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I know that to find the composition in a two phase region of a binary diagram we have to draw the isotherm and to look at the intersection with phase boundaries.

enter image description here

But why? Is there a demonstration?

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I will assume that the y-axis in your graph is temperature, and that the pressure of the system is fixed.

Consider the physical system: a mixture of two species in two phases is at equilibrium. Gibbs' phase rule tells us that we have two degrees of freedom, which can be chosen from the set $\{P, T, X_B^{liq}, X_B^\text{solid}\}$. By fixing $P$ (first degree of freedom) and letting $T$ (second degree of freedom) vary over a prescribed range, we produce the functions $X_B^{liq} = X_B^{liq}(T)$ and $X_B^\text{solid} = X_B^\text{solid}(T)$, and this is exactly what is represented by a phase diagram.

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  • $\begingroup$ I understand this but for $\ X_{b} $ fixed and T fixed (the green dot), why $\ X_{b}^{liquid} $ and $\ X_{b}^{solid} $ gives us the composition in this precise case? I don't know if I'm clear $\endgroup$ – Hugues Sep 25 '17 at 8:25
  • $\begingroup$ Actually, here you probably have $P$ and $T$ fixed, and $X_B$ just tells you whether or not you're at equilibrium. $X_B$ is not one of the degrees of freedom available because it doesn't tell you anything about phase compositions. For $P$ and $T$ fixed, I have argued that there is only one possible value of $X_B^\text{liq}$ and $X_B^\text{solid}$, and this can be graphically found by looking at the intersection of the curves with the line $T = T^*$, where $T^*$ is your known temperature. $\endgroup$ – a-cyclohexane-molecule Sep 25 '17 at 14:40
  • $\begingroup$ The last statement is a simple property of functions: if we have a function $f(x)$, then $f(5)$ is the intersection of $f(x)$ with the line $x=5$. $\endgroup$ – a-cyclohexane-molecule Sep 25 '17 at 14:44
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    $\begingroup$ A critical thing to keep in mind is that the red dot gives the composition of the liquid phase, and the blue dot gives the composition of the solid phase, but neither curve gives you the relative amounts of each phase, i.e. how much liquid there is compared to solid. To get that, you need the lever rule, which is really just a restatement of the formula for weighted averages. $\endgroup$ – Curt F. Sep 26 '17 at 20:16

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