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Suppose our system is at constant temperature. Total free energy is
|A(N1,N2,V1,V2,T) = A1(N1,V1,T) + A2(N2,V2,T)||(1)|
We want to minimize f(x,y) under conditions g(x,y)=0. Construct a new function
and minimize this as function of independent variables x, y, . Minimizing by we obtain g(x,y)=0, i.e.
The result is --I'd better make a square lot.
To minimize free energy (1) under conditions (2), we minimize a new function
The derivatives here are just (minus) pressures! We obtained:
In coexisting phase P, T, are equal (as well as other thermodynamic fields).
Let's forget about phase 2. The condition for N1 for phase 1 can be obtained from minimization a function
with external pressure and chemical potential P0 and .In equilibrium is -potential (we already derived this in previous lectures ).
The extremum should be a minimum second derivatives are positive!
Suppose we have an extensive variable xi. It has conjugated field
In equilibrium Xi is constant in all phases, and the matrix with
is positive (in equilibrium S has maximum, and A minimum!). We can obtain this by minimizing
with external value Xi(0)
© 1997 Boris Veytsman and Michael Kotelyanskii