## Method of Lagrange multipliers

In order to understand the new computational method, a mathematical detour into the theory of Lagrange multipliers is necessary.

Consider a function F(n1, n2, ). The values of n1, n2, at which F is a minimum are to be determined. The system is subject to the following constraints:

Where F, V and W are specified functions of the mole numbers of all species, ni. The differential of F is:

The differential is a minimum so dF is set equal to zero. The coefficients of dni are:

d ni

Taking the differentials of V and W:

dV = Vnidnj + vn2dn2 + = 0 and dW = wnjdni + wn2dn2 + = 0 (9.65)

where

Multiply dV by Xv and dW by Xw, the Lagrange multipliers, and add to dF:

(fni + XvVnl + XwWnl)dni + (fn2 + XvVn2 + XwWn2)dn2 +...= 0 (9.67)

Since all dni are arbitrary, their coefficients must be zero:

with constraints given by Equation (9.62). These equations are solved for Xv, Xw and n1, n2

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