Solution Thermo(2)

CPE553 CHEMICAL ENGINEERING THERMODYNAMICS 4/7/2013

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SOLUTION THERMODYNAMICS: THEORY

• THE IDEAL GAS MIXTURE MODEL

• FUGACITY AND FUGACITY

COEFFICIENT: PURE SPECIES

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The molar volume of an ideal gas is V = RT/P. All ideal gases, whether pure or mixed, have the same molar volume at the same T and P.

The partial molar volume of species i in an ideal gas mixture is found from eq. (11.7) applied to the volume; superscript ig denotes the ideal gas state:

where the final equality depends on the equation

This means that for ideal gases at given T and P the partial molar volume, the pure species molar volume, and the mixture molar volume are identical:

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, ,, ,

/

j jj

ig

ig

i

i i iT P n nT P n

nV nRT P RT n RTV

n n P n P

i jjn n n

ig ig ig

i i

RTV V V

P

(11.20)

Partial pressure of species i in an ideal gas mixture is define as the pressure that species i would exert if it alone occupied the molar volume of the mixture.

where yi is the mole fraction of species i. The partial pressures obviously sum to the total pressure.

Gibbs’s theorem statement:

This is expressed mathematically for generic partial property

by the equation:

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1, 2, ..., ii iig

y RTp y P i N

V

A partial molar property (other than volume) of a constituent species in an

ideal gas mixture is equal to the corresponding molar property of the

species as a pure ideal gas at the mixture temperature but at a pressure

equal to its partial pressure in the mixture.

ig ig

i iM V

, ,ig ig

i i iM T P M T p

(11.21)


2

The enthalpy of an ideal gas is independent of pressure; therefore

More simply,

where is the pure species value at the mixture T and P.

An analogous equation applies for and other properties that are

independent of pressure.

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, , ,ig ig ig

i i i iH T P H T p H T P

ig

iU

(11.22) ig ig

i iH H

ig

iH

The entropy of an ideal gas does depend on pressure, and by eq. (6.24),

Integration from pi to P gives

Substituting this result into eq. (11.21) written for the entropy yields

or

where is the pure-species value at the mixture T and P.

, , lnig ig

i i i iS T p S T P R y

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, , lnig ig

i i iS T P S T P R y

ln const Tig

idS Rd P

, , ln ln lnig ig

i i i i

i i

P PS T P S T p R R R y

p y P

lnig ig

i i iS S R y

(11.23)

ln const T

i i

P P

ig

i

p p

dS R d P

ig

iS

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ig ig

i p

dT dPdS C R

T P (6.24)

, ,ig ig

i i iM T P M T p

(11.21)

For the Gibbs energy of an ideal gas mixture,

the parallel relation for partial properties is

In combination with eqs. (11.22) and (11.23) this becomes

or

The summability relation, eq. (11.11), with eqs. (11.22), (11.23), and (11.24) yields

lnig ig ig

i i i iG G RT y

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ig ig igG H TS

ig ig ig

i i iG H TS

lnig ig ig

i i i iG H TS RT y

(11.24)

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ig ig

i i

i

H y H

lnig ig

i i i i

i i

S y S R y y

lnig ig

i i i i

i i

G y G RT y y

(11.25)

(11.26)

(11.27)

(11.22) ig ig

i iH H

lnig ig

i i iS S R y

(11.23)

Property change of mixing for ideal gas is defined as:

From eq. (11.25), enthalpy change of mixing for ideal gas is zero.

From eq. (11.26), entropy change of mixing for ideal gas is:

Because 1/yi > 1, this quantity is always positive, in agreement with the

second law.

From eq. (11.27), Gibbs free energy change of mixing for ideal gas is:

0ig ig

i i

i

H y H

8

8

1lnig ig

i i i

i i i

S y S R yy

0ig ig

i i

i

M y M

lnig ig

i i i i

i i

G y G RT y y


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An alternative expression for the chemical potential results when

in eq. (11.24) is replaced by an expression giving its T and P dependence.

This comes from eq. (6.10) written for an ideal gas:

Integration gives

where , the integration constant at constant T, is a species-dependent

function of temperature only.

Eq. (11.24) is now written as

where the argument of the logarithm is the partial pressure.

Application of the summability relation, eq. (11.11), produces an expression

for the Gibbs energy of an ideal gas mixture:

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ig

i

ig

iG

ln const Tig ig

i i

RTdG V dP dP RTd P

P

lnig

i iG T RT P

i T

(11.28)

lnig ig

i i i iG T RT y P

(11.29)

lnig

i i i i

i i

G y T RT y y P (11.30)

dG VdP SdT (6.10)

lnig ig ig

i i i iG G RT y

(11.24)

Ideal gases and ideal gas mixtures have analytical and well defined

equations but their applications are limited. They cannot be used to

describe the behavior of real fluids (which deviate significantly from ideal

gas behavior).

The concept of fugacity was introduced so that the ideal gas mixture

equations could be used for real fluids.

This chapter will concentrate on the definitions and formula of fugacities.

The next chapter will discuss on the applications of fugacities in vapor

liquid equilibrium.

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The origin of the fugacity concept resides in eq. (11.28), valid only for pure species i in the ideal gas state.

The concept of fugacity was introduced in order for eq. (11.28) to be valid for pure species, real fluid at constant temperature.

Thus, for a real fluid, an analogous equation that defines fi, the fugacity of pure species i is written:

This new property fi, with units of pressure, replaces P in eq. (11.28). If eq. (11.28) is a special case of eq. (11.31), then:

and the fugacity of pure species i as an ideal gas is necessarily equal to its pressure.

Subtraction of eq. (11.28) from eq. (11.31), both written for the same T and P, gives

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lni i iG T RT f

ig

if P

(11.31)

(11.32)

lnig ii i

fG G RT

P

lnig

i iG T RT P (11.28)

A residual property is simply the difference of a system property to the

property if the system behaves as an ideal gas (or ideal gas mixture)

In previous eqn., Gi – Giig is the residual Gibbs energy, Gi

R; thus

where the dimensionless ratio fi/P has been defined as fugacity coefficient,

given by the symbol i:

Eq. (11.34) apply to pure species i in any phase at any condition.

As a special case they must be valid for ideal gases, for which GiR = 0, i = 1,

and eq. (11.28) is recovered from eq. (11.31).

Eq. (11.33) may be written for P = 0, and combine with eq. (6.45):

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lnR

i iG RT

ii

f

P

(11.33)

(11.34)

0

1 const TR

PG dPJ Z

RT P (6.45)

0 0lim limln

R

ii

P P

GJ

RT

R

igM M M (6.41)


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In connection with eq. (6.48), the value of J is immaterial, and is set equal to

zero.

and

The identification of ln i with GiR/RT by eq. (11.33) permits its evaluation by

the integral of eq. (6.49)

0 0lim lim 1i

iP P

f

P

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0 0

1R

P P

P

S Z dP dPT Z

R T P P

(6.48)

0 0limln limln 0i

iP P

f

P

0

1 const TR

PG dPZ

RT P

0

ln 1 const TP

i i

dPZ

P

(6.49)

(11.35)

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Fugacity coefficients (and therefore fugacities) for pure gases are evaluated

by eq. (11.35) from PVT data or from a volume explicit equation of state.

For example, when the compressibility factor is given by eq (3.38),

Because Bii is a function of temperature only for a pure species, substitution

into eq. (11.35) gives

1 iii

B PZ

RT

14

1PV BP

ZRT RT

(3.38)

14

0

ln const TP

iii

BdP

RT

ln iii

B P

RT

(11.36)

There are a few methods for the determination of fugacity coefficient of

pure species:

◦ Compressibility factor method, useful for academic purpose

◦ Generalized correlations (e.g. Lee-Kesler), easy and practical

◦ Derived from Equation of States (e.g. Virial, etc.), useful especially in computer

simulation

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Evaluation of fugacity coefficients through cubic equations of state (e.g. the

Van der Waals, Redlich/Kwong, and Peng Robinson eqs) follows directly

from combination of eqs. (11.33) and (6.66b):

where

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1 lnRG

Z Z qIRT

(6.66b)

ln 1 lni i i i i iZ Z q I (11.37)

bP

RT

a Tq

bRT

(3.50)

(3.51)

1 1ln

1

bI

b

(6.65b)

lnR

i iG RT (11.33)

where:

a and b are positive constant

and are pure numbers, same

for all substances


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This eqn. written for pure species i (for the van der Waals equation,

Ii = βi/Zi).

Application of eq. (11.37) at a given T and P requires prior solution of an

equation of state for Zi by eq. (3.52) for a vapor phase or eq. (3.56) for a

liquid phase.

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1

ZZ q

Z Z

1 Z

Z Z Zq

(3.56)

(3.52)

Eq. (11.31) may be written for species i as saturated vapor and as a saturated

liquid at the same temperature:

By difference,

This eqn applies to the change of state from saturated liquid to saturated

vapor, both at temperature T and at the vapor pressure Pisat.

lnl l

i i iG T RT f

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lni i iG T RT f (11.38a)

(11.38b)

lnl ii i l

i

fG G RT

f

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According to eq. (6.69), Giv - Gi

l = 0; therefore:

where fisat indicates the value for either saturated liquid or saturated

vapor.

Coexisting phases of saturated liquid and saturated vapor are in

equilibrium; eq. (11.39) therefore expresses a fundamental principle:

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l sat

i i if f f

G G (6.69)

(11.39)

For a pure species coexisting liquid and vapor phases are

in equilibrium when they have the same temperature,

pressure and fugacity.

An alternative formulation is based on the corresponding fugacity

coefficients:

whence

This equation, expressing equality of fugacity coefficients, is an equally valid

criterion of vapor/liquid equilibrium for pure species.

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satsat ii sat

i

f

P

l sat

i i i

(11.40)

(11.41)


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The fugacity of pure species i as a compressed liquid may be calculated

from the product of easily evaluated ratios:

Ratio (A) is the vapor phase fugacity coefficient of pure vapor i at its

vapor/liquid saturation pressure, designated isat. It is given by eq. (11.35),

written,

In accord with eq. (11.39) ratio (B) is unity.

Ratio (C) reflects the effect of pressure on the fugacity of pure liquid i.

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sat l sat li i i il sati

i isat sat l sat

i i i i i

A B C

f P f P f Pf P P

P f P f P

0

ln 1 const TsatiPsat

i i

dPZ

P

(11.42)

The basis of its calculation is eq. (6.10), integrated at constant T to give

Another expression for this difference results when eq. (11.31) is written

for both Gi and Gisat; subtraction then yields

The two expressions for Gi – Gisat are set equal:

Ratio (C) is then

Substituting for the three ratios in the initial equation yields

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sati

Psat l

i i iP

G G V dP

lnsat ii i sat

i

fG G RT

f

1ln

sati

Pli

isat Pi

fV dP

f RT

1

expsati

lP

liil sat P

i i

f PV dP

RTf P

1exp

sati

Psat sat l

i i i iP

f P V dPRT

(11.43)

dG VdP SdT (6.10)

lni i iG T RT f (11.31)

Because Vil, the liquid phase molar volume, is a very weak function of P at

temperatures well below Tc, an excellent approximation is often obtained

when Vil is assumed constant at the value for saturated liquid. In this case,

The exponential is known as Poynting factor. Data required for application

of this equation:

◦ Values of Ziv for calculation of i

sat by eq. (11.42). These may come from

an equation of state, from experiment, or from a generalized

correlation.

◦ The liquid phase molar volume Vil, usually the value for saturated liquid.

◦ A value for Pisat.

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exp

l sat

i isat sat

i i i

V P Pf P

RT

(11.44)

If Ziv is given by eq. (3.38), the simplest form of the virial equation, then

and eq. (11.44) becomes

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sat

i1 and expsat

ii ii ii

B P B PZ

RT RT

exp

sat l sat

ii i i isat

i i

B P V P Pf P

RT

(11.45)


7

For H2O at a temperature of 300oC (573.15K) and for pressures up to

10 000 kPa (100 bar) calculate values of fi and i from data in the steam

tables and plot them vs. P.

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Solution:

Eq. (11.31) is written twice:

First, for a state at pressure P;

Second, for a low pressure reference state, denoted by *,

Both for temperature T:

Subtraction eliminates Гi (T), and yields

By definition Gi = Hi – TSi and Gi* = Hi* - TSi*; substitution gives

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* *ln and lni i i i i iG T RT f G T RT f

*

*

1ln i

i i

i

fG G

f RT

*

*

*

1ln i i i

i i

i

f H HS S

f R T

(A)

The lowest pressure for which data at 300oC (573.15K) are given in the steam table is 1 kPa. Steam at these conditions is for practical purposes an ideal gas, for which fi* = P* = 1 kPa. Data for this state provide the following reference values:

Hi* = 3076.8 J g-1 Si* = 10.3450 J g-1 K-1

Eq. (A) may now be applied to states of superheated steam at 300oC (573.15K) for various values of P from 1 kPa to the saturation pressure of 8592.7 kPa.

For example, at P = 4000 kPa and 300oC (573.15K):

Hi = 2962.0 J g-1 Si = 6.3642 J g-1 K-1

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Values of H and S must be multiplied by the molar mass of water (18.015 g

mol-1) to put them on a molar basis for substitution into eq. (A):

Thus the fugacity coefficient at 4000 kPa is

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3611.00.9028

4000

ii

f

P

-1 -1

-1 -1

* -1 -1

18.015gmol 2962.0 3076.8 Jgln 6.3642 10.3450 Jg K

8.314 Jmol K 573.15K

8.1917

i

i

f

f

*3611.0i

i

f

f

*3611.0 3611.0 1 kPa 3611.0 kPai if f

28


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Similar calculations at other pressures

lead to the values plotted in Fig. 11.3

at pressures up to the saturation

pressure Pisat = 8592.7 kPa. At this

pressure,

Hi = 1345.1 J g-1

Si = 3.2552 J g-1 K-1

Substitution in eq. (A) yields

29

6738.9 kPa

and

0.7843

sat

i

sat

i

f

According to eqs. (11.39) and (11.41),

the saturation values are unchanged by

condensation.

Although the plots are therefore

continuous, they do show discontinuities

in slope.

Values of fi and i for liquid water at

higher pressures are found by

application of eq. (11.44), with Vil equal

to molar volume of saturated liquid

water at 300oC:

3 -1

3 -1 -1

25.29cm mol 10000 8592.7 kPa0.7843 8592.7kPa exp 6789.8 kPa

8314cm kPamol K 573.15Kif

30

6789.8 /10000 0.6790i if P

30

3 -1 -1

3 -1

1.404cm g 18.015gmol

25.29 cm mol

l

iV

At 10 000 kPa, for example, eq. (11.44) becomes

The fugacity coefficient of liquid water at these condition is

Such calculations allow completion of Fig.

11.3, where the solid lines show how fi and i

vary with pressure.

The curve for fi starts at the origin, and

deviates increasingly from the dashed line for

an ideal gas (fi = P) as the pressure rises.

At Pisat there is discontinuity in slope, and the

curve then rises very slowly with increasing

pressure, indicating that the fugacity of liquid

water at 300oC (573.15K) is a weak function

of pressure.

This behavior is characteristic of liquids at

temperatures well below the critical

temperature.

The fugacity coefficient i decreases steadily

from its zero pressure value of unity as the

pressure rises. Its rapid decrease in the liquid

region is a consequence of the near-

constancy of the fugacity itself.

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31

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

Chemical Engineering Thermodynamics. Seventh Edition. Mc

Graw-Hill.

32


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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(2)

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