I am having a little trouble understanding your questions. Can you explain your questions further?

On the face of it your questions seem to revolve around the concept of a single electrode potential. If so it is important for you to be aware that this s a field of of long standing controversy. I have published on this topic, but before I get into a long explanation that might not even be relevant to your question I think it is important that I fully understand what you are asking.

Dear Dr. Rockwood; Than you for your attention to my question. I should ask the two following questions:

1- How the work potential of an anode (for example graphite) calculate?

2- Are the chemical potential and the work potential of an electrode the same?

Hossein,

It is probably not possible to give a satisfactory answer in one post. However, let's start with a definition or two. When dealing with ions or electrons one does not speak of chemical potential but rather electrochemical potential. There are historical reasons for this that we can perhaps get into later, but let's just say that the electrochemical potential of an ion relates to the partial molar Gibbs free energy of an ion, including all contributions to the free energy.

I know that this is not much information to get started with, but it is an absolutely crucial concept, and confusion surrounding this issue has been the cause of much of the historical problems in dealing with electrode potentials.

I recommend that you find my paper Phys. Rev. A 33, 554 – Published 1 January 1986 and try to understand it. Feel free to ask questions. Don't be discouraged if it seems confusing for a while. It is a topic that has historically been confusing to many.

An important issue discussed in the Phys Rev paper is the relationship between the work function of a conductor and the electron vapor pressure of the conductor. There is an essentially exact relationship that one can derive from the Fermi-Dirac distribution. Also , the electron vapor pressure in vacuum near the surface of the electrode metal is different from the electron vapor pressure at some other region of space if there are electrostatic fields present in space.

Another concept to ponder is what happens when two dissimilar metals are connected by a wire. The answer is that the Fermi levels equilibrate between the two levels. This occurs via the transfer of a small amount of charge (electrons) between the metals. Thus, the two metals are not electrically neutral at that point. If you think about the value of the work functions of the two metals (i.e. the difference between the vacuum level as it would be determined near the surface of the metal and the Fermi level of the metal) The potential difference in space in the gap between the two metals is given by the difference in the Fermi levels if the two metals are electrically connected, e.g. by a wire. This is well-known, and may sound esoteric at this point, but this concept turns out to be fundamental to the understanding of the slingle electrode potential problem.

Also, I see that you are at a physics institute. Please be aware that I discuss this problem from a combined point of view, i.e. drawing concepts from chemical thermodynamics and physics in a way to produce a coherent and rigorous treatment of the problem. To do this it is essential to understand certain chemical thermodynamic concepts, such as the meaning and utility of partial molar quantities, such as the partial molar Gibbs free energy.

Also, it is customary in physics to use the term "chemical potential" when discussing the Fermi level, whereas in chemical thermodynamics the correct term is "electrochemical potential." To repeat what I said above, there are some historical reasons for this, but to elaborate on this point a little further, the electrochemical potential includes all contributions to the Gibbs free energy (or more properly, the partial molar Gibbs free energy) whereas in the "chemical potential" of an ion or electron there is an attempt to subtract out the electrostatic part. However, as Gibbs showed more than a century ago, the electrostatic potential across a phase boundary is thermodynamically meaningless, so any attempt to subtract off the electrostatic part of the partial molar Gibbs free energy (to get a chemical potential) results in a thermodynamically meaningless quantity.

It is unfortunate that there is somewhat of a discrepancy between what the physicist means by "chemical potential" and what the chemical thermodynamicist means by "chemical potential" when dealing with ions, but it is just something we will need to deal with, and if you will be so kind as to use the term "electrochemical potential" when referring to the partial molar Gibbs free energy of an ion or electron it will make it possible to communicate more clearly.

Again, please be aware that you have put your foot into a conceptual minefield that has been source of controversy for almost a century. It is possible to find your way through this conceptual minefield, but it won't be easy, and I believe I can help you if you are truly serious about it.

There is another paper that may be helpful. published in ChemPhysChem Volume 16, Issue 9 June 22, 2015 Pages 1978–1991. The paper focuses on something called the single ion activity or single ion activity coefficient, and you don't need to worry about that problem for now, but that paper may be useful to you because contains a description of what happens when a charged particles equilibrate between phases, and in particular what happens to the electrostatic potential between points just outside of two phases that have equilibrated. One can do this by considering a set of elementary thermodynamic processes. It also contains a discussion of the relationship between chemical potential and electrochemical potential, including a discussion of the historical context of why some have made a distinction between chemical potential and electrochemical potential, along with some discussion of how this history has lead to conceptual confusion in the field. The bottom line is that the term "electrochemical potential" should never have been introduced, but now that it is in use we are stuck with it due to historical precedent.

Let me add one more thought, this one to partially address your question #2. Using the term "electrochemical potential" instead of "chemical potential", the work function is very nearly the same thing as the electrochemical potential. There are some subtleties involved, some trivial, like the relationship between volts and Joules, and some less trivial, such as reference points on energy scales, but as a first conceptual approximation you can think of electrochemical potential and work function as as being more or less the same thing. Just be mentally prepared to refine this concept in the light of future knowledge that you will gain in dealing with these problems.