Phase Diagrams and Solidification
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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! - 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] |