Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Stirling's approximation

Stirling's approximation is:

ln N! = NlnN - N, for large N

The entropy,

S = k lnw

where w is the number of possible configurations for a system.

For a mechanical mixture w = 1 as the only arrangement is A atoms on A sites and B atoms on B sites.

For a solid solution of A and B containing xAN A atoms and xBN B atoms the value of w is calculated as follows

 N! {xAN}!{(1 - xA)N}!

Assuming that the thermal entropy of the system remains unchanged when A and B go into solution

ΔSmix =
 k ln N! -k ln1 {xAN}!{(1 - xA)N}!
= k [ln N! - ln {xAN}! - ln {(1 - xA)N}!]
= k [ N ln N - N - xAN ln xAN + xAN - (1 - xA)N ln (1 - xA)N + (1 - xA)N]
= kN [ln N - 1 - xA ln xAN + xA - (1 - xA) ln (1 - xA)N + (1 - xA)]
= kN [ln N - xA ln {xAN} - (1 - xA) ln {(1 - xA)N}]
= kN [ln N - xA ln xA - xA ln N - (1 - xA) ln (1 - xA) - ln N + xA ln N
= kN [- xA ln xA - (1 - xA) ln (1 - xA)]
= kN [- xA ln xA - xB ln xB]