Solution Thermodianamicco

CHE553 Chemical Engineering Thermodynamics 4/2/2015

1

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:

3

THE IDEAL GAS MIXTURE MODEL

, ,, ,

/

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:

4

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.

5

, , ,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

6

, , 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

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

7

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)

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

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


3

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

iig

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.

Fugacity is a property related to chemical potential. Chemical potential cannot directly measured. Thus fugacity takes the place of chemical potential.

Fugacity measure the tendency of a substance to prefer one phase (liquid, solid, gas) over another. The phase with the lowest fugacity will be the most favorable. Therefore, fugacity is useful for predicting the phase state of multi-component mixtures at various temperatures and pressures.

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.

10

FUGACITY AND ITS IMPORTANCE

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

11

FUGACITY AND FUGACITY COEFFICIENT: PURE SPECIES

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):

12

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)


4

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

13

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)

R R RS H G

R RT RT (6.47)

RH

RT

RG

RT

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 the second virial coefficient, 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)

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

15

DETERMINATION OF FUGACITY COEFFICIENT

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

16

FUGACITY COEFFICIENTS FROM THE GENERIC CUBIC EQUATION OF STATE

1 lnRGZ Z qI

RT (6.66b)

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

bP

RT

a TqbRT

(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


5

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.

17

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 Pi

sat.

lnl l

i i iG T RT f

18

VAPOR/LIQUID EQUILIBRIUM FOR PURE SPECIES

lni i iG T RT f (11.38a)

(11.38b)

lnl ii i l

i

fG G RT

f

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:

19

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.

20

satsat ii sat

i

f

P

l sat

i i i

(11.40)

(11.41)


6

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 i

sat. 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.

21

FUGACITY OF A PURE LIQUID

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 Gi

sat; 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

22

sati

Psat l

i i iP

G G V dP

lnsat ii i sat

i

fG G RT

f

1ln

sati

Pliisat P

i

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 Vi

l 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.

23

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

24

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.

25

EXAMPLE 11.5 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

26

* *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

27

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

28

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


8

Similar calculations at other pressures lead to the values plotted in Fig. 11.3 at pressures up to the saturation pressure Pi

sat = 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 Vi

l 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

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

30

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.

31

Smith, J.M., Van Ness, H.C., and Abbott, M.M. 2005. Introduction to Chemical Engineering Thermodynamics. Seventh Edition. Mc Graw-Hill.

32

REFERENCE


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PREPARED BY: NORASMAH MOHAMMED MANSHOR FACULTY OF CHEMICAL ENGINEERING, UiTM SHAH ALAM. [email protected] 03-55436333/019-2368303

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Solution Thermodianamicco