z0 and Z1 given in Table E.1- E.4 (Appendix E).

Another method, and the one adopted by Lee and Kesler to extend their correlation to fugacity coefficients, is based on an equation of state.

Eq. (11.66) may also be written,

Table E.13 – E.16 present values for these quantities as derived from the Lee/Kesler correlation as function of Tr and Pr.

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0 1ln ln ln (11.66)

0 0 1 1

0 0ln 1 and ln

r rP Pr r

r r

dP dPZ Z

P P

0 1 (11.67)

Estimate from eq. (11.67) a value for the fugacity of 1-butene vapor at 200oC (473.15K) and 70 bar.

Solution:

From Table B.1,

Tc = 420 K Pc = 40.43 bar = 0.191

Tr = T/Tc = 473.15/420 = 1.127

Pr = P/Pc = 70/40.43 = 1.731

By interpolation in Table E.15 and E.16 at these conditions,

0 = 0.627 and 1 = 1.096

Equation (11.67) then gives

= (0.627)(1.096)0.191 = 0.638

and f = P = (0.638)(70) = 44.7 bar

16


5

A useful generalized correlation for ln results when the simplest form of

the virial equation is valid.

Eqs. (3.61) and (3.63) combine to give

Substitution in eq. (11.65) and integration yield

where

This equation provides reliable values of for any nonpolar or slightly polar

gas when applied at conditions where Z is approximately linear in pressure.

E.g. H2, N2, O2, CO2, CH4, C2H4

Figure 3.14 serves as a guide to its suitability.

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0 11 r

r

PZ B B

T

0 1

ˆ1 1

ˆ

r

r

BP PZ B

RT T

B B B

(3.61)

(3.63)

0 1 0 1ln or expr r

r r

P PB B B B

T T

17

0 1

1.6 4.2

0.422 0.1720.083 and 0.139

r r

B BT T

(11.68)

18

The general expression for calculation of from the second virial coefficient data is given by eq. (11.64).

Values of the pure species virial coefficients Bkk, Bii, etc., are found from the generalized correlation represented by eqs. (3.62), (3.63), (3.65), and (3.66).

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ˆln k

1ˆln 22

k kk i j ik ij

i j

PB y y

RT

(11.64)

0 1

ˆ

ˆ

c

c

BPB

RT

B B B

0

1.6

1

4.2

0.4220.083

0.1720.139

r

r

BT

BT

(3.62)

(3.63)

(3.65)

(3.66)

The cross coefficients Bik, Bij, etc., are found from an extension of the same correlation.

For this purpose, eq. (3.63) is rewritten in the more general form

where

and B0 and B1 are the same functions of Tr as given by eqs. (3.65) and (3.66).

Thus,

20

0 1ˆij ijB B B

ˆ ij cij

ij

cij

B PB

RT

(11.69a)

(11.69b)

0 1

0 1

ij cij

ij

cij

cij

ij ij

cij

B PB B

RT

RTB B B

P


6

The combining rules for calculation of ij, Tcij, and Pcij are

In eq. (11.71), kij is an empirical interaction parameter specific to an i-j

molecular pair.

When i=j and for chemically similar species, kij = 0. Otherwise, it is a small

positive number evaluated from minimal PVT data or in the absence of data

set equal to zero.

When i=j, all equations reduce to the appropriate values for a pure species.

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1/ 2

31/ 3 1/ 3

11.702

1 11.71

11.72

11.732

11.742

i j

ij

cij ci cj ij

cij cij

cij

cij

ci cj

cij

ci cj

cij

T T T k

Z RTP

V

Z ZZ

V VV

When ij, these equations define a set of interaction parameters having no

physical significance.

Reduced temperature is given for each ij pair by Trij ≡ T/Tcij.

For a mixture, values of Bij from eq. (11.69b) substituted into eq. (11.61) yield

the mixture second virial coefficient B, and substituted into eq. (11.64) [eq.

(11.63) for a binary] yield values of .

22

ˆln i

ˆ ij cij

ij

cij

B PB

RT (11.69b)

i j ij

i j

B y y B (11.61)

1ˆln 22

k kk i j ik ij

i j

PB y y

RT

(11.64)

2

1 11 2 12ˆln

PB y

RT

2

2 22 1 12ˆln

PB y

RT

(11.63a)

(11.63b)

Estimate and by eqs. (11.63) for an equimolar mixture of methyl ethyl

ketone(1)/toluene(2) at 50oC (323.15 K) and 25 kPa. Set all kij = 0.

Solution:

From Table B.1, the required data are as follows:

Values in the last row have been calculated by eqs. (11.70) – (11.74).

2̂

23

ij Tcij/K Pcij/bar Vcij/cm3 mol-1 Zcij ij

11 535.5 41.50 267 0.249 0.323

22 591.8 41.06 316 0.264 0.262

12 562.9 41.30 291 0.257 0.293

1̂

23

The values of Trij, B0, B1 and Bij calculated for each ij pair by eqs. (3.65), (3.66)

and (11.69), are as follows:

ij Trij B0 B1 Bij/cm3 mol-1

11 0.603 -0.865 -1.300 -1378.6

22 0.546 -1.028 -2.045 -1873.9

12 0.574 -0.943 -1.632 -1610.4

24

0

1.6

1

4.2

0.4220.083 3.65

0.1720.139 3.66

rij

cij

r

r

TT

T

BT

BT

24

ˆ ij cij

ij

cij

B PB

RT

0 1ˆij ijB B B

0 1

0 1

ij cij

ij

cij

cij

ij ij

cij

B PB B

RT

RTB B B

P

(11.69b)

(11.69a)

R = 83.14 cm3 bar mol-1 K-1


7

Calculating 12 according to its definition gives:

Eq.(11.63) then yield:

Thus,

25

3 -1

12 12 11 222 2 1610.4 1378.6 1873.9 31.7 cm molB B B

22

1 11 2 12

22

2 22 1 12

25ˆln 1378.6 0.5 31.7 0.01288314 323.15

25ˆln 1873.9 0.5 31.7 0.01748314 323.15

PB y

RT

PB y

RT

1 2ˆ ˆ0.987 and 0.983

R = 8314 cm3 kPa mol-1 K-1

The chemical potential for ideal gas mixture model is given by eq. (11.24)

For an ideal solution,

Partial volume,

By eq. (11.4), ; thus,

, lnid id

i i i iG G T P RT x

26

, lnig ig ig

i i i iG G T P RT y

(11.24)

(11.75)

,

idid i i

i

TT x

G GV

P P

/i iTG P V

id

i iV V (11.76)

26

As a result of eq. (11.19),

By eq. (11.5),

Because , substitution by eqs. (11.75) and (11.77) yield

or

The summability relation, eq. (11.11), applied to the special case of an ideal

solution, is written:

27

,

lnid

id i ii i

PP x

G GS R x

T T

lnid

i i iS S R x (11.77)

ln lnid

i i i i iH G RT x TS RT x

id id id

i i iH G TS

id

i iH H (11.78)

id id

i i

i

M x M

Application to eqs. (11.75) – (11.78) yields

28

ln 11.79

ln 11.80

11.81

11.82

id

i i i i

i i

id

i i i i

i i

id

i i

i

id

i i

i

G x G RT x x

S x S R x x

V x V

H x H


8

From eqs. (11.46) and (11.31),

Subtraction yields the general equation:

For the special case of an ideal solution,

Comparison with eq. (11.75) gives:

This equation, known as the Lewis/Randall rule, applies to each species in an ideal solution at all conditions of temperature, pressure and composition.

29

ˆln

ln

i i i

i i i

T RT f

G T RT f

ˆlni i i iG RT f f

ˆ id

i i if x f (11.83)

(11.46)

(11.31)

ˆlnid id id

i i i i iG G RT f f

, lnid id

i i i iG G T P RT x

(11.75)

It shows that the fugacity of each species in an ideal solution is proportional

to its mole fraction.

Division of both sides of eq. (11.83) by Pxi gives

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.

Raoult’s law is based on assumption of ideal solution behavior for the liquid

phase, the same system that obey Raoult’s law form ideal solution.

Ideal solutions are liquid phases whose molecules are of similar size and

which are of similar chemical nature.

30

ˆ

ˆ

id

i i i

i i

id

i i

f x f

Px Px

(11.84)

ˆˆ 11.52

11.34

ii

i

ii

f

y P

f

P

Residual properties are used to measure the deviation of real fluid

properties to ideal gas properties. Although residual properties can be

used for all fluid (both gases and liquids), it is more convenient to measure

the deviation of real solution (liquid mixtures) to that of ideal solution.

This is accomplished through the use of excess properties.

Hence, excess properties are analogous to residual properties (residual

for deviation from ideal gas and excess for deviation from ideal solution).

Excess properties is defined as:

31

E idM M M 11.85

Eq. (11.46) may be written as

In accord with eq. (11.83) for an ideal solution, this becomes

By difference,

The left side of this equation is the partial excess Gibbs energy ; the

dimensionless ratio on the right side is the activity coefficient of species i in

solution, i.

By definition,

32

ˆlni i iG T RT f

lnid

i i i iG T RT x f

ˆlnid i

i i

i i

fG G RT

x f

E

iG

ˆi

i

i i

f

x f (11.90)

ˆ id

i i if x f (11.83)


9

So,

These equation establish a thermodynamic foundation for the activity

coefficient, as a factor introduced into Raoult’s law to account for liquid

phase non-idealities.

Comparison with eq. (11.51) shows that eq. (11.91) relates i to

exactly as eq. (11.51) relates to .

For an ideal solution, and therefore i = 1.

A further relation results from combination of eq. (11.91) with eq. (11.88)

Elimination of the final term by eq. (11.75) and rearrangement gives

This equation is an alternative definition of the activity coefficient.

33

lnE

i iG RT (11.91)

ln id

i i iRT G G

lni i i i iG G RT x (11.92)

E id

i i iM M M (11.88)

, lnid id

i i i iG G T P RT x

(11.75)

E

iG

ˆlnR

i iG RT (11.51)

ˆi

R

iG

0,E

iG

For ideal gas mixture model,

For ideal solution model,

For non-ideal solution,

34

lnig ig

i i iG RT y (11.24)

lnid

i i iG RT x (11.75)

lni i i iG RT x (11.92)

From summability relation (eq. 11.11),

Substitution of eq. (11.91),

These relation is very important in determining activity coefficient, i.

From Gibbs/Duhem eqn. at constant T and P (eq. 11.14):

Substitution of eq.(11.91),

For constant T and P,

This eqn. is very important to check the thermodynamic consistency of

binary liquid-vapor equilibrium.

35

E E

i i

i

G x G

ln lnE

E

i i i i

i i

GG RT x or x

RT

0E

i i

i

x dG

ln 0i i

i

x d RT

ln 0i i

i

x d

lnE

i iG RT (11.91) (11.99)

(11.100)

Smith, J.M., Van Ness, H.C., and Abbott, M.M. 2005. Introduction to

Chemical Engineering Thermodynamics. Seventh Edition. Mc

Graw-Hill.

36


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PREPARED BY:

MDM. NORASMAH MOHAMMED MANSHOR

FACULTY OF CHEMICAL ENGINEERING,

UiTM SHAH ALAM.

[email protected]

03-55436333/019-2368303

Solution Thermo(3)

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