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 x_{A}N A atoms and x_{B}N B atoms the value of w is calculated as follows
N! 

{x_{A}N}!{(1  x_{A})N}! 
Assuming that the thermal entropy of the system remains unchanged when A and B go into solution
ΔS_{mix}  = 


=  k [ln N!  ln {x_{A}N}!  ln {(1  x_{A})N}!]  
=  k [ N ln N  N  x_{A}N ln x_{A}N + x_{A}N  (1  x_{A})N ln (1  x_{A})N + (1  x_{A})N]  
=  kN [ln N  1  x_{A} ln x_{A}N + x_{A}  (1  x_{A}) ln (1  x_{A})N + (1  x_{A})]  
=  kN [ln N  x_{A} ln {x_{A}N}  (1  x_{A}) ln {(1  x_{A})N}]  
=  kN [ln N  x_{A} ln x_{A}  x_{A} ln N  (1  x_{A}) ln (1  x_{A})  ln N + x_{A} ln N  
=  kN [ x_{A} ln x_{A}  (1  x_{A}) ln (1  x_{A})]  
=  kN [ x_{A} ln x_{A}  x_{B} ln x_{B}] 