Fugacity, Ideal Solutions, Activity, Activity Coefficient

Chapter 7

Why fugacity?

Equality of chemical potential is the fundamental criterion for phase and chemical equilibrium.

It is difficult to use, because it approaches -∞ as the concentration approaches 0.

Fugacity approaches 0 as the concentration approaches 0.

)(ln TBfRTg iiii

As chemical potential approaches -∞, RT ln(f) must also approach -∞

Since RT is a constant, ln(f) must approach -∞, and so f must approach 0

The units of chemical potential are kJ/mole – so RT was needed to make the units work out, since natural logs are dimensionless

)(ln TBfRTg iiii

Most practical calculations do not include B(T)

)(ln TBfRTg iiii

RT

TBn

TSPVU

f

i

nPTi

ij

)()(

exp,,

)(ln TBfRTg iiii

We know that if phase 1 and phase 2 are in equilibrium then:

)2()1(ii

)2()1(ii ff Our new criterion for

physical equilibrium

Pure substance fugacities

gg i For pure substances…

)(ln TBfRTg

)(ln TBfRTg

Take the derivative

dBfTdfdTRdg )ln(ln

At constant temperature….

And recall that…

vdPsdTdg

fRTdvdP ln

Combine and rearrange

RT

v

P

f

T

ln

P

1 And for an

ideal gas…

But…

PP

P

T

1ln

TT P

f

P

P

lnln

For all gases, as the pressure approaches zero, the gas approaches ideal behavior, so…

Which implies that for an ideal gas, f=p

0 as 1lim

PP

f

You can check out the rest of the math in Appendix C, however…

2

*ln

RT

hh

T

f

P

The * indicates an

ideal gas

P

dPRTP

f0

1

exp

PdP

P

z0

1-exp

zP

RT

zvv

P

RT

11

1 residual volume

zP

RT

zvv

P

RT

11

1 residual volume

For an ideal gas,

Z=1 for an ideal gas, so 1-z=0

So what is the fugacity? For an ideal gas it is identical to the

pressure It has the dimensions of pressure Think of it as a corrected pressure,

and use it in place of pressure in equilibrium calculations

Calculations often include the ratio of fugacity to pressure, f/P – called the fugacity coefficient

Notice that f/P approaches 1 as the value of Pr goes down, and the value of Tr goes up—in other words, as the gas approaches ideality

Fugacity of liquids and solids

Follows the same equations as gases, but in practice are calculated differently

The fugacity of gases and liquids is usually much less than the pressure

The Fugacity of Pure gases

Illustrate this process with example 7.1, page 134

Estimate the fugacity of propane gas at 220 F and 500 psia

The easiest way is to use figure 7.1

We need Pr and Tr

CRr P

PP 812.0

)7.14*9.41(

500

psi

psi

CRr T

TT 022.1

R 6.665

R 680

812.0rP

022.1rT

76.0f

Now let’s calculate f, instead of using a chart

If you have reliable PvT measurements, like the steam tables, you could use them – which would give the most reliable values

For this problem, PvT measurements were used to create Figure 7.2 – a plot of z values. (Pv=zRT)

The data are also presented in Table 7.A, so we don’t have to read a graph

Table 7.A – Volume residuals of Propane at 220 F

Pressure, psia Z RT/P * (1-z)

0 1.0 3.8

100 0.9462 3.923

200 0.8873 4.110

300 0.8221 4.324

400 0.7787 4.582

500 0.6621 4.929

Recall that…

P

dPRTP

f0

1

exp

changes with pressure, but not as strongly as z, so we can approximate as a constant

RT

P

P

f exp 747.0exp 67.679*73.10

500*25.4

I shamefully left out units, in the interest of space – see page 135

We could also use an equation of state (EOS) to find the fugacity

The ideal gas law would tell us that pressure and fugacity are equal – which is obviously a really bad estimate for this case

Let’s try the “little EOS” from Chapter 2 Equations 2.48 – 2.50

Little EOS Pv=zRT

10 zzz

6.10 422.0

083.01rr

r

TT

Pz

2.41 172.0

139.0rr

r

TT

Pz

Eq 2.48

Eq 2.49

Eq 2.50

P r dPP

Tf0

rPP )( /T

exp cr

Pr dP

P

Tf0

rr )( /TPexp

Substitute into Eq. 7.9

PdP

P

z

P

f0

1-exp

)(exp r

r

r TfT

P

Do the integration

Now we can substitute in the values of Pr and Tr=0.771

One more approach…Use Thermodynamic tables

Recall that…

)(ln TBfRTg Rewrite this equation for two different states

1212 lnln ffRTgg 1

2lnf

fRT

(If you keep the temperature constant, and vary the pressure)

1

212 ln

f

fRTgg

RT

gg

f

f 12

1

2 exp

RT

gg

PP

Pf

P

f 12

21

11

2

2 exp

Divide both sides by P1P2 and rearrange

At very low pressures, f and P are equal

RT

gg

P

P

P

f 12

2

1

2

2 exp But only at very low pressures

RT

gg

P

P

P

f 12

2

1

2

2 exp

Now plug in values from a table of thermodynamic values – pg 137

Choose P1 as 1 psia

lbmol

lbm

RlbmolBtu

lbmBtu

Rpsia

psia

P

f062.44*

67.679987.1

54.173528.1554exp

500

1

2

2

740.02

2 P

f

There are obviously several ways to find fugacity, depending on what you know and how accurate you need to be

Use a fugacity coefficient table like Figure 7.1

Use PvT data, to find z , then use equation 7.9

Use an equation of state Use thermodynamic tables and

equation 7.11

Fugacity of Pure Liquids and Solids

We could compute the fugacity just like we did in the previous example

It is impractical for mathematical reasons

See examples 7.2 and 7.3

Fugacity of Solids and Liquids

Does not change appreciably with pressure

Normally we approximate the fugacity of pure solids and liquids as the pure component vapor pressure

We’ll return to this in Chapter 14 when we study cases where this approximation is no longer valid

Fugacities of species in mixtures

)(ln TBfRTg iiii

RT

v

P

f i

T

i

ln

0 as 1lim

P

Px

f

i

i

See Appendix C

Fugacities of species in mixtures

2

*ln

RT

hh

T

f ii

P

i

P

ii

i dPRTPx

f0

1

exp

PdP

P

z0

1-exp

These equations are the same as the pure component equations, except that P has been replaced with the partial pressure, xiP

See Appendix C

Mixtures of Ideal Gases

P

RTnV T True only for ideal gases

P

RTn

nv T

nPTii

j,,

But R, T and P are constant, so…

jnPTi

Ti n

n

P

RTv

,,

1

and 0 ii vP

RT

All of which leads to…

PyPf iii

For mixtures of ideal gases, the fugacity of each species is equal to the partial pressure of that species

Can we extend this concept to ideal solutions?

Ideal solutions are like ideal gases Neither exist in nature – real gases and

solutions are much more complicated Many gases and solutions exhibit

practically ideal behavior It is often easier to work with deviations

from ideal behavior, rather than work directly with the property of interest The compressibility factor z is an example Activity coefficient is similar for liquids

Ideal solutions

The definition of an ideal solution is that …

0ii xff

Usually fi0 is defined as the pure

component fugacity – though this is not the only choice

Any ideal solution has the following properties

0iii fxf

iii xRTgg ln0

00 ii vv

iii xRss ln0

00 ii hh

True for gases, liquids and solids

Any ideal solution has the following properties

0iii fxf

iii xRTgg ln0

00 ii vv

iii xRss ln0

00 ii hh

Tells us there is no volume change with mixing

Any ideal solution has the following properties

0iii fxf

iii xRTgg ln0

00 ii vv

iii xRss ln0

00 ii hhTells us that there is no heat of mixing

Any ideal solution has the following properties

0iii fxf

iii xRTgg ln0

00 ii vv

iii xRss ln0

00 ii hh

There is always an entropy change associated with mixing

Which means there is a Gibbs free energy change with mixing

Activity and Activity Coefficients Fugacity has dimensions of pressure

This can cause problems if we don’t pay strict attention to units

Often we want a non-dimensional representation of fugacity – which leads to the activity

When we deal with non-ideal solutions we’ll want a measure of departure from ideal – like z – which leads to the activity coefficient

Activity 0i

ii f

fa

Activity is defined as the ratio of the fugacity of component i, to it’s pure component fugacity

Recall that for an ideal gas…

Pyf ii

Recall that for an ideal solution…

0iii fxf

Usually – but not necessarily, chosen as the pure component fugacity

Activity 0i

ii f

fa

0i

ii f

fx ia

For an ideal solution of either gas or liquid, activity is equivalent to mole fraction

Rearrange to give…

Activity Coefficient

i

ii x

a

0ii

i

fx

f

The activity coefficient is just a correction factor on x, which converts it to a

iii ax

i

ii x

a

0iii faf 0

iii fx

Why bother?

Both are dimensionless They lead to useful correlations of

liquid-phase fugacities. The normal chemical equilbrium

statement – the law of mass action – is given in terms of activities (The law of mass action is the definition

of the equilibrium constant, K)

For now…

Activity is rarely used for phase equilibria

It will show up again in Chapters 12 and 13

Activity coefficient is useful to us now!!

For a pure species or for ideal solutions

Activity = mole fraction and.. Activity coefficient, =1 We could redefine an ideal solution as

one where equals 1 This is like defining an ideal gas as

one where z=1

Activity coefficients

For real solutions activity coefficients can be either greater or less than 1

Typically range between 0.1 and 10

From Appendix C (the general case)

RT

vv

Pii

xT

i

i

0

,

ln

2

0

,

ln

RT

hh

Tii

xP

i

For an ideal solution

0

0

Eq. 7.31

Eq. 7.32

Example 7.4

At 1 atm pressure the ethanol-water azeotrope has a composition of: 10.57 mole% water 89.43 mole% ethanol

The composition is the same in the vapor and liquid phases

The temperature is 78.15 C

Example 7.4 cont

At this temperature the pure component vapor pressures are: Water 0.434 atm Ethanol 0.993 atm

Estimate the fugacity and activity coefficients in each phase

For an azeotrope…xethanol=yethanol and xwater=ywater

At one atm it is a reasonable assumption that both the water and ethanol behave as ideal gases

Thus the fugacity in the gas phase is equal to the partial pressure

atmf gas

ethanol8943.0

atmf gas

water1057.0

atmyethanol 1*

atmywater 1*

Because we are at equilibrium the gas phase fugacity equals the liquid phase fugacity

atmf liquid

ethanol8943.0

atmf liquid

water1057.0

0ethanolethanol fa

0waterwater fa

If this was an ideal solution, a would be equal to x – but it’s not, so…

ethanolethanolxethanola

waterawaterwater x

We use the pure component vapor pressure as the standard fugacity

atmfethanol 8943.0

atmfwater 1057.0

0ethanolethanolPa

0waterwaterPa

If this was an ideal solution, a would be equal to x – but it’s not, so…

ethanolethanolxethanola

waterawaterwater x

0ethanolethanol fa

0waterwater fa

Remember, I’m using Pi0 to indicate

the pure component vapor pressure – but Dr. deNevers is using p

Thus

0

ethanolethanol

phasegasethanol

ethanol Px

f

0

waterwater

phasegaswater

water Px

f

atm

atm

993.0*8943.0

8943.0

atm

atm

244.0*1057.0

1057.0

007.1

31.2

Thus

0

ethanolethanol

phasegasethanol

ethanol Px

f

0

waterwater

phasegaswater

water Px

f

007.1

31.2

This solution is not ideal, as shown by the fact that the activity coefficients are not 1

Fugacity coefficient, Poynting Factor and Alternative Notation

The ratio of fugacity to pressure is called the fugacity coefficient

Often the symbol, , is used Older literature Usually refers only to pure component fugacities

and pressures Sometimes the symbol, , is used

Newer literature Can refer to both pure components and mixtures

Older usage

PP

ff

ipurei *

0

P*

Pxf iiii *

And since PxP iii a, the activity

, the activity coefficient corresponds to deviation from ideality due to mixing in the gas or liquid

, the fugacity coefficient, corresponds to deviation from ideal gas behavior

Modern usage

For pure species iis exactly the same as i, and accounts only for the departure from ideal gas behavior

For mixtures accounts for not only the deviation from ideal gas behavior, but also for nonideal mixing behavior.

i

iii ˆ

Modern Usage

In older usage, , is called the fugacity of component i.

In more modern usage it is often represented as , and called the partial fugacity – like partial pressure

Obviously, it can’t be a partial molal property, because fugacity is not an extensive property

if

if

Poynting Factor

P

ii

i dPRTPx

f0

1

exp

i

atm

ii pdPRT

Pxf

9503.0

0

0 1

exp

satiii PFpf **0

Estimating Fugacities of Individual Species in Gas Mixtures

If you don’t have an ideal gas – how do you estimate the fugacity of a gas? Use reliable PvT data Use an equation of state

Example 7.5

Example 7.5

Table 7.F presents Volume residuals () for a mixture of methane with n-butane, at 220 F

Use this data to find the fugacity of methane and the fugacity of n-butane in a mixture that is: 78.4 mole% methane 21.6 mole% n-butane

This corresponds to a mixture that is 50 wt% methane

We’ll need the partial molal volume residual,

Find its value at each of the pressures in table 7. F, which will allow us to integrate using the trapezoid rule

methane

P

ii

i dPRTPy

f0

1

exp

Volume Residual

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100

Mole % Methane

Vo

lum

e R

es

idu

al,

ft3

/lbm

ol

Tangent Line, at 78.4% methane

At 220 F and 100 psia

0.6 ft3/lbmole

6.6 ft3/lbmole

This is the partial molal volume residual of methane

Use the method of tangent intercepts

Repeat the process at the other pressures, to create Figure 7.10 avg

P

ii

i dPRTPy

f0

1

exp

961.068073.10

290exp 3

3

RPy

f

lbmoleRftpsia

lbmoleftpsia

i

i

784.0*1000*961.0961.0 psiaPyf ii

psiafi 753

Substitute

Fugacities from an EOS for Gas Mixtures

PvT data are only available for a small number of gas mixtures

For other cases, using a reliable EOS is a good (though not as accurate) alternative

P

ii

i dPRTPy

f0

1

exp

P

i

i dPz

RTPy

f0

P

1-1exp

What equation to use?

This is easier for the graphical procedure, used with PvT data

This is easier if we are going to use an EOS

But we have a problem

The equations of state from chapter 2 are for singe pure species

For mixtures we need a mixing rule Usually semi-empirical For a mixture of a and b, at some T and

P, we can find the pure component values of z and use them in our mixing rule to get a combination value of z

Any mixing rule will have the form…

PTbabamix yyzzfz ,),,,(One possibility is…

nnbb

naamix zxzxz /1/1

Lewis and Randall fugacity rule

bbaamix zxzxz which is just a weighted average

It is equivalent to an ideal solution of non-ideal gases

If n=1

This is not rigorously correct, but is a useful approximation, especially at pressures less than a few atmospheres

Example 7.6

Compares the Lewis Randall fugacity rule results to those found with PvT data

I’ll let you work through it on your own

Lewis Randall Rule

Used because it is simple It is the next step in complexity after

the ideal gas law Unfortunately, sometimes gases exist

in states for which we can not compute i

Gases can exist as a mixture at conditions where they would normally be liquids if they were pure

This makes finding the fugacity coefficient hard

You’ll need to use an equation of state, because you can’t compare it to some non-existent gas

See Example 7.7

Other mixing rules

The Lewis Randall rule is only one of many possibilities

They’ll be introduced in Chapters 9 and 10

Summary

1. Fugacity was invented because chemical potential is awkward

2. For pure gases we correlate and compute f/P based on either measured PvT data, or an appropriate EOS

Summary cont

3. For pure liquids and solids we usually compute the fugacity from the vapor pressure. The effect of increases above the vapor pressure are small.

4. Ideal liquids are like gases, an approximation

5. Activity and activity coefficient are nondimensional representations of fugacity

Summary Cont

6. Fugacity, activity, and activity coefficient are calculated – they can not be measured

7. For mixtures, we calculate fugacity directly from PvT data, or from an EOS

• When we use an EOS we need to use a mixing rule

Summary cont.

8. The simplest mixing rule is the Lewis and Randall rule – an ideal solution of nonideal gases

9. For mixtures of liquids we’ll have to wait for chapters 8 and 9 for appropriate mixing rules