# Talk:Fugacity

## Comment

Definition of Fugacity

Fugacity is a pressure-like property, that allows us to apply the same equations from the perfect gas to a real gas.

In a perfect gas, the molecules are so distant apart, such that collisions only occur between molecules and walls.

However, in a real gas, interference between molecules does happen, and statistical properties such as entropy can no longer be linearly added together (eg: say, for a two fluid mixture).

The fugacity definition solves the problem, as it directly relates to the compressibility factor (Z), which quantifies how far our assumptions are from the perfect gas, pV = ZRT. In turn, it represents a way to "correct" the idealized pressure, so as to remain able to use the classic equations.

The compressibility factor can be found in charts for different substances.

The beauty and generality of it lies on that, for different substances, if the whole chart is normalized regarding a critical pressure (easily identified by an inflection point), all charts (of a given molecule shape) will look alike.

Contributed by E.Hwang, March 22nd, 2006.

## Too many definitions

The above definition is clear enough. To me this means that fugacity is a theoretical (or imaginary) property of gases diamensionally the same as pressure, the magnitude of which is such that substituting its value for pressure gives correct relationships for the ideal gas law. Why do we complicate it so that no one can understand it?

This is not the same as what is said under 'Detail', that is Fugacity is the change of pressure required etc. A change of pressure is dimensionless and the dimension of fugacity ought to be [mass]/[distance][time]*[time]. Perhaps someone will tell me if I am totallly wrongLouisBB 22:19, 31 March 2007 (UTC)

## 1) definitions not specific ; 2) is the sign in the formula for ${\displaystyle \mu }$ correct?

There should be a clear mathematical formula defining fugacity. I found none (admittedly, the page is more than disorderly now) so I added one following from the definition of chemical potential in terms of fugacity. So far I have seen chemical potential appear first... And I believe there should be a minus sign in this formula [should be ${\displaystyle f=\exp(-\mu /k_{B}T)}$] but I found no reference so I left an expression following from the above "definition of ${\displaystyle \mu }$". If anyone can confirm there is a minus sign please correct it. I think the whole thermodynamics section should be rewritten anyway since it is much more customary to defime ${\displaystyle \mu }$ as the primary variable for a grand canonical ensemble.

There is no minus sign. You are probably tricked by the form that the partition function is usually written down, where the Boltzmann weight is given by ${\displaystyle e^{-\beta (E-\mu N)}}$. It is a common misconception to misread this as the sign of the beta-mu exponent being negative. However, it rearranges to ${\displaystyle e^{-\beta (E-\mu N)}=e^{-\beta E}\cdot e^{+\beta \mu N}=e^{-\beta E}f^{N}}$, where ${\displaystyle f:=e^{+\beta \mu }}$ (and ${\displaystyle \beta =1/k_{\rm {B}}T}$ as usual).--134.76.88.179 (talk) 10:01, 18 June 2012 (UTC)

## definition

${\displaystyle \mu }$ - chemical potential is partial molar derivation of Gibbs energy

fugacity (or its increment) was originally by Lewis defined as follows: d${\displaystyle \mu _{i}}$ = RT d Ln(fi) - this is original definition

Lewis also stated that fugacity of some component in ideal solution is equal of partial pressure of this component. Then fugacity coefficient was defined because formal integration between 2 states real and ideal solution with same T,p, and composition.

as f/(f*) where f* = p*, is fugacity odf i in ideal solution equal to its partial pressure. —Preceding unsigned comment added by 212.5.210.202 (talk) 10:51, 21 September 2007 (UTC)

Isn't fugacity (or the partial fugacity of each component) equal for each phase at equilibrium? That's a major reason it's useful and should be included... the article mentions that its useful but doesn't mention why. See [1]

Also a useful way to calculate fugacity is in terms of the Compressibility factor Z: ${\displaystyle f=P*\int {\frac {Z-1}{P}}dP}$[2]. Best, Mattb112885 (talk) 02:41, 19 February 2009 (UTC)

## Units?

It is stated that the units of fugacity are the same as for pressure. But later the defined quantity ${\displaystyle f}$ is unitless as it is an exponential of a fraction of chemical potential (energy per added particle) ${\displaystyle \mu }$ and ${\displaystyle k_{B}T}$ (thermal energy). This should be explained.

## Major problems

This article hasn't seen a critical review in some time and as a result it has issues with format, lack of references, and validity of the the content. For example:

${\displaystyle f\approx {\frac {P^{2}{\bar {V}}}{RT}}}$
"This formula allows us to calculate quickly the fugacity of a real gas at ${\displaystyle P}$,${\displaystyle T}$, given a value for V (which could be determined using any equation of state), if we assume ${\displaystyle \Phi }$ is constant between 0 and ${\displaystyle P}$."

I may be wrong but "f" is an empirically determined value based on a gases V at a given P and T. There is no way V could be determined equation of state. I would be happy to be corrected but it seems as if the author got lost in the theory and math. In the process the author forgot the intended use of fugacity and its relationship to the real world. It seems like this disconnect is pervasive through out the article.

Fugacity is used to describe the pressure behavior of real gases more accurately than the ideal gas law. Since pressure is also the most common term to describe the concentration of gases fugaicty is also used to describe a gases effective concentrations in a mixture. For a gas the "fugacity coefficient" is essentially the same as "activity" for a condensed state. Fugacity has a relation ship to liquids and solids but it largely mediated by the liquids and solids relationship to gases. This whole subject should be predicated on describing real world gases more accurately than is allowed by the ideal gas law. After that many of the other applications could be described, then after that perhaps someone could have a field day with equations.--OMCV (talk) 05:13, 12 March 2010 (UTC)

I agree, and I think your edits are great. The article was conceptually very muddy. The article has definitely needed a significant overhaul for a long time. It ought to first describe fugacity in broad, general, and conceptual terms that have meaning in the real world, and then move toward more precise and technical material. As to the above Taylor expansion approximation, I don't think it's necessary at all in the article and I agree with taking it out. It's good for textbooks, but Wikipedia has a different purpose than a textbook. COGDEN 19:02, 12 March 2010 (UTC)
Thanks for your estimate of my previous edits. I've gone a bit further with the material and I hope you like what I've done. I must admit that I'm not qualified to review the mathematical derivations that I've left in the text, I would have been better qualified as an undergraduate than I am now after years in the lab. They look right but I'm not sure. At times the associated language makes me I wonder if the authors goal is develop fancy models that grow out or first principles rather than to describe the real world. Then I wonder the same thing about Peter Atkins so maybe its best I keep to synthetic chemistry.--OMCV (talk) 03:20, 14 March 2010 (UTC)

## Φ - broken definition?

Came to the page wondering what fugacity was, and have a reasonable idea now, thanks. But am confused on one aspect - either I'm misunderstanding or the bit about Φ is broken. It on the one hand states that for an ideal gas Φ is 1.0, but then also has the equation ${\displaystyle {\bar {V}}={\frac {RT}{P}}+\Phi }$ which is my problem - surely for an ideal gas ${\displaystyle {\bar {V}}={\frac {RT}{P}}}$ which would mean that Φ is 0 for an ideal gas, not 1... ? 194.81.223.66 (talk) 13:10, 11 November 2010 (UTC)

Fixed. The problem was a little more subtle. It is not capital Φ = V - RT/P which is 1, but rather small φ = f/P further down which is 1. Dirac66 (talk) 03:39, 1 February 2011 (UTC)

## Not real-world pressure but effective pressure

The intro repeatedly describes the fugacity as the real or "real-world" pressure of a gas. This leaves the incorrect impression that if one measures the pressure of a real gas, one obtains a value equal to the fugacity. Not true - if one measures the pressure, one obtains a value equal to ... the pressure. For example nitrogen at 0oC and a pressure of 100 atm has a fugacity of 97.03 atm (Atkins and de Paula 8th edn, p.112), and measurement of the mechanical pressure with a pressure gauge gives the value 100 atm (which is different from the ideal pressure nRT/V).

So what does 97.03 atm refer to? The fugacity is an effective pressure, meaning that the species' chemical potential depends on the fugacity of a real gas in the same way that it would depend on pressure for an ideal gas. In the example, the chemical potential of real nitrogen at 100 atm equals that of ideal nitrogen at 97.03 atm. One could say that fugacity is a pressure corrected to describe the real-world chemical potential and the real-world law of mass action.

This point is clear in the article's mathematical development, the current intro is likely to confuse the reader before s/he gets to the math. I intend to revise the intro and eliminate the mentions of "real-world" pressure. Dirac66 (talk) 18:11, 1 February 2011 (UTC)

to Dirac66. "Ideal nitrogen" is not good expression. I suppose "Real nitrogen at 100 atm has chemical potential (molar Gibbs free energy) 0.9703 times lesser than that of ideal gas at 100 atm due to nonideality of nitrogen gas: intermolecular attraction forces at 100 atm". Please see Atkins, de Paula, 8th. edition, p.111, Fig. 3.24., p.112. With best regards — Preceding unsigned comment added by 193.40.243.251 (talk) 11:41, 6 October 2012 (UTC) to Dirac 66. Correction: ... has chemical potential increase ln 0.9703 times lesser than that of ideal gas ... . With best regards — Preceding unsigned comment added by 193.40.243.251 (talk) 12:00, 6 October 2012 (UTC)

Thank you. I have now replaced ideal nitrogen by the phrase if nitrogen were an ideal gas, and added a brief paragraph to say that the difference in chemical potential is due to nonideality and attractive forces. Dirac66 (talk) 15:19, 7 October 2012 (UTC)

## Liquid phase fugacity

This article reads like fugacity only applies to the vapor phase. Fugacity can be applied to the liquid phase (or even solid phase) as well. Amazingly, the equation for the fugacity of a pressurized solid/liquid doesn't appear to be on Wikipedia, so I have added it as a new section in this article. Obviously, this section (as well as the article) could use some sprucing up by some devoted soul. Stieltjes (talk) 20:27, 28 August 2012 (UTC)

Do you have sources for fugacity in the liquid or solid phase? I have three physical chemistry books (Atkins and de Paula, Engel and Reid, Laidler and Meiser) which discuss fugacity in the gas phase, but none of them mentions fugacity in liquids or solids. Instead they discuss (thermodynamic) activity, which would be related to fugacity but not quite the same thing. Dirac66 (talk) 21:59, 28 August 2012 (UTC)
Yes, Prausnitz et al. Molecular Thermodynamics of Fluid Phase Equilibria 3rd Ed. pp41. I'm not sure how to add this reference since it has reflist in the markup?? Anyway, it is important to note that fugacity being defined for the liquid phase is a key requirement for a lot of real vapor-liquid-equilibrium done by chemical engineers. Makes sense to me that physical chemist would be interested in activity and highlight it, but it certainly makes sense to define fugacity for a liquid or solid phase.
Also, equality of individual component fugacities in the respective phases is a requirement for equilibrium. Again, this reinforces the sensibility of a definition of fugacity in non-gaseous phases but seems important to point out in this article in its own right.Stieltjes (talk) 23:45, 28 August 2012 (UTC)

## Original paper of Lewis

The 1908 paper is on osmotic pressure. I added his 1901 article where the phrase was first introduced. Feel free to double-check if this really is the first one.

## Ethane graph

For the graph at the end of the fugacity of ethane, the caption says that the temperature is +60°F, but on expanding the figure we see that both the graph title and the file name say T = -60°F. Which is correct please? Also what is the source for the data which is graphed? Dirac66 (talk) 00:22, 8 March 2013 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Fugacity/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 I think it's a horrible explanation. Pressure-like property?? As far as I know pressure is a physical property. You can measure, apply and observe it. On the other hand fugacity is a measured but discrete property.

Last edited at 22:33, 12 April 2007 (UTC). Substituted at 15:41, 29 April 2016 (UTC)

## Deletion of helpful numerical example

I disagree with the recent deletion by Rock Magnetist of nitrogen as a numerical example. In my teaching experience when introducing new quantities, it helps students to provide a numerical example. This is even more true when several related quantities are involved, as here fugacity, fugacity coefficient, chemical potential and activity. Perhaps all the values cluttered up the lede somewhat, so I will restore them as a separate section after the lede.

Also when altering text the references should he rechecked. Atkins and de Paula p.112 was originally provided as a source for the value f = 97.03 atm for nitrogen at p = 100 atm. The current version uses it as a source for the definition of fugacity in terms of chemical potential, but Atkins and de Paula do not mention chemical potential in that chapter. On p.111 they define fugacity in terms of molar Gibbs free energy (for a pure gas), and on p.112 the values for nitrogen are given. I will fix this too. Dirac66 (talk) 01:40, 14 May 2018 (UTC)

I don't see how it makes anything more clear, but placing in the body instead of the lead is certainly an improvement. I think an even better place for it would be after the Definition.
Thanks for cleaning up my citation errors. It would be nice if ref. 2 could also be combined with ref. 1 instead of having two separate editions of the same work. RockMagnetist (DCO visiting scholar) (talk) 05:20, 14 May 2018 (UTC)
The numerical example is intended as a quick explanation of the meaning of fugacity values. Readers who are so inclined can then go through the more complete definition.
And I have now referred all mentions of Atkins' book to the same (8th) edition which I have. I will not object if someone wants to look up the pages in the most recent edition. Dirac66 (talk) 10:47, 14 May 2018 (UTC)
In its current location, the numerical example uses "molar Gibbs energy", "ideal gas", and the constant R before any of those terms has been described. RockMagnetist (DCO visiting scholar) (talk) 15:53, 14 May 2018 (UTC)

## Latest equation dimensionally incorrect plus sign problem

The latest equation added is :${\displaystyle \left({\frac {\partial \ln f}{\partial T}}\right)_{P}=-{\frac {S_{\mathrm {m} }}{R}}<0.}$

However since molar entropy has the same units as the gas constant, the right-hand side is dimensionless while the left side has units K-1.

Also the same edit says that "Since the entropy is positive, ln f increases with pressure." Yes, ln f increases with pressure but here the right-hand side is negative which would imply the opposite.

Clearly there is an error here, but I am not certain what was intended here so will not attempt to fix it myself. Dirac66 (talk) 19:48, 22 May 2018 (UTC)

Just a really sloppy reading of the source on my part. Thanks for bringing it to my attention. RockMagnetist (DCO visiting scholar) (talk) 21:45, 22 May 2018 (UTC)
The equation now looks like ${\displaystyle \left({\frac {\partial (T\ln f)}{\partial T}}\right)_{P}=-{\frac {S_{\mathrm {m} }}{R}}<0}$. However, there are still two problems with it. First, the logarithm of a non-dimensionless value is a dangerous thing and has to be dealt with carefully. Specifically, a partial derivative of log(f) is fine because one can always make f dimensionless by dividing by some unit of pressure and then notice that this extra factor vanishes from the result of differentiation. This, however, is not the case when one differentiates T*log(f) since the result of differentiation will have the term log(f) which is not well-defined outside differentiation!
Secondly, the r.h.s. is not correct. Physically, if I have some degrees of freedom in the real gas which does not affect the pressure (e.g., intra-molecular degrees of freedom that do not interact with translational degrees of freedom), then these degrees do not contribute to fugacity (or real-gas pressure), but they obviously contribute to entropy. Thus, it cannot be an absolute entropy in the r.h.s., but rather it must be some difference. I tried to derive the exact differential of fugacity in TP coordinates, which resulted in ${\displaystyle RTd(\ln f)=-\left[S_{m}(T,P)-S_{m,id}(T,P)+R\ln \left(f(T,P)/P\right)\right]dT+V_{m}(T,P)dP}$. This expression does obviously reproduce the partial derivative with respect to pressure, but as for the derivative with respect to volume it produces ${\displaystyle \left({\frac {\partial (T\ln(f/P))}{\partial T}}\right)_{P}=-\left[S_{m}(T,P)-S_{m,id}(T,P)\right]/R}$. This expression looks similar to what is now in the article, but it is still different. Did not want to edit before discussing. One suggestion might be to actually add this exact differential (and its derivation) to the article. Velizhan (talk) 19:59, 14 October 2018 (UTC)
@Velizhan: Good catch. It appears that in the source for this equation, the author incorrectly substituted from an earlier equation. I have corrected it. RockMagnetist (DCO visiting scholar) (talk) 01:05, 16 October 2018 (UTC)

## That ethane figure

It has been five years since Dirac66 asked for a source for the data, and none has been forthcoming. It also has that little box saying "Chart area" and there is no explanation of the breakpoint at saturation pressure. Is it time to delete the figure? RockMagnetist (DCO visiting scholar) (talk) 17:32, 28 May 2018 (UTC)

Yes, I think it is time. The figure was inserted on 28 August 2012 by Stieltjes who last edited Wikipedia on 20 September 2012 and never reacted to my request of 8 March 2013. So we have to judge it on its merits without expecting further input from Stieltjes. Since it is unsourced and not properly explained, I will delete it now. If someone wants to restore it, please add a source for the data and an explanation of the breakpoint as requested. Dirac66 (talk) 00:26, 29 May 2018 (UTC)

## ln P

I wonder how you want to take the logarithm of apressure

${\displaystyle d\mu =RTd\ln P.}$

of course you have to take the ratio to standard pressure first

${\displaystyle d\mu =RTd\ln(P/p^{0}).}$ Ra-raisch (talk) 21:48, 14 November 2018 (UTC)
When it's derivatives, that's not necessary, since
${\displaystyle d\ln P={\frac {dP}{P}}.}$
RockMagnetist (DCO visiting scholar) (talk) 02:17, 15 November 2018 (UTC)
ok, correct, the drivative does it. Ra-raisch (talk) 16:21, 14 June 2019 (UTC)