An Alternative View of Fugacity Leon L. Combs P. 0. Drawer CH, Mississippi State University, Mississippi State, MS 39762
The concept of fugacity originates o ' m the desire to determine the pressure dependence of the chemical potential of a real gas. Years of teaching physical chemistry have led me to note two conceptual problems with the normal derivation of fugacity equations. This article notes the problems and offers a solution. Traditional Derivation From a fundamental equation of thermodynamics
. . .fugacity measures the Gihhs energy of a real gas in the way that pressure measures the Gibhs energy of an ideal gas. Atkins (4) calls fugacity the "effective pressure". Barrow (2) says
. . .fugacity is a pressurelike quantity that is used ta treat
the free energy of nonideal gases.
dG=-SdT+Vdp
Levine (5) states
one obtains, a t constant temperature &=V,dp
Different texts attempt various definitions of fugacity. Castellan (3) states
(1)
Using the equation of a n ideal gas for V,
substituting it into eq 1,and integrating between 1bar and a n arbitrary pressure p , we get This equation is valid only for an ideal gas because the ideal gas equation of state was used for V, in eq 1. However for a real gas, most physical chemistry texts (the texts by Noggle ( I ) and Barrow (2) provide more details than most books) state that eq 3 is modified for real gases by replacing pressure with fugacity f to give The standard approach to relate f to some measurable property quickly yields
Equation 5 can then be solved for In f or for ln (flp) = In y, where y is the fugacity coefficient. The equation can also be
. .fugacity f, plays the same role in a nonideal gas mixture as the partial pressure Pj in an ideal gas mixture. These statements certainly have validity, but students still ask "OK, but what is fugacity?" Also, most chemical engineers and some chemists become very proficient a t using fugacity plots, but they still do not have a n intuitive understanding of fugacity. Conceptual Problems The Meaning of Effective Pressure
Conceptually, there are two problems with eq 5. One problem occurs when the students are told that fugacity is a n "effective pressure", and then they try to understand what that means. The p's on both sides of eq 5 are the experimental pressure, not the pressure of an ideal gas. So sincep is the real pressure, what does fbeing the "effective pressure" mean? Ifeq ti is used to deterrnincf: and thcn thc students try to understand why f 1s called the "effiaive ~rexsure"from that equation, co&sion also arises. quat ti on 6 seems to say that f i s p with corrections made for size and intermolecular forces (through the virial coefficients). But p is the real pressure, not the ideal gas pressure. So what does i t mean to correct the real pressure for size and intermolecular forces? A Mathematical Discrepency
where z is the compressibility or compression factor
which can be obtained from the virial series
z = I + B ' ~ + ~ ? ... + where B is the second virial coefficient, C is the third virial coefficient, etc. If one knows enough virial coefficients to give a n acceptable degree of accuracy, then eq 6 can be easily integrated to give f a t anyp in terms of the virial coefficients. But the introduction of fugacity is from eq 4. Thus, it appears that fugacity replaces pressure for a real gas. 218
Journal of Chemical Education
The second ~ r o b l e marises when the students look closely a t the derivation of eq 5. They see that the lefbhand side orieinated from settine the lower limit of oressure integration to be 1bar in the &ations leading tdthe in f and l n p terms of eqs 3 and 4. The right-hand side has zero a s its lower limit of pressure integration. This observation is confusing because it is not mathematically correct. Not all students will observe this discrepancy, particularly since eq 3 usually occurs several pages before eq 4, a s in the text by Adamson (6). An excellent text by Reid (7)removes the mathematical inconsisteney, but he does not address the conceptual confusion. Both of these confusions can be eliminated by a n alternative approach to deriving eq 5. Alternative Derivation The alternative derivation begins by substituting eq 2 into eq 1, and integrating between pa and p , rather than between 1 bar a n d p to obtain for an ideal gas
[I
~ ~ ~ - p IV;dp ! ~ = =
~
~
l
n
~
PO
Now, for a real gas
P
~-P"=Jv,dp PO
(11)
The next step would be to substitute an equation of state for V, into eq 11 and integrate. However, there is no equation of state that is known with accuracy for all gases and that is valid over any pressure range. Also, it would still be desirable if eq 11 could be rearranged to a form that is similar to the simple form of eq 3, so that equilibrium constants and other properties of real systems can be readily obtained. Thus, in order to obtain an equation like eq 3 without using a lower limit of 1 bar in the integration, a definition is introduced so that the right-hand side of eq 11 becomes
where f is the fugacity a t p, and f" is the fugacity a t pa, which becomes p when po -+ 0. By definition, this states that if we knew V, exactly in terms ofp, then the result of the integration would be the right-hand side of eq 12 in analogy with the ideal gas derivation of eq 3. The next step is to substitute eq 12 into eq 11, and then to subtract the blocked portion of eq 9 from eq 11 to give
Now as pa approaches zero, f" will approachpo,and the first two terms of eq 14, will cancel so that eq 14 becomes ea 5. This route to ea 5 removes both the conce~tual roblems mentioned above. There is no ambig& wit6 the choice of lower inteaation limits, and it is clear that the concept of fugacity comes from correcting V,, not p. Equation 4 can then be obtained by substituting eq 12 into eq 11, and lettingf" be 1 bar as a standard fugacity, in the sense that uo is defined as u whenf is eaual to 1 bar. Thus, conceptually, fugacity resilts from using an exact eauation of state for V,... in the left side of ea.12.. which corrects for deviations from ideality due to size and intermolecular forces. Since this equation of state is not known, it is necessary to make the stated definition, and determine the fugacity from the right hand of eq 5 or 6. The driving force to introduce the definition is to remain with the simple form of eq 4 so that uncomplicated formulas for equilibrium expressions will be obtained. Conce~tuallv.two things become clear: The correction has bee; made'for the nonideal uolurne of the gas, and fugacity is just a term introduced to make this volume correction in a simple form. Thus, it is also clear why fugacity will d e ~ e n dupon sizes and intermolecular forces. It also then seems preferable to say that fugacity, in itself. has no nhvsical meanine. However. when we know the fug&ity, we'ck calculate eqkibrium cbnstants of real gas systems, as well as other properties of real gases. There are many functions in science that have no physical meaning by themselves, but from them we can obtain perhaps all the information possibly available about a system. An exam~leis the wavefunction w for some svstem. It is much hette; to affirm at the beginning that the function itself has no physical meaning. Thus, you can eliminate conceptual problems before they can be formulated.
Conclusion An alternative derivation of the equations to determine fugacity has been presented. This dekvation seems better than the traditional method, and it can be presented as another approach to help students develop a better understanding of the concept of fugacity. Literature Cited 1. Noggle, J. H.Pkysiml Chemi~lry;Little, Bmwn a n d c o m p a n y : Boston, 1985;pp 27& 281. 2. Barrow.G.M.Phrsim1 Chamlslry,5th ed: M e G r a w , N Y k 1988:pp23LLZ43. 3. Castellan, G. W .Physlcol Chamisfry, 3rd ed: Addison Wesley: Reading, 1983:p 215, 4. Atkhs, P.W Physical Chemistry, 4 t h ed; W. H . Freeman: New York. 1990: pp 12& 126.
Volume 69 Number 3 March 1992
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