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Suppose our system is at constant temperature. Total free energy is
A(N_{1},N_{2},V_{1},V_{2},T) = A_{1}(N_{1},V_{1},T) + A_{2}(N_{2},V_{2},T) | (1) |
(2) |
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.To minimize free energy (1) under conditions (2), we minimize a new function
(3) |
In coexisting phase P, T, are equal (as well as other thermodynamic fields).
Let's forget about phase 2. The condition for N_{1} for phase 1 can be obtained from minimization a function
with external pressure and chemical potential P_{0} 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 x_{i}. It has conjugated field
In equilibrium X_{i} 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 X_{i}^{(0)}© 1997 Boris Veytsman and Michael Kotelyanskii