Appendix 7: he chemical potential and the Gibbs fact H. Matsuoka I. hermal equilibrium between two systems that can exchange both heat and particles Consider two systems that are separated by a wall which allows heat and particles to go through it. F example, two metallic pieces in contact with each other can exchange both heat and conduction electrons. On the macro level, in the final equilibrium state i he al internal energy and the al mole number must be conserved: U f U f U const n f n f n const ii Accding to the nd law, the al entropy given by S U,U,V,V,n,n S U,V, n S U U,V,n n must be maximum at U U f and n n f : 0 S U S U U U f,n n f U U f,n n f S U U U f,n n f f f S U S U U U f,n n f U U f,n n f. 3 Also, we must have 0 S n U U f,n n f S n U U f,n n f S n U U f,n n f. Define the chemical potential f each system by µ S n, U,V which leads to the extended version of the fundamental equation: ds du P dv µ dn.
We then get S n U U f,n n f f µ f f µ f S n U U f,n n f. 4 By solving the four equations through 4 simultaneously, we can find U f, U f, n f, and n f. he conditions f equilibrium f systems in thermal and diffusive contact: and f f f µ µ f f U f f,v,n U f,v,n f µ U f,v, n f µ U f,v, n he chemical potential Just as heat flows from a high-temperature system to a low-temperature system, particles flow from a high chemical potential system to a low chemical potential system. o see this, consider two systems initially at the same temperature i.e., but at different chemical potentials, µ > µ. ds S U du S n dn S U S - U., / du S n S - n., / dn du µ µ dn µ Since µ > µ, in der f S to increase ds > 0, dn < 0 n must decrease so that particles flow from system to system. µ dn
3 II. he Gibbs fact as a probability a statistical weight A system in thermal equilibrium with a heat and particle reservoir at and µ Macro level: he system: U, V, N; at and µ he heat bath: U B,V B, N B ; at and µ U << U B, V << V B, and N << N B he al energy: he al particle number: U U U B >> U N N N B >> N In equilibrium: he temperature of the system the temperature of the heat bath he chemical potential of the system the chemical potential of the heat bath µ S bath U bath and S bath N U bath U B bath N bath N B µ Micro level: he system: in quantum state s he heat bath: in quantum state r bath he al system: in quantum state s ˆ with energy with energy E r ˆ N,r N ˆ bath { } with U E r N ˆ bath and N N ˆ N ˆ bath >> N ˆ >> In equilibrium: W sys U sys,v,n sys W bath U U sys,v B, N N sys at U sys U and N sys N. N bath >> is maximum Since the overwhelming majity of the quantum states f the al system belongs to the W sys U, V, NW bath U U, V B, N N states, U is well approximated by an average of over all the quantum states s { N ˆ,r N ˆ } that satisfies U bath E r N ˆ and bath N N ˆ N ˆ bath : U s N ˆ W bath U, N N ˆ ˆ N W bath U s N ˆ, N N ˆ
4 We approximate W bath U W bath U s N ˆ, N N ˆ, N N ˆ exp S k bath U B,N N ˆ, -. / exp S k bath U U 0 B 7 5 exp S k bath U B, N B 7 B U s ˆ k B N 6S bath 6U bath as follows:, N N e S bath U B,N B / k B e U µn / k B s N ˆ µ N ˆ exp k B N N ˆ, -. 4 3 U s ˆ 6S bath N k U bath U B B 6N bath N bath N B N N ˆ, 7 -. 7 where we have assumed U s N ˆ << U, U << U, ˆ <<U U and N << N N, which follows from N << N and N << N. Since the majity of the quantum states f the al system are those cresponding to the equilibrium state the W sys U, V, NW bath U U, N N We finally get W bath U N sys, N N ˆ states, then W sys U sys,v, N sys W bath U U sys,v B, N N sys U sys W sys U, V, N W bath U U, N N e S sys U,N /k B e S bath U B,N B / k B U where P defined as / k B e U S sys µn N ˆ s N ˆ P s N ˆ s N ˆ exp s N ˆ µ N ˆ k B s
5 P s e US sys µn / k B exp k B exp k B e / k B µ N ˆ µ N ˆ can be interpreted as the probability f finding the system in eigenstate. In the second line, we have used the definition f the thermodynamic potential: U S sys µn. We can also derive the relation between the Gibbs sum and the thermodynamic potential by noting that the probability P must be nmalized: when we add up the probability f all the eigenstates, we should find the al sum to be one. from which we get N ˆ P e / k B N ˆ,V, µ e / k B exp s µ N ˆ, k B, V,µ e / k B, V,µ k B ln. We also get exp s N ˆ µ N ˆ P s k B.