Hello Cristina,

Thermodynamics is usually applicable only to systems at equilibrium. At equilibrium, the chemical potential of a solute, for example, dissolved in a solvent should be the same everywhere. In the case where there is a spatial variation in the amount of solute, the difference in the chemical potentials between the areas or high concentration and lower concentrate can be thought of as 'driving' the diffusion of the solute because such a system is not at equilibrium.

However, the situation is somewhat more complex than what I have just stated because there is no preferred direction for the movement of atoms or molecules in the process of diffusion, for example. Consider a one dimensional concentration difference in the x-direction, and take a thin slice of the volume of thickness, delta-x, and cross-sectional area, A; we call this the reference slice. Solute atoms or molecules in this thin reference slice move equally to the right or the left, there is no preferred direction. If that is the case, then how does diffusion occur? The party line answer is that it is the adjacent thin slices that determine the net flows of solute atoms and molecules. Consider the slices on either side. If the slice to the right of the reference slice has a zero concentration of solute, then there will be a net flow of solute molecules into this adjacent slice. If the slice to the left of the reference has a slightly higher solution concentration, then there will be net flow into the reference slice because more atoms or molecules will enter the reference slice from the left slice than leave from the reference slice. This picture is hard to visualize, and I probably have not given it a clear explanation.

For an ideal solution, i.e., one where the mole fraction of the solvent is close to unity (1), the solute atoms or molecules behave like ideal gas atoms or molecules against the background of the solvent atoms or molecules, and the expression for the chemical potential can be written as a function of the mole fraction or the concentration (either as a molality or a molarity) - and temperature and pressure; molality is usually preferred because it does not vary with temperature because it is calculated on the basis on the amount (mass or moles) of solvent, while molarity is calculated on the basis of the volume of solution. In other words, in dilute solutions it is the difference in the number of solute atoms or molecules that 'drives' the flow. For more concentrated solutions, one can use the concepts of activity (liquids) or fugacity (gases) to measure deviations from ideal behavior due to interactions between the solute and solvent.

In the end, all measurements come down to directly measurable quantities such as mole fractions, concentrations, temperatures, and pressures. Chemical potential is usually inferred from these measurements, and not measured directly.

Regards,

Tom Cuff

Dear Tom,

Thanks a lot for your extended and interesting answer. The picture you described is very helpful to visualize the process.

It makes sense that measurable quantities are being used for the calculations. However, why the difference in temperature and pressure are ignored in the two formulas indicated in my post, if the chemical potential also depends on them? Aren't we ignoring their influence? Or are they somehow included in the coefficient of mass transfer "k" or the coefficient of diffusion "D"?

Thanks again,

Best regards,

Cristina

Hello Cristina,

You have answered your own question. For example, the coefficient of diffusion does indeed depend on temperature and pressure, especially in the case of gases. In the case of liquids and solids, pressure plays only a small role, but temperature still has a very strong effect. See the following textbooks:

Frank P. Incropera, David P. DeWitt; Fundamentals of Heat and Mass Transfer, 2nd Ed.; John Wiley & Sons, Inc.; 1985; pp. 718-719, Section 15.1.4 - Mass Diffusion Coefficient.

R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot; Transport Phenomena; John Wiley & Sons, Inc.; 1960; pp. 504-508, Section 16.3 - Temperature and Pressure Dependence of Mass Diffusivity.

Note, temperature effects on diffusion are even important in electrical phenomena such as the diffusion of electrons or holes through a semiconductor, for example. See the following solid state physics textbook for an extremely terse and opaque discussion of the concept of chemical potential - I always hesitate to recommend this book to anyone because it is difficult to read and the index is very incomplete,

Charles Kittel; Introduction to Solid State Physics, 4th Ed.; John Wiley & Sons, Inc.; 1971; p. 244.

Regards,

Tom

Hello Tom,

Thanks again for your answer and all the recommended bibliography. I'll have a look at it, especially the mysterious Charles Kittel discussion :)

Do you know if the dependance of the diffusion coefficient D on temperature is exactly the dependence of the chemical potential on temperature, or does it include other phenomena?

My problem is that I do not even see the direct conversion between chemical potential gradient and concentration gradient. The relationship is logarithmic. If "m" is the chemical potential and "x" the molar fraction:

mi = mi0 + RTln(x); (only ideal, otherwise fugacities and activities must be included in the equation)

How do they go from (mi2 - mi1 ) to (xi2 - xi1 )?

There are even situations in which mass transfer occurs against the concentration gradient between phases.

Best regards,

Cristina

Hello Cristina,

Here is my best answer to your first question. The diffusion coefficient can be used in two ways, depending on the system. Case #1: in normal diffusion, the process is either a steady state process (temperature, pressure, and concentration vary with position, but are independent of time) or a transient process (temperature, pressure, and concentration vary with location and time), which may evolve with time into either a steady state or equilibrium state. Case #2: in self-diffusion, the process is an equilibrium process (temperature, pressure, and concentration are independent of location and time). An example of case #2, would be if we could tag some of the water molecules in a container of water - at constant temperature and pressure - and follow their movements. It is unlikely that the temperature dependence of the two case would be identical. As far the temperature dependence of the chemical potential is considered, thermodynamics only deals with equilibrium situations: the initial state(s) and the final state have to be equilibrium states.

I am not exactly sure I understand what you mean by your second question, but let me try and paraphrase it. Please, let me know if am wrong or - the more distant possibility - that I am right. Are you asking how, from the expression you gave for the chemical potential, someone can determine the final equilibrium state by knowing the initial equilibrium state(s)? If that was your question, then my answer is that, in general, you cannot know, i.e., a priori (without experience - experiment, in this case), you usually cannot even qualitatively determine the final equilibrium state. In general, theory is what you might call a guideline, sort of like the pirate's code. This is true of all theories, e.g., quantum mechanics does not allow us to quantitatively calculate phase change parameters. Newton was vexed until the end of his days because his universal law of gravity could not predict the path of our closest neighbor, the Moon. It turns out that the Moon, is influenced almost equally by both the Earth and the Sun, i.e., the path of the Moon is a three-body problem not a two-body problem, and there is no general closed form solution for N>2 body problems. Water and ethanol are miscible in all proportions, but the mixing of these two substances produces highly unexpected behavior, heat evolution and volume contraction. If you want to know how something evolves, do the experiment and then try to fit the theory to the data, not the other way around.

With regard to your last statement about counter intuitive mass transfer can you give us a real world example?

Regards,

Tom

Hi Tom,

Thanks a lot again for your answer!

I do not have a specific real world example. I was just wondering how accurate is to use the gradient of concentration instead of the gradient in chemical potential in mass transfer problems. Most books and theories use the concentration gradient (Fick's law, convective mass transfer equation, etc.), but usually it is not explained that the real driving force is the chemical potential.

Maybe a more specific question would be clearer: could you physically derive the concentration gradient (c1-c2) from the chemical potential gradient (m1-m2) (being c the concentration and m the chemical potential)?

Best regards

Hello Cristina,

Given that there are 5 other people following this discussion, it surprises me that no one else has decided to join us in the deep end of pool. Perhaps they all hear, in the background, the ominous music from the movie "Jaws".

With respect to your first paragraph, I believe you have, again, answered your own question. You are trying to get a unified way to view mass transport in the light of thermodynamics, but you mentioned Fick's law. Fick's law is just that, a phenomenological law, which works in many but not all instances. Fick's law is not a fundamental theory, it is just a way of summarizing a large subset of observations about mass transport. Of course, one could say the same thing about the discipline of thermodynamics.

But thermodynamics has a special place in science because it appears to transcend time and space, and people's conception of reality. For example, quantum mechanics, which is a relatively new discipline, contradicts many concepts found in classical mechanics, but it still obeys the first (energy conservation) and second (entropy) laws of thermodynamics; quantum mechanics probably also obeys the zeroth (thermal equilibrium) law of thermodynamics, which counter intuitively was only explicitly stated well after the first and second laws had become well established. Unlike Fick's law, the laws of thermodynamics are thought to have no exceptions. And like yourself many people have tried to unify mass transport and thermodynamics. The sub-discipline of non-equilibrium thermodynamics (a.k.a., the thermodynamics of irreversible processes) is devoted to this quest; think of the Onsager relations linking forces and fluxes, for example. If all this sounds like a complicated mess, you are right, and it is reflected in your textbooks.

Remember what George Box said, "All models are wrong, but some are useful." But do not become cynical and say, "Yeah, it is just a model." That is like saying, "Yeah, it is just air ...", but you need air to breathe just as you need models to think.

Let me give you an example of a counter intuitive mass transport process. The cells in the human body accumulate potassium against a concentration gradient, i.e., the concentration of potassium in cells is higher than in the surrounding fluid even though the cell membrane is permeable to potassium, sodium, etc., i.e., it is leaky with respect to these cations, and to a host of anions.

I will defer answering the question posed in your second paragraph for two reasons: 1) the hope that someone else will answer it, and 2) my wife is calling me to lunch.

Regards,

Tom

Hello Cristina,

I realize that I forgot to provide some references to justify what I have been saying.

Self-diffusion:

The following two articles are how I first learned about self-diffusion. They are articles that are mercifully short, direct, and free from the usual technical jargon.

J. H. Wang; Self-Diffusion and Structure of Liquid Water. I. Measurement of Self-Diffusion of Liquid Water with Deuterium as Tracer; Journal of the American Chemical Society; Vol. 73; No. 2; February 1951; pp. 510-513.

Jui Hsin Wang; Self-Diffusion and Structure of Liquid Water. II. Measurement of Self-diffusion of Liquid Water with O18 as Tracer; Journal of the American Chemical Society; Vol. 73; No. 9; September 1951; pp. 4181-4183.

Fick's Law:

The proof that Fick's law is a phenomenological law that does not employ the concept of chemical potential comes from the following articles by Adolph Eugen Fick.

Adolf Fick; Ueber Diffusion; Annalen der Physik; Vol. 170 (Complete Series), Vol.94 (4th Series); No. 1; 1855; pp. 59-86.

Adolph Fick; On Liquid Diffusion; The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science; Vol. 10 (4th Series); No. 63; July 1855; 30-39.

Fick's papers, which used the standard partial differential equation that is employed in current textbooks appeared (1855) before the concept of the Gibbs free energy function or it molar equivalent, the chemical potential, were invented in the following papers (1869-1878).

F. Massieu; Sur les fonctions caractéristiques des divers fluides; Comptes Rendus Hebdomadaires des Séances de l’Académie des Science (Paris); Vol. 69; No. …; 1869; pp. 858-862.

F. Massieu; Addition au précédent Mémoire sur les fonctions caractéristiques; Comptes Rendus Hebdomadaires des Séances de l’Académie des Science (Paris); Vol. 69; No. …; 1869; pp. 1057-1061.

J. W. Gibbs; Graphical Methods in the Thermodynamics of Fluids; Transactions of the Connecticut Academy of Arts and Sciences; Vol. 2; No. 2; April-May 1873; pp. 309-342.

J. W. Gibbs; A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces; Transactions of the Connecticut Academy of Arts and Sciences; Vol. 2; No. 2; December 1873; pp. 382-404.

J. W. Gibbs; On the Equilibrium of Heterogeneous Substances, First Part; Transactions of the Connecticut Academy of Arts and Sciences; Vol. 3; No. 1; October 1875 – May 1876; pp. 108-248.

J. W. Gibbs; On the Equilibrium of Heterogeneous Substances (Concluded); Transactions of the Connecticut Academy of Arts and Sciences; Vol. 3; No. 1; May 1877 – July 1878; pp. 343-524.

Note, I got these citations for Massieu and Gibbs and an explanation of their significance from the following book.

Gilbert Newton Lewis, Merle Randall; Thermodynamics and the Free Energy of Chemical Substances; McGraw-Hill Book Company, Inc.; 1923; p. 158, see footnotes #1 & #2.

It is important to note that Fick acknowledged that he got his idea for the partial differential equation (PDE) he used from Jean Bapiste Joseph Fourier's treatise on heat conduction in solids: Théorie analytique de la chaleur (1822). Sometime after Fick co-opted Fourier's PDE, Lord Kelvin (William Thomson) employed this same equation from the same source, Fourier, to model the Atlantic submarine cables used to send telegraphic messages between Ireland and Newfoundland (circa 1860s). All three researchers essentially modeled three very different phenomena (heat conduction, mass transport, and electrical signal propagation in cables) the exact same way. Without this history, there is no timeline, without the timeline there is no context, and without the context there can be no understanding of the relationship among these models.

Basically, Fick and Kelvin used analogy to justify their use of the same PDE as Fourier, however, there is a danger in the use of analogy - the problem of false equivalence. For example, Kelvin's PDE was incomplete because it only employed distributed resistance and capacitance, which in heat conduction in solids have the following duals: thermal conductivity and thermal heat capacity, respectively. Real signal cables, though, also have distributed inductance, which has no real dual in heat conduction, at least, not that we know of yet. In the case of diffusion in mass transport, the coefficient of diffusion may be a function of the concentration at high concentrations, which does not happen in the case of heat transport or electrical signals.

Counter Intuitive Mass Flow:

The example I gave of cells accumulating potassium ions against a concentration gradient is a very famous example of this kind of phenomena. The following book gives a historical treatment of the discovery and elucidation of this phenomena in nerve cells both human and animal.

Alan J. McComas; Galvani's Spark; Oxford University Press; 2011; pp. 33-37.

Another reasonably clear description of this phenomena but minus the historical background is the following book.

John C. Eccles; The Understanding of the Brain; McGraw-Hill Book Company; 1973; pp. 1-32, Chapter 1 - Neurones, Nerve Fibers, and the Nerve Impulse.

The Slice Model:

In my first message, I spend the second paragraph discussing the fact that there is no preferred direction in diffusion, for example. I have never particularly liked this model for the following reason. In an equilibrium process, it is true that there is no preferred direction, but, as you mentioned, in a non-equilibrium process the difference in the chemical potential produces a preferred direction. Therefore, there cannot be slices next to one another that have different concentrations because this model only works with systems at equilibrium, where there are, in fact, no concentration differences.

I think the questions and observations you have posed are very good. When I first attended university, I was so clueless that I could not even formulate intelligent questions. I have never been a fanboy of the way modern textbooks explain concepts since they almost never provide a historical background, which I need to understand the evolution of the concepts they are trying explain. Anyway, this is my attempt at pushing back what Bruce Springsteen referred to, in one of his songs, as ".. the darkness at the edge of town ...".

Regards,

Tom

Dear Tom,

thank you very much for all the references, comments and reflexions :)

Here I found a document that relates the cocentration gradient in the Fick law with the chemical potential gradient, but, as you said, I think it is obtained using analogy...

https://www.uio.no/studier/emner/matnat/kjemi/KJM5120/v09/undervisningsmateriale/Defects-and-transport-2009-Ch5-Diffusion.pdf

Regards,

Cristina

Sorry I saw this just today. I have a different view and perhaps a correct one. Fick's law is valid only for dilute and ideal solutions. Otherwise we must use the Darken's equation to take into account the effect of change in activity on flux. Else the mass transfer or diffusion coefficient will be erroneous by, sometime by several factors. It can be easily shown that Fick's law is a special case of Darken's equations for the case of a dilute or ideal solution.

Brahma Deo Dear Prof. Deo, I appreciate your note which points out the importance of activity coefficient gradient in the evaluation of a kinetic analysis. As you mentioned, the partial differential of ln gamma_i wrt ln x_i is indeed not a constant during the integration in the Eq. (12), in general. I agree with you on this point. Therefore, unless we have an accurate information not only the gamma_i as a function of x_i, but also variation of x_i at each time, the consideration of the partial derivative makes us difficult to perform an accurate calculation. I think this is one of some reasons why we use the "mass transport" coefficient a bit implicitly, not using the diffusion coefficient which is more rigorous approach. I would appreciate it if you give your opinion about this.

Dear Prof. Kang,

An example is given in Chapter 9 Part 3 which I am attaching with this mail. I shall explain further.

First determine gama from activity compostion plot: Example: say for a fixed CaO/SiO2 ratio in CaO-FeO-SiO2 system which iso activity lines are available). It is best to plot ln gamma as function of ln Xi: Dtermine the equation of obtained curve through second order polynomial, then differentiate this equation (for the curve) with respect to ln Xi which will give the value of slope ln gama versus ln xi in simple terms of Xi. On substituting the value of Xi the slope ln gama/lnxi can be obtained for that Xi. This is a simplified procedure (approximation). But as you point out, it is easily possible to integrate the total equation also including the polynomial equation obtained. I shall try to send you the integrated form in a few days. Let me try. Else the approximate procedure explained above works. Depending upon the system you shall perhaps find that the value of slope is some times 3 so my worry is that the mass transfer coefficient determined from experiment will have this factor of 3 , and more annoying is that the factor will vary with Xi too. Kindly let me know if this way of doing things is acceptable.

Dear Prof. Kang,

But I agree with you that it may not be possible to obtain ln gamma versus Xi for all systems of study. In steel making, however, the area I work in, actually the data is available fora a lot of systems.

Only thing is that in the absence of gama term the overall mass transfer coefficients reported become case specific as they contain the gamma term also for that experiment (hidden effect). Already the area of reaction is vague in fast reacting systems. To add to this the changing surface tension, interfacial tension coupled with viscosity changes, cause further difficulties. As a result in steel making models nothing is reproducible. Then, there is another aspect that BOF steel making process in a non-linear dynamical system (actually a lot of chaos in it, as proven by Lyapunov exponent). People do not even check for the presence of Chaos in their system because then the whole modelling and process control approach changes. Do share you experience please. Thanks a lot. my email id is [email protected] perhaps you may like to share some papers with me. I would welcome separately, though.

Brahma Deo Dear Prof. Deo, thank you for your kind reply to my question, and your excellent chapter! This should be also very useful in understanding thermodynamics of steelmaking. Thank you for providing it to us.

Coming back to our discussion, I think the integration is possible, if the gamma_i is given as a function of x_i at a given temperature. But what I would like to mention is, if we have diffusivity or mobility as an intrinsic property of the diffusing species, and if we have the composition dependence of the gamma_i, then the Darken's approach should be better than a simple diffusion equation using the mass transfer coefficient. But during the integration in the Eq. (11), x_i is also a function of time t, which varies during the integration by t, which subsequently influence on the gamma_i. The x_i (= C_i/C) is linked to the LHS of the intergration i the Eq. (12). This seems the integration to be not a simple matter. If the partial differential is a constant (within a short time and little variation of the x_i), then we can use the Eq. (12). Hopefully, many cases we deal with fall in this range.

I agree with you that whether or not the partial differential is a constant, the term (1 + "the partial differential") plays an important role in the kinetic equation. Unfortunately, so far many peoples are using the kinetic equation using "mass transfer coefficient" without considering what you are pointing out. I consider this in such a way that the mass transfer coefficient (mostly obtained from experiments) reported in literature already possess two terms (k" and the (1 + "the partial differential")) by invisible way. So, I think we do not need to multiply the (1 + "the partial differential") term to the reported mass transfer coefficient. If we have, however, the k" term (containing mobility/diffusivity term or "coupling" term), then we should multiply the (1 + "the partial differential") term, I think.

I am wondering whether the above my thoughts are reasonable or not. Your comment and insight would be much appreciated :-)

" we do not need to multiply the (1 + "the partial differential") term to the reported mass transfer coefficient".

1. The only way way out is to treat the slope term constant during a small time step in kinetics.

2. But I would suggest that perhaps you may like to mention the value of the (1+ +slope term) because it will certainly be different for different cases/systems. Some times this value is very low and in some cases very high too (by factor of 7-10). Further, the slag composition rapidly changes in BOF during the blow FeO in slag comes down from 35% at start of blow to merely around 6-10 percent during the middle blow period (I am presuming that you are in this field!) So imagine the situation there. All depends on the activity composition diagram. I can send you a paper on dissolution of MgO in slags which shows the order of error in reported values (paper available on my Research Gate portal, kindly see, else I will send to you). The slope term is also temperature dependent and for that an approximate (simple) solution is T1 ln gama1= T2 ln gama2. Of course slag models can also be used. But more we include the larger is the error band (all errors being additive). I do not know what is window of error in FACTSAGE model results.

Brahma Deo Thank you. I will find the paper, and read it soon. Anyway, raising a concern regarding the term of activity coefficient in the kinetic analysis is well appreciated. Thanks a lot for this nice discussion!

Dear Prof. Kang,

A very good and comprehensive treatment of diffusion is given in

The trouble with diffusion available

on

Article The Trouble with Diffusion

Materials Research

Print version ISSN 1516-1439On-line version ISSN 1980-5373

Mat. Res. vol.5 no.3 São Carlos July/Sept. 2002

https://doi.org/10.1590/S1516-14392002000300002

The Trouble with Diffusion

R.T. DeHoff* and Nagraj Kulkarni

Brahma Deo Thanks a lot for your kind information of such a nice paper!

We are concerned with two types of applications, usually: those involving mass transfer and then chemical reaction at interface and those involving dissolution. We have already exchanged our views about the first.

As regards the second, i.e. dissolution, may I suggest a good paper by Dejmek and Ward (file attached) which clearly describes the role of activity and activity coefficients and the assumptions made in the equations developed by them. I hope this paper too will be useful.

Subsequent to this there is good work (based on Dejmek's work) by LU XiaoHua et al. (also attached).

I find that the simplifying assumptions which the researchers make in Chemistry are very close. But if we implement them in our work (for our systems) then they may become too stretched (beyond acceptance) because the extent of non-ideal behaviour in our metal-slag systems is sometimes large (can not be ignored) . Good things is that data is available to describe that. I also fell in the same trap, but I guess we can always improve and decide what is best for us.

I am not sure if I am allowed to attach such papers, but it is for our internal discussion only, so I am doing it.

I am going to revisit my earlier work and see what modifications I can make or propose, specially in the multi-component mixed transport control theory which many researchers are using to make models (this paper (Ohguchi, Deo, Robertson) is on my research gate website and you may like to have a look at and perhaps suggest how things can be remedied or modified. I shall look forward to your comments.

my email id is bdeo at iitbbs.ac.in

So, this is my last submission as an answer to the query raised by Cristina Ricart.

Wow, I am just amazed by the way You two gentleman have Carried out the discussion so respectfully. interesting discussion too.

Please refer to answers of other researchers. I may give just simple ideas. Mostly, they can replaced repectively, but concentration is the way that we familiar with. In one case like phase transformation, potiential should be used, because mass is not drived by gap of composition. Take things simple.