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.

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 Most practical calculations do not include B(T)

We know that if phase 1 and phase 2 are in equilibrium then: Our new criterion for physical equilibrium

Pure substance fugacities For pure substances…

Take the derivative And recall that… At constant temperature….

Combine and rearrange And for an ideal gas… But…

Which implies that for an ideal gas, f=p For all gases, as the pressure approaches zero, the gas approaches ideal behavior, so…

You can check out the rest of the math in Appendix C, however… The * indicates an ideal gas

For an ideal gas, a=0 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

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 a=RT/P * (1-z) 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… I shamefully left out units, in the interest of space – see page 135 a changes with pressure, but not as strongly as z, so we can approximate a as a constant

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 Eq 2.48 Eq 2.49 Eq 2.50

Substitute into Eq. 7.9 =0.771 Do the integration Now we can substitute in the values of Pr and Tr =0.771

One more approach… Use Thermodynamic tables Recall that… Rewrite this equation for two different states (If you keep the temperature constant, and vary the pressure)

Divide both sides by P1P2 and rearrange At very low pressures, f and P are equal But only at very low pressures

Now plug in values from a table of thermodynamic values – pg 137 Choose P1 as 1 psia

Use a fugacity coefficient table like Figure 7.1 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 See Appendix C

Fugacities of species in mixtures See Appendix C These equations are the same as the pure component equations, except that P has been replaced with the partial pressure, xiP

Mixtures of Ideal Gases True only for ideal gases But R, T and P are constant, so… 1 and

All of which leads to… 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 … Usually fi0 is defined as the pure component fugacity – though this is not the only choice

Any ideal solution has the following properties True for gases, liquids and solids

Any ideal solution has the following properties Tells us there is no volume change with mixing

Any ideal solution has the following properties Tells us that there is no heat of mixing

Any ideal solution has the following properties Which means there is a Gibbs free energy change with mixing There is always an entropy change associated 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 Activity is defined as the ratio of the fugacity of component i, to it’s pure component fugacity Recall that for an ideal gas… Recall that for an ideal solution… Usually – but not necessarily, chosen as the pure component fugacity

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

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

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, g =1 We could redefine an ideal solution as one where g 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) Eq. 7.31 For an ideal solution 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

Because we are at equilibrium the gas phase fugacity equals the liquid phase fugacity If this was an ideal solution, a would be equal to x – but it’s not, so…

We use the pure component vapor pressure as the standard fugacity If this was an ideal solution, a would be equal to x – but it’s not, so… Remember, I’m using Pi0 to indicate the pure component vapor pressure – but Dr. deNevers is using p

Thus

Thus 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, n, is used Older literature Usually refers only to pure component fugacities and pressures Sometimes the symbol, f, is used Newer literature Can refer to both pure components and mixtures

Older usage g, the activity coefficient corresponds to deviation from ideality due to mixing in the gas or liquid n, the fugacity coefficient, corresponds to deviation from ideal gas behavior And since a, the activity

Modern usage For pure species fi is exactly the same as ni, 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.

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

Poynting Factor

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 (a) 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

At 220 F and 100 psia Use the method of tangent intercepts 6.6 ft3/lbmole This is the partial molal volume residual of methane 0.6 ft3/lbmole Tangent Line, at 78.4% methane

Repeat the process at the other pressures, to create Figure 7.10 aavg=0.290

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

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… One possibility is…

Lewis and Randall fugacity rule If n=1 which is just a weighted average It is equivalent to an ideal solution of non-ideal gases 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 ni

This makes finding the fugacity coefficient hard 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 Fugacity was invented because chemical potential is awkward For pure gases we correlate and compute f/P based on either measured PvT data, or an appropriate EOS

Summary cont For pure liquids and solids we usually compute the fugacity from the vapor pressure. The effect of increases above the vapor pressure are small. Ideal liquids are like gases, an approximation Activity and activity coefficient are nondimensional representations of fugacity

Summary Cont Fugacity, activity, and activity coefficient are calculated – they can not be measured 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. The simplest mixing rule is the Lewis and Randall rule – an ideal solution of nonideal gases For mixtures of liquids we’ll have to wait for chapters 8 and 9 for appropriate mixing rules