Homework Help: Thermodynamics, cannot reach the answer (Reichl's book related)

1. Jun 17, 2013

fluidistic

1. The problem statement, all variables and given/known data
Consider a binary mixture of particles of types 1 and 2 whose Gibbs free energy is given by $G=n_1 \mu _1 ^0 (P,T)+n_2 \mu _2 ^0 (P,T) +RT n_1 \ln x_1 +RTn_2 \ln x_2 +\lambda n x_1x_2$.
Where $n=n_1+n_2$. And $x_1$ and $x_2$ are the mole fractions of particles 1 and 2 respectively.
The book on page 159 states that the conditions $\mu _1 ^I=\mu _1 ^{II}$ and $\mu _2 ^I=\mu _2 ^{II}$ where the upperscript denotes the phase. I have no problem to understand that those conditions must be fulfilled for equilibrium when there's a coexistance of phases.
But now comes the part where I struggle. From the condition $\mu _1 ^I=\mu _1 ^{II}$, I'm supposed to find that $RT \ln x_1 ^I + \lambda (1-x_1 ^I)^2=RT \ln x_1 ^{II}+\lambda (1-x_1 ^{II})^2$.
But I don't reach this.
2. Relevant equations
$\left ( \frac{\partial g}{\partial n_1} \right ) _{T,P,n_2} =\mu _1$.
$x_2=1-x_1$.

3. The attempt at a solution
$\mu _1 =\left ( \frac{\partial g}{\partial n_1} \right ) _{T,P,n_2}=\mu _1 ^0 (P,T)+RT \ln x_1 + \lambda x_1 (1-x_1)$.
So if I use the condition $\mu _1 ^I = \mu _1^{II}$, I'd get $RT\ln x_1 ^I + \lambda x_1 ^I (1-x_1 ^I )=RT\ln x_1 ^{II} + \lambda x_1 ^{II} (1-x_1 ^{II} )$. Which differs from the equation I'm supposed to find.
I know I simply have to derivate g=G/n with respect to $n_1$ while keeping P, T and $n_2$ constant, but I'm simply failing at that apparently.
I've rechecked the algebra many times, I really don't see where I go wrong. My friend told me he reached the good result, so I know I've went wrong somewhere.

Thank you for any help.

2. Jun 17, 2013

TSny

x1 and x2 are not constants. They depend on n1 and n2.

3. Jun 17, 2013

fluidistic

Thank you TSny, I overlooked this.