Free energy of a binary alloy. We have an alloy that is made of two types of ato

Question

Free energy of a binary alloy. We have an alloy that is made of two types of atoms A and B, which sit on a regular lattice. There N atoms total, N_A atoms of A, and N_B = N - N_A atoms of B. The multiplicity of the system is given by g(N_A, N) = N!/(N - N_A)! N_A! and the entropy can be approximated by sigma(c) = -n[(1 - C)LN(1 - C) + C LN C], where x = N_A/N is the concentration of species A. Assuming that there is an energy E_A and an energy E_B associated with atoms A and B respectively, and the system is at a constant temperature tau calculate a) the partition function, b) the average energy U, and c) the Helmholtz free energy F.

Explanation / Answer

a) partition function =   [(N !)/(N - NA)!] * [ln(c)/ln(c - 1)]

b) average energy , U = sigma * [(N !)/(N - NA)!] * [ln(c)/ln(c - 1)]

c) Helmholtz free energy F = sigma * [(N !)/(N - NA)!] * [ln(c)/ln(c - 1)] * (NA/N)

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