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. E

    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. Dario56

    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. G

    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. chikchok

    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.
  5. B

    Calculating particle numbers in diffusive equilibrium of a battery

    So I think I have the principles mixed up here because I'm getting kind of "circular" answers. ## N = N_1 + N_2## ##dN## = 0 bc/ particle number fixed so ##dN_1 = -dN_2## ##F = cN^2 = c(N_1 + N_2)^2## In diffusive equilibrium, free energy would be minimized and chemical potentials equal... $$...
  6. Kaguro

    Chemical potential and fugacity

    I am trying to learn statistical physics. While learning MB statistics, my textbook defined chemical potential as ##\mu = (\frac{\partial F}{\partial N})_{V,T}##. That's nice. Later when I started on Quantum statistics, my textbook described all three distribution functions via: ##n_i =...
  7. J

    Thermodynamics problem relating to Chemical Potentials

    For part (a), I used this formula where where the i's represent the substance being used and mu_i^0 represents some reference potential. However, to my knowledge this simply calculates the change in chemical potential from one state to another which is not of much help in finding the relative...
  8. E

    Meaning of small triangle in chemical potential

    It is given that the solution is ideal, i.e. that we can take ##\gamma_A = 1##. I wondered what that small triangle signifies in the second definition? Thanks!
  9. QuarkDecay

    B Chemical Potential μ in Solids

    There is the equation: μ= Eu +Eg/2 +3/4kβTln(mu/mc) Eg is the band gap, but I don't understand what Eu stands for and how we can calculate it? Could it be the valence band?
  10. cahill

    I Change in chemical potential energy from nuclear decay

    I have been amateur reading about beta decay. The example given for electron capture was krypton-81 into bromine-81. Going from a noble gas to a halogen gives rise to a big change in chemical potential energy. How is this energy accounted for in the equations of the reactant particles and...
  11. SchroedingersLion

    Batteries: Is the current a diffusion current?

    Greetings, I realized that I don't understand a fundamental fact of common Li-ion batteries. During the charging process, electrons are forcefully extracted from the cathode and pushed into the anode. Charge balance then yields a flow of positive Li ions from the cathode to the anode (through...
  12. W

    Thermal Physics: Fermi Gas and chemical potential

    Hi all, I have an issue trying to understand the following paragraph from Blundell's book. How, exactly, does the definition of ##\mu_0 = E_F## "make sense"? In the sentence after 30.21, it seems to say that the mean energy for a system with ##N## particles differs from that of a system with...
  13. L

    I What is the relation between chemical potential and the number of particles?

    Chemical potential is defined as the change in energy due to change in the number of particles in a system. Let we have a system which is defined by the following Hamiltonian: $$H = -t \sum_i^L c_i^\dagger c_{i+1} + V\sum_i^L n_i n_{i+1} -\mu \sum_i^L n_i$$ where ##c^\dagger (c)## are creation...
  14. steroidjunkie

    Semiconductor chemical potential and the density of holes

    Homework Statement [a) Show that chemical potential in intrinsic semiconductor at T = 0 lies exactly in the middle of the energy gap. At what temperature is the chemical potential equal ##(3/4)(E_c + E_v)##, where ##E_c## and ##E_v## are energies od the bottom of the conduction band and of the...
  15. P

    I Chemical potential on a solid and its vapor pressure

    I have been reading the book "Nanostructures and Nanomaterials" by G. Cao and Y. Yang, and was intrigued by the following passage in page 33: "Assuming the vapor of solid phase obeys the ideal gas law, for the flat surface one can easily arrive at: μv − μ∞ = −kTlnP∞, where μv is the chemical...
  16. amjad-sh

    A Spin chemical potential

    Can somebody give me some insights related to spin chemical potential ? I searched on google but didn't get satisfied. What it means when two spin groups of electrons (group A with spin up and group B with spin down)have different spin chemical potential?
  17. Pushoam

    Relation between chemical potential and S,V,T,P

    Homework Statement Homework EquationsThe Attempt at a SolutionChemical potential is defined as ## \mu = Gibbs potential per particle ##. So, is the system consists of N particles, ##\mu = \frac { G } {N } ##. Now, dG = VdP – SdT ## \frac {dG } { N } = \frac { VdP } { N } - \frac { SdT }...
  18. F

    Standard chemical potential and activity of asphaltene

    Hi , this is my equation : μA= μA0 + RTlnaA A is the asphaltene in toluene solution . i want to calculate μA in different temperatures. where can i get μA0 and aA? tables? other equations? μA0 is the standard chemical potential of asphaltene and aA is the asphaltene activity in solution.
  19. V

    B Negative chemical potential

    Is possible and what does it mean if a chemical potential is negative? I mean that for boson it means that in environment is "needed" boson (photon) and is possible to create him. Is it true? And what about for fermions? Could it mean that it is pleasent for environment to creat some fermion? Or...
  20. W

    Thermodynamic Identity: Chemical Potential

    Homework Statement Homework Equations Thermodynamic Identity The Attempt at a Solution While I was able to work out the problem with the help of the hint, I couldn't completely understand the implication of said hint. The hint suggests that the equations for Chemical Potential in a process...
  21. M

    Chemical potential using Boltzmann equation

    Homework Statement I must calculate chemical potential using the Boltzmann equation in relaxation time approximation $$f=f^0-\tau v_z^2 \partial f^0/\partial z,$$ where ##f^0## is given as $$f^0 = 2(\frac{m}{2\pi\hbar})^3 \frac{1}{\exp{\beta(z)(\frac{mv^2}{2}-\mu(z))}+1}$$ I have to consider...
  22. RicardoMP

    I Sommerfeld Expansion & Chemical Potential

    Hi! I'm trying to show how the chemical potential depends on the temperature and I'm advised to use the Sommerfeld expansion. I'm using it on the density of charge n=\int^{+\infty}_{-\infty} \rho(\epsilon)n_Fd\epsilon , which gives n=\int^{\mu}_{0} \rho(\epsilon)d\epsilon...
  23. D

    Determine Enthelpy, Entropy, and Chemical Potential

    Homework Statement Hello, i am given an isothermal transition for nitrogen, N2, where temperature is constant at 700K, p1=1bar, and p2=100bar. For this problem i am not allowed to use any equations of states, such as Benedict-Webb-Rubin, or Beattie-Bridgeman. Rather i am given only Cp data...
  24. H

    Diffusion down chemical potential gradients

    Ignoring cross-diffusion, diffusive mass fluxes down chemical potential gradients can be described by the equation (I am working from de Groot and Mazur's 1984 text on non-equilibrium thermodynamics): \frac{\partial C_k}{\partial t} = L_{kk}\frac{\partial (\mu_k-\mu_n)}{\partial x} where C_k...
  25. Mayan Fung

    Finding chemical potential with given thermodynamic relation

    Homework Statement Suppose you are given the following relation among the entropy S, volume V , internal energy U, and number of particles N of a thermodynamic system, where A is a constant.: $$ S = A(NVU)^{\frac 1 3} $$ Find the chemical potential μ(T,P) Homework Equations $$ \frac μ T =...
  26. L

    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. M

    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. Einj

    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. Crush1986

    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. Coffee_

    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. C

    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. It's me

    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. A

    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. T

    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. L

    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. ChristineMarie

    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. D

    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. B

    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. C

    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. alpha358

    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. A

    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. D

    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. Useful nucleus

    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. A

    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. A

    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. A

    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. T

    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. I

    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. G

    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. U

    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...