Chemical potential

The chemical potential μ\mu is an energy difference that determines the amount of energy that needs to be spent to add a particle to a system. Despite being called a potential, it cannot be defined up to a constant as it is already a difference of two energies.

It is important in the description of the grand canonical ensemble, where the number of particles is variable, but also generally describes the energy of adding a particle to a system like the electron shell of an atom.

It can be expressed in terms of the internal energy, entropy, Helmholtz free energy and Gibbs free energy as

μ=(∂E∂N)S,V=−T(∂S∂N)E,V=(∂A∂N)V,T=(∂G∂N)P,T\mu=\left( \frac{ \partial E }{ \partial N } \right)_{S,V}=-T\left( \frac{ \partial S }{ \partial N } \right)_{E,V}=\left( \frac{ \partial A }{ \partial N } \right)_{V,T}=\left( \frac{ \partial G }{ \partial N } \right)_{P,T}

If μ>0\mu>0, then if NN increases (i.e. particles are added), so do AA and GG.

Connection to the first law of thermodynamics

In a system of NN particles that exchanges particles and is subject to a chemical potential μ\mu, the first law of thermodynamics is extended to

dU=TdS−PdV+μdNdU=TdS-PdV+\mu dN

since energy now also depends on the number of particles. This is true for all energy functions, not just the internal energy. The Helmholtz free energy variation is

dA=−PdV−SdT+μdNdA=-PdV-SdT+\mu dN

and the Gibbs free energy one is

dG=dA+d(PV)=−PdV+μdV+PdV−VdP−SdT=VdP−SdT+μdNdG=dA+d(PV)=-PdV+\mu dV+PdV-VdP-SdT=VdP-SdT+\mu dN

The chemical potential is the Lagrange multiplier that governs the number of particles.