# What is Chemical potential: Definition and 96 Discussions

In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. The molar chemical potential is also known as partial molar free energy. When both temperature and pressure are held constant, chemical potential is the partial molar Gibbs free energy. At chemical equilibrium or in phase equilibrium the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free energy is at a minimum.In semiconductor physics, the chemical potential of a system of electrons at zero absolute temperature is known as the Fermi energy.

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1. ### I How to see this form for the chemical potential of an ideal gas?

In Chapter 13.2 of his text, Callen states that the chemical potential with respect to the ##j##th component of an ideal gas can be written as $$\mu_j = RT \left[\phi_j(T) + \ln P + \ln x_j \right].$$ He states this outright and doesn't prove it, and I am trying to do so now. Based on what has...
2. ### Concept of Absolute Thermodynamic Activity

In the textbook Electrochemical Systems by Newman and Alyea, Chapter 14: The definition of some thermodynamic functions, chemical potential of component (ionic or neutral) is written as a function of absolute activity: $$\mu_i=RT\ln(\lambda_i)\tag1$$ where ##\lambda_i## is the absolute activity...
3. ### Calculate chemical potential

Hi Unfortunately, I can't get on with the following task. The system looks like this it is divided in such a way that the same number of particles is present in each ##\epsilon## section. I am now to determine the energy ##E(P_h,V_h,N)## at the height h using the energy ##h=0## i.e...
4. ### I Chemical potential and Fermi level

Does fermi level (in metals) depend on the density of states? I am asking this because from fermi-dirac distribution it seems like that fermi level is non-dependent of DOS, but there is chemical potential in fermi-dirac distribution, which is said to be dependent of DOS.

26. ### Chemical potential at equlibrium

What is the value of chemical potential of a substance at equilibrium? According to following article it is zero: https://www.uni-muenster.de/Physik.AP/Demokritov/en/Forschen/Forschungsschwerpunkte/mBECwatfratcp.html But I have seen in many articles that it is the change in chemical potential...
27. ### Defect formation energy

When determining the formation energy associated with a point vacancy in say a monoatomic crystal - when the total energies of both perfect and defected crystals are known - how exactly is the chemical potential determined? Formation Energy should be given by expression...
28. ### Euclidean correlators for finite chemical potential

Hello everyone, my question is about Euclidean correlators (say a 2-pt function to be specific) in presence of non-zero chemical potential. The question in particular is: is it still true that the Minkowski time ordered 2-pt function can be simply obtained from the Euclidean one by analytic...
29. ### Chemical Potential Homework: Ideal Monatomic Gas

Homework Statement I just have a question about chemical potential for ideal monatomic gas. I see that by definition \mu = \frac{\partial U}{\partial N} Homework Equations \mu = \frac{\partial U}{\partial N} The Attempt at a Solution I was wondering why it is wrong to use U=3/2NkT take...
30. ### Chemical potential question

The chemical potential is defined as ##µ=\frac{\partial E}{\partial N}## while keeping entropy and volume constant. Well how to keep the volume constant while adding particles is obvious but what about the entropy? Obviously adding a particle at a certain energy ##E_{p}## changes the total...
31. ### Chemical potential equilibrium cosmology

In Dodelson's "Introduction to Modern Cosmology" at p. 61 he introduces a non- equilibrium number density $$n_i = g_i e^{\mu_i/T} \int \frac{d^3p}{(2\pi)^3} e^{-E_i/T}$$ and an equilibrium number density $$n_i^{(0)} = g_i \int \frac{d^3p}{(2\pi)^3} e^{-E_i/T},$$ from which it follows that the...
32. ### Chemical potential of water using the Van der Waals model

Homework Statement Obtain the chemical potential of water as a function of temperature and volume using the Van der Waals model. Homework Equations μ=∂U∂N The Attempt at a Solution I don't really understand how to do this at all. Any help would be greatly appreciated.
33. ### Chemical Potential: Pressure, Fugacity, & More

Chemical potentials can be written with an an ln term which has pressure, fugacity, concentration, activity coeff times concentration, mole fraction, and mole fraction times activity coeff. Is there any source that makes sense of this?
34. ### Relationship between pressure & chem potential in ideal gas

Homework Statement (Excerpted from a longer, multipart problem but essentially) Show that for an ideal gas, $$\frac{\partial p}{\partial T}\bigg)_\mu = \frac{S}{V}.$$ Homework Equations • The ideal gas law, of course $$pV = Nk_{\rm B}T$$ • Pressure, temperature, and chemical potential...
35. ### Thermodynamics: ideal gas chemical potential

Can someone explain me why we could put pi, μi and μiο in this equation: and we get this: (this is The chemical potential of a component in a perfect mixture of ideal gases wher pi is partial pressure)
36. ### Deriving Gibb's Free energy

Homework Statement Derive an equation for the change in free energy, ΔGmixing, when ideal gases with the same temperature and pressure, are mixed. Homework Equations ΔGmixing = nRT∑(xi)ln(xi) (∂/∂T(G/T))p = -H/(TxT) The Attempt at a Solution Pi = xiPi* μi = Gi,m μ = (∂G/∂n) at constant T...
37. ### Thermodyamics - Chemical Potential

Homework Statement Given dE=TdS-PdV+\mu dN, PV=NT and Cv=\frac{3}{2}N Find E Homework Equations dE=TdS-PdV+\mu dN, PV=NT and Cv=\frac{3}{2}N The Attempt at a Solution That part that is confusing me is the chemical potential, not sure what to do with it. Finding the energy for the ideal...
38. ### Thermodynamics Chemical Potential

Homework Statement Consider a monatomic ideal gas that lives at a height z above sea level, so each molecule has potential energy mgz in addition to its kinetic energy.(a) Show that the chemical potential is the same as if the gas were at sea level, plus an additional term mgz:μ(z) = -kT ln...
39. ### Why is the chemical potential negligible at equilibrium?

I am currently reading cosmology and I read the statement "... chemical potential which is nearly always negligible when processes are in equilibrium." So the current understanding I got of chemical potential is through the law: $$dE = - pdV + TdS + \mu dN.$$ Hence for processes which occur at...
40. ### Thermodynamics: ideal gas chemical potential pressure dependence

I don‘t understand one step in derivation of ideal gas chemical potential. Generally Gibbs free energy is: (1) (2) We observe that: (3) From equation (3) we make differential equation and integrate it: (4) (5) We get Gibbs free energy dependence on pressure...
41. ### Crystal Lattice of Spin-1 Particles: Chemical Potential?

Homework Statement A crystal lattice consists of a spin 1 particle at each lattice point. Spin 1 particles can have z-components of magnetic moment that take on the values +μZ, 0, and -μZ. In an external magnetic field B, each spin can have an energy U = -μZB, so the possible energies are...
42. ### Chemical potential for van de Waarls gas

Hello, In my book, we have two equations to find the chemical potential µj=(dF/dNj) T, V and N constants and µj=-kT ln (ζj/Nj) We calculated the potential of a Van der Waals gas using each of these equations, but the result is not the same, i want to understand why ?
43. ### Bounds on the chemical potential of electrons

In a semiconductor, is the chemical potential of electrons limited to take values only between the valence band maximum and conduction band minimum? Are there circumstances where it can cross these bounds?
44. ### Entropy, chemical potential, temperature

For a thermodynamic system there exists a function called entropy S(U,N,V) etc. We then define for instance temperature as: 1/T = ∂S/∂U μ = ∂S/∂N etc. When taking these partial it is understood that we only take the derivative of S wrt the explicit dependece on U,N etc. right? Because...
45. ### Half-Filled Band: Explaining Zero Chemical Potential

In an exercise I look at a square lattice and consider a model where the energy band of the electron is half filled and temperature is zero. I am then supposed to explain why a half filled band corresponds to zero chemical potential. For me the most meaningful definition of the chemical...
46. ### What is chemical potential?

I have seen various treatments of the chemical potential. Basically it all amounted to a definition of it as a quantity maximizing the entropy for systems allowed to exchange particles. Now this quantity has become quite important in some models I am studying for current flow, specifically an...
47. ### Chemical Potential and delta G

Homework Statement Consider a container with two chambers of the same size separated by a fixed membrane in the middle (permeable only to the ideal solute, but not the solvent). One chamber initially contains 1M of solute, and the other contains 0.5M of solute. Write an equation for the...
48. ### Chemical Potential: Heat Energy Carried by Particles

in class, we just learned that the chemical potential was μ=∂G/∂N i'm looking in my book and it says that μ=-T(∂S/∂N) this looks a lot like the partial of heat energy with respect to N. is my interpretation correct? (assuming my interpretation was correct): so qualitatively, the...
49. ### Why particles flow from high chemical potential to low

Homework Statement Why does the equation (d2G/dNi2)T,P=(dμ/dNi)T,P>0 tells you that particles will flow from high chemical potential to region of low chemical potential at const. temperature and pressure? Thank you all. Homework Equations dG=VdP-SdT+μdNi The Attempt at a Solution...
50. ### Variation of chemical potential with T and P

So the expression for Gibb's free energy is: dG = -SdT + VdP + μdN, Here, we see that the Gibb's free energy changes with temperature (dT), change in pressure (dP) and change in chemical potential (as a result of change in particle number). My question is: we know chemical potential...