Binaires.tex

\chapter{Mélanges Binaires - Équilibres de Phase}


\section{Binaires Liquide/Vapeur (Systèmes fermés)}
\subsection{Variance}
\noindent \underline{Phase liquide unique :} système \textbf{divariant}, on trace donc soit des diagrammes isobares, soit des diagrammes isothermes.

\begin{center}
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\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
\clip(-3,-0.98) rectangle (3,5.62);
\draw (-1,0)-- (-1,2);
\draw (0,2)-- (-1,2);
\draw (-1,0)-- (1,0);
\draw (1,0)-- (1,2);
\draw (0,2)-- (1,2);
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\draw (-1,5)-- (0,5);
\draw (1,2)-- (1,5);
\draw (1,5)-- (0,5);
\draw (-0.7,1.58) node[anchor=north west] {$Liq$};
\draw (-0.7,4.06) node[anchor=north west] {$Vap$};
\draw (1.44,4.06) node[anchor=north west] {$y_1 ,  y_2$};
\draw (1.4,1.6) node[anchor=north west] {$x_1 ,  x_2$};
\draw (-3,2.5) node[anchor=north west] {{\LARGE $(\Sigma)$}};
\end{tikzpicture}
\end{center}


\noindent S'il y a \underline{miscibilité nulle} à l'état liquide, le système est \textbf{monovariant} : à P fixée, T et la composition du système (triphasique) sont imposées.

Remarque : Miscibilité partielle $\rightarrow$ $v=1$.
\subsection{Solutions liquides idéales}
\paragraph{Isothermes :}
En notant $P^*$ la pression de vapeur saturante, on a :
$$P_{1_{(T)}}=x_1P_{1_{(T)}}^* \hspace{1cm} P_{2_{(T)}}=x_2P_{2_{(T)}}^* \hspace{1cm} P=P_1 + P_2 $$
$$ \text{(1)}\hspace{.5cm} P(T)=P_{2_{(T)}}^* + x_1 \left( P_{1_{(T)}}^* - P_{2_{(T)}}^* \right) \hspace{1.5cm} P_{(T)}(x_1) : \text{courbe d'ébullition}$$
\\
\noindent En général, $x_i$ en abscisse.

$$x_1=\displaystyle\frac{ P_{(T)} - P_{2_{(T)}}^*}{ P_{1_{(T)}}^* - P_{2_{(T)}}^*}=\displaystyle\frac{ P_{1_{(T)}}}{P_{1_{(T)}}^*}=\displaystyle\frac{ y_1 P_{(T)}}{P_{1_{(T)}}^*}$$

\noindent Soit donc : $$P_{1_{(T)}}^*\left(P_{(T)} - P_{2_{(T)}}^* \right)=y_1 P_{(T)} \left(P_{1_{(T)}}^* - P_{2_{(T)}}^* \right)$$
$$ P_{(T)}=\displaystyle\frac{P_{1_{(T)}}^*P_{2_{(T)}}^*}{P_{1_{(T)}}^*-y_1 \left(P_{1_{(T)}}^* - P_{2_{(T)}}^* \right)}$$

\noindent $P(y_1)$ est la courbe de rosée. En général, $y_i$ en abscisse.

\begin{center}
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\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
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\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
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\draw [color=ffqqqq] (-2,0)-- (7,6.48);
\draw (2.6,3.4) node[anchor=north west] {$\text{{\Large L+V}}$};
\draw (0.42,5.58) node[anchor=north west] {$\text{{\Large L}}$};
\draw (4.37,0.8) node[anchor=north west] {$\text{{\Large V}}$};
\draw [dash pattern=on 2pt off 2pt] (0.78,2)-- (0.77,0.41);
\draw [dash pattern=on 4pt off 4pt] (-2,2)-- (3.81,2);
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\draw [dash pattern=on 1pt off 2pt on 4pt off 4pt] (0.78,2)-- (0.78,3.68);
\draw (0.4,4.4) node[anchor=north west] {$I$};
\draw (0,2.8) node[anchor=north west] {$E$};
\draw (0.7,1.3) node[anchor=north west] {$R$};
\draw (0.77,0.42)-- (0.76,-0.5);
\draw (0.93,-0.1) node[anchor=north west] {$F$};
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\draw (-2.96,1.32) node[anchor=north west] {$P_r$};
\draw (-2.94,0.39) node[anchor=north west] {$P_2^*$};
\draw (7.21,6.98) node[anchor=north west] {$P_1^*$};
\draw (-2.8,8) node[anchor=north west] {$\text{P}$};
\draw (7.21,8) node[anchor=north west] {$\text{P}$};
\draw (3.8,2.1) node[anchor=north west] {$B$};
\draw [color=ffqqqq](8.06,-0.4) node[anchor=north west] {${x_1}$};
\draw [color=qqqqff](8.06,-1) node[anchor=north west] {$y_1$};
\begin{scriptsize}
\draw [color=black] (0.78,2)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
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\end{scriptsize}
\end{tikzpicture}
\end{center}

\noindent Cas étudié : $P_{1_{(T)}}^* > P_{2_{(T)}}^*$, soit 1 plus volatil que 2. En $E$ apparait la 1$^{ere}$ bulle de vapeur. Sur la courbe d'ébullition, l'abscisse de $B$ donne la composition de la 1$^{ere}$ bulle. $$ x_1=\displaystyle\frac{n_{1,l}}{n_{1,l}+n_{2,l}}$$

\paragraph{Isobares : }
\noindent La courbe d'analyse thermique ($\theta=f(t)$) permet de tracer le diagramme isobare. En $A$, il y a apparition de la 1$^{ere}$ bulle; en $D$, il y a disparition du liquide. $\theta_{eb1}>\theta_{eb2}$, 2 est plus volatil que 1.
\begin{center}
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\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
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\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
\draw [color=ffqqqq](8.04,-0.5) node[anchor=north west] {$x_1$};
\draw [color=qqqqff](8.04,-1) node[anchor=north west] {$y_1$};
\draw [->] (-14,-2) -- (-14,8);
\draw [->] (-15,-1) -- (-5,-1);
\draw (-5,-0.68) node[anchor=north west] {$t$};
\draw (-14.2,9) node[anchor=north west] {$\theta$};
\draw (-2.17,9) node[anchor=north west] {$\theta$};
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\draw (-3.22,7.7) node[anchor=north west] {$\theta_{eb1}$};
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\draw [shift={(-0.56,10.01)},color=ffqqqq]  plot[domain=5.15:5.3,variable=\t]({1*7.94*cos(\t r)+0*7.94*sin(\t r)},{0*7.94*cos(\t r)+1*7.94*sin(\t r)});
\draw [shift={(1.4,7.77)},color=ffqqqq]  plot[domain=5.22:5.76,variable=\t]({1*5*cos(\t r)+0*5*sin(\t r)},{0*5*cos(\t r)+1*5*sin(\t r)});
\draw [shift={(114.24,-74.12)},color=ffqqqq]  plot[domain=2.5:2.51,variable=\t]({1*134.44*cos(\t r)+0*134.44*sin(\t r)},{0*134.44*cos(\t r)+1*134.44*sin(\t r)});
\draw [shift={(10.72,-13.46)},color=qqqqff]  plot[domain=2.07:2.2,variable=\t]({1*19.73*cos(\t r)+0*19.73*sin(\t r)},{0*19.73*cos(\t r)+1*19.73*sin(\t r)});
\draw [shift={(38.01,-61.57)},color=qqqqff]  plot[domain=2.04:2.09,variable=\t]({1*75.04*cos(\t r)+0*75.04*sin(\t r)},{0*75.04*cos(\t r)+1*75.04*sin(\t r)});
\draw [shift={(-9.87,34.3)},color=qqqqff]  plot[domain=5.16:5.23,variable=\t]({1*32.13*cos(\t r)+0*32.13*sin(\t r)},{0*32.13*cos(\t r)+1*32.13*sin(\t r)});
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\draw [shift={(0.36,0.47)},color=qqqqff]  plot[domain=2.1:3,variable=\t]({1*2.38*cos(\t r)+0*2.38*sin(\t r)},{0*2.38*cos(\t r)+1*2.38*sin(\t r)});
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\draw [shift={(-9.59,5)}] plot[domain=2.65:3.14,variable=\t]({1*3.98*cos(\t r)+0*3.98*sin(\t r)},{0*3.98*cos(\t r)+1*3.98*sin(\t r)});
\draw [shift={(-7.21,4.21)}] plot[domain=2.15:2.63,variable=\t]({1*5.49*cos(\t r)+0*5.49*sin(\t r)},{0*5.49*cos(\t r)+1*5.49*sin(\t r)});
\draw [shift={(-3.78,0.2)}] plot[domain=2.76:3.08,variable=\t]({1*6.73*cos(\t r)+0*6.73*sin(\t r)},{0*6.73*cos(\t r)+1*6.73*sin(\t r)});
\draw [shift={(0.94,-1.92)}] plot[domain=2.58:2.74,variable=\t]({1*11.9*cos(\t r)+0*11.9*sin(\t r)},{0*11.9*cos(\t r)+1*11.9*sin(\t r)});
\draw [shift={(15.37,-0.11)}] plot[domain=2.76:2.88,variable=\t]({1*23.44*cos(\t r)+0*23.44*sin(\t r)},{0*23.44*cos(\t r)+1*23.44*sin(\t r)});
\draw [shift={(-4.52,-0.69)}] plot[domain=2.65:3.11,variable=\t]({1*3.2*cos(\t r)+0*3.2*sin(\t r)},{0*3.2*cos(\t r)+1*3.2*sin(\t r)});
\draw [shift={(-1.76,-1.59)}] plot[domain=2.27:2.67,variable=\t]({1*5.29*cos(\t r)+0*5.29*sin(\t r)},{0*5.29*cos(\t r)+1*5.29*sin(\t r)});
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\draw (-8.26,6.5) node[anchor=north west] {$D$};
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\begin{scriptsize}
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\end{scriptsize}
\end{tikzpicture}
\end{center}

On a donc deux diagrammes, avec ici 1 plus volatil que 2 :
\begin{center}
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\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
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\draw [color=ffqqqq] (-2,0)-- (7,6.48);
\draw (2.52,3.3) node[anchor=north west] {$\text{L+V}$};
\draw (0.41,5.62) node[anchor=north west] {$\text{L}$};
\draw (4.35,1.25) node[anchor=north west] {$\text{V}$};
\draw (-2.94,0.42) node[anchor=north west] {$P_2^*$};
\draw (7.22,7.04) node[anchor=north west] {$P_1^*$};
\draw (-2.68,8.23) node[anchor=north west] {$\text{P}$};
\draw (7.19,8.23) node[anchor=north west] {$\text{P}$};
\draw [color=ffqqqq](8.05,-0.3) node[anchor=north west] {$x_1$};
\draw [color=qqqqff](8.05,-0.7) node[anchor=north west] {$y_1$};
\draw [->] (10.01,-2) -- (10.01,8);
\draw [->] (9.01,-1) -- (20.01,-1);
\draw [->] (19.01,-2) -- (19.01,8);
\draw (13.57,3.6) node[anchor=north west] {$\text{L+V}$};
\draw (12.08,2.23) node[anchor=north west] {$\text{L}$};
\draw (16.06,6.21) node[anchor=north west] {$\text{V}$};
\draw (8.85,6.45) node[anchor=north west] {$\theta_{eb2}$};
\draw [color=ffqqqq](20.05,-0.3) node[anchor=north west] {$x_1$};
\draw [color=qqqqff](20.05,-0.7) node[anchor=north west] {$y_1$};
\draw (19.0,1.3) node[anchor=north west] {$\theta_{eb1}$};
\draw (19.12,8.23) node[anchor=north west] {$\theta$};
\draw (9.3,8.23) node[anchor=north west] {$\theta$};
\draw [shift={(6.63,-9.76)},color=qqqqff]  plot[domain=0.98:1.36,variable=\t]({1*16.12*cos(\t r)+0*16.12*sin(\t r)},{0*16.12*cos(\t r)+1*16.12*sin(\t r)});
\draw [shift={(4.71,-12.39)},color=qqqqff]  plot[domain=0.741:0.973,variable=\t]({1*19.38*cos(\t r)+0*19.38*sin(\t r)},{0*19.38*cos(\t r)+1*19.38*sin(\t r)});
\draw [shift={(21.14,13.23)},color=ffqqqq]  plot[domain=3.719:4.19,variable=\t]({1*13.27*cos(\t r)+0*13.27*sin(\t r)},{0*13.27*cos(\t r)+1*13.27*sin(\t r)});
\draw [shift={(18.82,10.04)},color=ffqqqq]  plot[domain=4.23:4.731,variable=\t]({1*9.36*cos(\t r)+0*9.36*sin(\t r)},{0*9.36*cos(\t r)+1*9.36*sin(\t r)});
\end{tikzpicture}
Courbe de \textcolor{blue}{Rosée}. Courbe d'\textcolor{red}{Ébullition}.
\end{center}
\subsection{Solutions liquides réelles à miscibilité totale}
On peut avoir des diagrammes isoP et isoT avec ou sans extremum. Prenons les isoT (il en ira de même pour les isoP).
\paragraph{Courbe sans extremum :}
Faible écart à l'idéalité, diagramme similaire à celui vu précédemment. Exemple : mélange Ph-H/Ph-CH$_3$.

\paragraph{Courbe avec extremum :}
Théorème de \bsc{Gibbs-Konovalov}
\begin{center}
Un extremum de température à pression constante, ou un extremum de pression totale, à température constante entraîne l'identité de composition des phases liquide et vapeur.
\end{center}
On appelle \textbf{Azéotropes} les mélanges qui correspondent à de tels extrema. Exemple d'\textbf{Azéotrope positif} :
\begin{center}
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\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
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\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
\draw (0.48,4.72) node[anchor=north west] {$\text{L+V}$};
\draw (0.05,6.51) node[anchor=north west] {$\text{L}$};
\draw (3.24,3.15) node[anchor=north west] {$\text{V}$};
\draw (-3.1,1.34) node[anchor=north west] {$P_2^*$};
\draw (7.2,6.6) node[anchor=north west] {$P_1^*$};
\draw (-2.8,8.18) node[anchor=north west] {$\text{P}$};
\draw (7.2,8.18) node[anchor=north west] {$\text{P}$};
\draw [color=ffqqqq](8.05,-0.17) node[anchor=north west] {$x_1$};
\draw [color=qqqqff](8.05,-0.8) node[anchor=north west] {$y_1$};
\draw [shift={(7.64,-2.61)},color=ffqqqq]  plot[domain=2.22:2.78,variable=\t]({1*10.29*cos(\t r)+0*10.29*sin(\t r)},{0*10.29*cos(\t r)+1*10.29*sin(\t r)});
\draw [shift={(-6.61,12.03)},color=qqqqff]  plot[domain=5.11:5.6,variable=\t]({1*11.96*cos(\t r)+0*11.96*sin(\t r)},{0*11.96*cos(\t r)+1*11.96*sin(\t r)});
\draw [shift={(5.82,-0.34)},color=ffqqqq]  plot[domain=1.68:2.21,variable=\t]({1*7.39*cos(\t r)+0*7.39*sin(\t r)},{0*7.39*cos(\t r)+1*7.39*sin(\t r)});
\draw [shift={(22.59,-10.48)},color=qqqqff]  plot[domain=2.37:2.5,variable=\t]({1*24.93*cos(\t r)+0*24.93*sin(\t r)},{0*24.93*cos(\t r)+1*24.93*sin(\t r)});
\draw [shift={(5.02,4.54)},color=ffqqqq]  plot[domain=0.63:1.58,variable=\t]({1*2.46*cos(\t r)+0*2.46*sin(\t r)},{0*2.46*cos(\t r)+1*2.46*sin(\t r)});
\draw [shift={(7.66,9.31)},color=qqqqff]  plot[domain=3.97:4.52,variable=\t]({1*3.38*cos(\t r)+0*3.38*sin(\t r)},{0*3.38*cos(\t r)+1*3.38*sin(\t r)});
\draw [shift={(5.01,6.55)},color=qqqqff]  plot[domain=0.68:2.42,variable=\t]({1*0.45*cos(\t r)+0*0.45*sin(\t r)},{0*0.45*cos(\t r)+1*0.45*sin(\t r)});
\draw (5.55,7) node[anchor=north west] {$\text{{\scriptsize L}}$};
\draw (5.9,6.8) node[anchor=north west] {$\text{{\scriptsize +}}$};
\draw (6.27,6.6) node[anchor=north west] {$\text{{\scriptsize V}}$};
\draw [->] (5,7) -- (6.19,7.01);
\draw [->] (5,7) -- (3.86,6.99);
\draw (4.6,8.1) node[anchor=north west] {$Z$};
\begin{scriptsize}
\draw [color=black] (5,7)-- ++(-2.5pt,-2.5pt) -- ++(5.0pt,5.0pt) ++(-5.0pt,0) -- ++(5.0pt,-5.0pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
Ecart positif à l'idéalité : $P_i>P_i(LR)$. Déstabilisation du mélange liquide par rapport au cas idéal. Exemple : eau/éthanol.
\\ \\
\textbf{Azéotrope négatif :} écart négatif à l'idéalité, $P_i<P_i(LR)$. Stabilisation du mélange liquide par rapport au cas idéal. Exemple : acétone/chloroforme.
\\ \\
En Z, la variance est de 1 : 6 paramètres, 5 relations. T est imposée, les autres paramètres sont fixes. 
\\ \\
A $T=c^{ste}$, pour la composition Z comme pour les corps purs, la pression totale reste constante tant que coexistent les 2 phases.

\subsection{Théorème des Moments Chimiques}
Permet d'atteindre les quantités de matière de chacune des phases.
\begin{center}
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\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.8cm,y=0.8cm]
\clip(-4.03,-2.27) rectangle (9.01,8.31);
\draw [->] (-2,-2) -- (-2,8);
\draw [->] (-3,-1) -- (8,-1);
\draw (-2.67,8.18) node[anchor=north west] {$\text{P}$};
\draw [color=ffqqqq](8.05,-0.17) node[anchor=north west] {$x_1$};
\draw [color=qqqqff](8.05,-0.8) node[anchor=north west] {$y_1$};
\draw [shift={(4.16,0.24)},color=ffqqqq]  plot[domain=1.72:2.62,variable=\t]({1*5.16*cos(\t r)+0*5.16*sin(\t r)},{0*5.16*cos(\t r)+1*5.16*sin(\t r)});
\draw [shift={(-1.27,8.74)},color=qqqqff]  plot[domain=5.03:5.65,variable=\t]({1*7.96*cos(\t r)+0*7.96*sin(\t r)},{0*7.96*cos(\t r)+1*7.96*sin(\t r)});
\draw (0.07,6.18) node[anchor=north west] {$\text{L}$};
\draw (2.09,1.13) node[anchor=north west] {$\text{V}$};
\draw (1.22,3.28) node[anchor=north west] {$\text{L+V}$};
\draw [dash pattern=on 5pt off 5pt] (-2,3.73)-- (6,3.73);
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\draw (3.1,-0.97) node[anchor=north west] {$\text{X}_1$};
\draw (-0.1,4.74) node[anchor=north west] {$A$};
\draw (4.99,3.8) node[anchor=north west] {$B$};
\draw (3.06,4.68) node[anchor=north west] {$M$};
\begin{scriptsize}
\draw [color=black] (3.37,3.73)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
$$ n_g=n_{1,g}+n_{2,g} \hspace{1cm} x_1=\displaystyle\frac{n_{1,l}}{n_{l}}$$
$$ n_l=n_{1,l}+n_{2,l} \hspace{1cm} y_1=\displaystyle\frac{n_{1,g}}{n_{g}}$$
On définit $X_1$ comme étant la fraction molaire globale :
$$ X_1=\displaystyle\frac{n_{1,g}+n_{1,l}}{n_{g}+n_{l}}=\displaystyle\frac{y_1 n_{g}+x_1 n_{l}}{n_{g}+n_{l}}$$
On en déduit (car $n_g\text{MB}=n_l\text{AM}$) : 
$$ n_g(y_1-X_1)=n_l(X_1-x_1)$$
Remarque : si en abscisse sont reportées les fractions massique $w_{1,g}$ et $w_{1,l}$, en posant $W_1=\displaystyle\frac{m_1}{m_1+m_2}$, on a :
$$ m_g(w_{1,g}-W_1)=m_l(W1-w_{1,l})$$

\subsection{Propriétés colligatives}
Ce sont des propriétés qui dépendent :
\begin{enumerate}
\item de la nature du solvant.
\item de la quantité de soluté.
\item mais \underline{\textbf{PAS}} de la nature du soluté.
\end{enumerate}
\subsubsection{Tonométrie}
\noindent État initial : solvant S à l'équilibre L/V $P_I=P_S^*(T)$. On opère à $T=C^{ste}$. On ajoute une petite quantité de soluté A peu volatil.
\\ \\
État final : $\text{S(l)}=\text{S(g)}$ à $P_F$
$$P_F\simeq P_S(LR)= P_S^*(T) x_S=P_S^*(T)(1-x_A)$$
La pression a diminué de $x_A P_S^*(T)$. Abaissement relatif : 
$$\displaystyle\frac{P_S^*(T)-P_F}{P_S^*(T)}=x_A= \displaystyle\frac{n_A}{n_A+n_S}\simeq \displaystyle\frac{n_A}{n_S}$$
On a : $n_A=\displaystyle\frac{m_A}{M_A}$. Soit donc :
$$ \displaystyle\frac{P_S^*(T)-P_F}{P_S^*(T)}=\displaystyle\frac{m_A}{n_S} \displaystyle\frac{1}{M_A} \longrightarrow M_A$$
\begin{center}
\definecolor{uuuuuu}{rgb}{0.27,0.27,0.27}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\clip(-3.55,-2.45) rectangle (9.28,8.7);
\draw [->] (-2,-2) -- (-2,8);
\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
\draw (6.8,7.04) node[anchor=north west] {$P_S^*(\text{T})$};
\draw (-3.2,8.23) node[anchor=north west] {$\text{P}$};
\draw (7.19,8.23) node[anchor=north west] {$\text{P}$};
\draw (7,6.07)-- (5.96,5.36);
\draw [dash pattern=on 2pt off 2pt] (5.96,5.36)-- (-2,0);
\draw [shift={(2.56,5.38)}] plot[domain=5.61:6.28,variable=\t]({1*3.4*cos(\t r)+0*3.4*sin(\t r)},{0*3.4*cos(\t r)+1*3.4*sin(\t r)});
\draw (8.16,-0.8) node[anchor=north west] {$x$};
\draw (6,-1.1) node[anchor=north west] {$1$};
\end{tikzpicture}
\end{center}
\subsubsection{Ébulliométrie}
\noindent État initial : solvant S à l'équilibre L/V à $T_I$. On opère à $P=C^{ste}$ : $P=P_S^*(T_I)=P_I$. On ajoute une petite quantité d'un soluté A peu volatil. 
\\ \\
État d'équilibre : $\text{S(l)}=\text{S(g)}$ à $T_F$.
$$ K_{(T_F)}^0=\displaystyle\frac{P_S}{P^0 x_S}=\displaystyle\frac{P_I}{P^0 x_S} \hspace{1cm} K_{(T_I)}^0=\displaystyle\frac{P_I}{P^0} $$

$$ \displaystyle\frac{\mathrm d\ln{K_{(T)}^0}}{\mathrm dT}=\displaystyle\frac{\Delta_{eb}H^0_{(S)}}{RT^2}  \hspace{1cm} \text{Soit :} \hspace{0.5cm} \int_{T_I}^{T_F} \! \mathrm d\left(\ln{K_{(T)}^0}\right)=\displaystyle\frac{\Delta_{eb}H^0_{(S)}}{R} \int_{T_I}^{T_F} \displaystyle\frac{\mathrm dT}{T^2}$$
\vspace{0.5cm}
$$ \ln{\left(\displaystyle\frac{K_{(T_F)}^0}{K_{(T_I)}^0}\right)}=\displaystyle\frac{\Delta_{eb}H^0_{(S)}}{R} \left(\displaystyle\frac{1}{T_I}-\displaystyle\frac{1}{T_F} \right)  \hspace{1cm} \text{Et donc :} \hspace{0.5cm} -\ln{x_S}=\displaystyle\frac{\Delta_{eb}H^0_{(S)}}{R T_I T_F} \left(T_F-T_I \right)$$
On pose $\Delta T = T_F-T_I > 0$. On a : $T_I T_F=T_{eb(s)}(T_{eb(s)}+\Delta T) \simeq T_{eb(s)}^2$. On a ainsi :
$$ x_A \simeq \displaystyle\frac{\Delta_{eb}H^0_{(S)} \Delta T}{R T_{eb(s)}^2}$$

\subsection{Distillation (Isobare)}

\subsubsection{Distillation simple}
Chauffage assez doux : proche de l'équilibre. T $\nearrow$, $x_A \searrow$. Vapeur toujours plus riche en A que le liquide, mais la composition en A $\searrow \ $. On peut ainsi obtenir le composé le moins volatil pur, mais avec un mauvais rendement.

\subsubsection{Distillation fractionnée en l'abscence d'azéotrope}
Colonne de distillation : lieu où se produisent plusieurs distillation (ébu/condensation) en continu. Différents types de colonnes : à plateaux (industrie), de \bsc{Vigreux} (labo du lycée), à remplissage (anneaux). 
 \\
Si la colonne est assez performante, en tête de colonne on peut recueillir le composé le plus volatil, dans un premier temps.

\begin{center}
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\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
\clip(-3.96,-2.38) rectangle (11.58,9.2);
\draw [->] (-2,-2) -- (-2,8);
\draw [->] (-3,-1) -- (9.99,-1);
\draw (-3.51,9.1) node[anchor=north west] {$\theta_{tête \ de \ colonne}$};
\draw (10.3,-0.5) node[anchor=north west] {$t$};
\draw (0.88,2.75)-- (2.76,2.75);
\draw (5.86,5.52)-- (7.74,5.52);
\draw (-2,1.02)-- (-0.52,1.02);
\draw [shift={(-3.33,4.72)}] plot[domain=5.365:5.845,variable=\t]({1*4.65*cos(\t r)+0*4.65*sin(\t r)},{0*4.65*cos(\t r)+1*4.65*sin(\t r)});
\draw [shift={(2.58,1.93)}] plot[domain=0.19:0.61,variable=\t]({1*6.28*cos(\t r)+0*6.28*sin(\t r)},{0*6.28*cos(\t r)+1*6.28*sin(\t r)});
\draw [shift={(2.79,2.13)}] plot[domain=0.35:1.63,variable=\t]({1*0.62*cos(\t r)+0*0.62*sin(\t r)},{0*0.62*cos(\t r)+1*0.62*sin(\t r)});
\draw [shift={(4.27,2.72)}] plot[domain=3.53:4.5,variable=\t]({1*0.98*cos(\t r)+0*0.98*sin(\t r)},{0*0.98*cos(\t r)+1*0.98*sin(\t r)});
\draw [shift={(4.16,2.49)}] plot[domain=4.54:6.09,variable=\t]({1*0.72*cos(\t r)+0*0.72*sin(\t r)},{0*0.72*cos(\t r)+1*0.72*sin(\t r)});
\draw [shift={(-11.48,6.44)}] plot[domain=6.035:6.14,variable=\t]({1*16.86*cos(\t r)+0*16.86*sin(\t r)},{0*16.86*cos(\t r)+1*16.86*sin(\t r)});
\draw [shift={(8.09,3.64)}] plot[domain=2.44:3.01,variable=\t]({1*2.92*cos(\t r)+0*2.92*sin(\t r)},{0*2.92*cos(\t r)+1*2.92*sin(\t r)});
\draw [dash pattern=on 2pt off 2pt] (0.88,2.75)-- (-2,2.75);
\draw [dash pattern=on 2pt off 2pt] (5.86,5.52)-- (-2,5.52);
\draw (-3.39,6.2) node[anchor=north west] {$\theta_{eb,B}$};
\draw (-3.39,3.35) node[anchor=north west] {$\theta_{eb,A}$};
\draw (5.9,6.57) node[anchor=north west] {$B \ pur$};
\draw (1,3.67) node[anchor=north west] {$A \ pur$};
\draw (-1.92,-1) node[anchor=north west] {$0$};
\end{tikzpicture}
\end{center}

\subsubsection{Distillation fractionnée en présence d'azéotrope}
On recueille  : 
\begin{itemize}
\item l'azéotrope
\item le corps pur en excès par rapport à la composition azéotropique
\end{itemize}

\subsection{Solutions liquides réelles à miscibilité nulle}
On part d'un mélange liquide contenant 2 phases liquides : A et B purs. On chauffe en agitant. $$ X_A=\displaystyle\frac{n_A}{n_A+n_B} $$
Le point $H$ est appelé \textbf{Hétéroazéotrope}.
\begin{center}
\definecolor{ffqqqq}{rgb}{1,0,0}
\definecolor{qqqqff}{rgb}{0,0,1}
\definecolor{qqzzqq}{rgb}{0,0.6,0}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
\clip(-15.21,-2.35) rectangle (9.3,9.07);
\draw [->] (-2,-2) -- (-2,8);
\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
\draw (8.09,-1.1) node[anchor=north west] {$X$};
\draw [color=qqzzqq](8.07,-0.1) node[anchor=north west] {$Y_A$};
\draw [->] (-14,-2) -- (-14,8);
\draw [->] (-15,-1) -- (-5,-1);
\draw (-5,-0.68) node[anchor=north west] {$t$};
\draw (-14.2,9) node[anchor=north west] {$\theta$};
\draw (-2.17,9) node[anchor=north west] {$\theta$};
\draw (-2.91,1) node[anchor=north west] {$\theta_H$};
\draw (7.25,2.1) node[anchor=north west] {$\theta_{ebA}$};
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\draw [color=qqzzqq](0.94,0.5) node[anchor=north west] {$A(l)+B(l)$};
\draw (6.78,9) node[anchor=north west] {$\theta$};
\draw (-14.25,0.25) node[anchor=north west] {{\small $X_A=0,2$}};
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\draw (-11.85,1.1) node[anchor=north west] {$D_{A(l)}$};
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\draw [shift={(-9.03,-0.52)}] plot[domain=2.77:3.14,variable=\t]({1*4.2*cos(\t r)+0*4.2*sin(\t r)},{0*4.2*cos(\t r)+1*4.2*sin(\t r)});
\draw [color=qqqqff] (-9.5,0.99)-- (-8.16,0.99);
\draw [color=ffqqqq] (-6.87,0.99)-- (-5.97,0.99);
\draw [shift={(-6.75,-0.51)},color=qqqqff]  plot[domain=2.64:3.15,variable=\t]({1*3.13*cos(\t r)+0*3.13*sin(\t r)},{0*3.13*cos(\t r)+1*3.13*sin(\t r)});
\draw [shift={(-3.46,-0.49)},color=ffqqqq]  plot[domain=2.73:3.14,variable=\t]({1*3.73*cos(\t r)+0*3.73*sin(\t r)},{0*3.73*cos(\t r)+1*3.73*sin(\t r)});
\draw [color=ffqqqq] (-5.97,0.99)-- (-5.69,1.51);
\draw [shift={(-19.75,7.79)},color=ffqqqq]  plot[domain=5.86:6.26,variable=\t]({1*15.4*cos(\t r)+0*15.4*sin(\t r)},{0*15.4*cos(\t r)+1*15.4*sin(\t r)});
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\draw [color=qqqqff](-10.9,0.25) node[anchor=north west] {{\small $X_A=X_H$}};
\draw [color=ffqqqq](-8.2,0.25) node[anchor=north west] {{\small $X_A=0,8$}};
\draw [dotted] (-14,0.99)-- (7,0.99);
\draw (-15,1.82) node[anchor=north west] {$\theta_H$};
\draw [dash pattern=on 2pt off 2pt] (-9.98,5.7)-- (-0.01,5.7);
\draw [dash pattern=on 2pt off 2pt,color=ffqqqq] (-5.69,1.51)-- (6.01,1.51);
\draw [color=qqzzqq] (-2.01,0.99)-- (6.99,0.99);
\draw [shift={(-7.62,-4.85)},color=qqzzqq]  plot[domain=0.465:1.124,variable=\t]({1*13.01*cos(\t r)+0*13.01*sin(\t r)},{0*13.01*cos(\t r)+1*13.01*sin(\t r)});
\draw [shift={(7.19,-7.27)},color=qqzzqq]  plot[domain=1.593:1.94,variable=\t]({1*8.86*cos(\t r)+0*8.86*sin(\t r)},{0*8.86*cos(\t r)+1*8.86*sin(\t r)});
\draw [color=qqzzqq](-1.19,3.87) node[anchor=north west] {$V+B(l)$};
\draw [color=qqzzqq](5.15,1.6) node[anchor=north west] {{\scriptsize  $V+A(l)$}};
\draw (-0.37,-0.96) node[anchor=north west] {$0,2$};
\draw [color=qqqqff](3.64,-0.99) node[anchor=north west] {$X_H$};
\draw [color=ffqqqq](5.63,-0.96) node[anchor=north west] {$0,8$};
\draw (-11.5,6.5) node[anchor=north west] {$D_{B(l)}$};
\draw [color=ffqqqq](-6.2,1) node[anchor=north west] {$D_{B(l)}$};
\draw [color=ffqqqq](-5.36,2.41) node[anchor=north west] {$D_{A(l)}$};
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\draw [color=ffqqqq](-7.2,1.8) node[anchor=north west] {$A$};
\draw [color=qqqqff](-10.06,3.3) node[anchor=north west] {$D_{A(l)B(l)}$};
\draw [->,color=qqqqff] (-8.96,2.48) -- (-8.16,0.99);
\begin{scriptsize}
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\end{scriptsize}
\end{tikzpicture}
En $A$, apparition de la 1$^{ere}$ bulle. En $D_{A(l)}$, disparition de $A(l)$.
\end{center}
\subsubsection{Applications}
\noindent \underline{$\star$ Hydrodistillation} (distillation simple, eau $+$ liquide non miscible à l'eau):
\\ \\ 
\noindent Lorsque l'on arrive à $\theta_H$, il y a apparition de la vapeur à la composition hétéroazéotropique ($\theta_H < 100°C$). Le distillat est composé de 2 phases que l'on sépare (ampoule à décanter). Avec ce type de distillation, on minimise les risques de dégradation thermique des composés organiques.
\\ \\
\noindent \underline{$\star$ Entrainement à la vapeur :}
\\ \\
\noindent Même principe, montage différent.
\\ \\ \\
\section{Binaires Liquide/Solide (Systèmes Fermés)}
\subsection{Variance}
\noindent Liquide : 1 seule phase. Le cas 1 seule phase solide est rare, $v=1$. Si miscibilité nulle à l'état solide, 2 phases solides, $v=0$ : la T d'équilibre est imposée par la nature de $(\Sigma)$, idem pour la composition.

\subsection{Miscibilité totale à l'état solide}
\begin{center}
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\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
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\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
\draw (3.07,6.27) node[anchor=north west] {$L$};
\draw [shift={(6.44,8.59)}] plot[domain=3.44:4.053,variable=\t]({1*8.83*cos(\t r)+0*8.83*sin(\t r)},{0*8.83*cos(\t r)+1*8.83*sin(\t r)});
\draw [shift={(-10.98,-10.16)},color=qqqqff]  plot[domain=0.832:1.064,variable=\t]({1*18.49*cos(\t r)+0*18.49*sin(\t r)},{0*18.49*cos(\t r)+1*18.49*sin(\t r)});
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\draw [shift={(-11.84,-10.74)},color=qqqqff]  plot[domain=0.6:0.82,variable=\t]({1*19.49*cos(\t r)+0*19.49*sin(\t r)},{0*19.49*cos(\t r)+1*19.49*sin(\t r)});
\draw [shift={(4.79,3.16)}] plot[domain=4.7:5.48,variable=\t]({1*3.18*cos(\t r)+0*3.18*sin(\t r)},{0*3.18*cos(\t r)+1*3.18*sin(\t r)});
\draw [shift={(6.71,-0.81)},color=qqqqff]  plot[domain=1.4:2.51,variable=\t]({1*1.72*cos(\t r)+0*1.72*sin(\t r)},{0*1.72*cos(\t r)+1*1.72*sin(\t r)});
\draw [shift={(4.81,0.69)},color=qqqqff]  plot[domain=3.68:5.53,variable=\t]({1*0.7*cos(\t r)+0*0.7*sin(\t r)},{0*0.7*cos(\t r)+1*0.7*sin(\t r)});
\draw (5.47,0.79) node[anchor=north west] {{\scriptsize $L$}};
\draw (5.81,0.96) node[anchor=north west] {{\scriptsize $+$}};
\draw (6.22,1.07) node[anchor=north west] {{\scriptsize $S$}};
\draw (4.25,0) node[anchor=north west] {$M$};
\draw [->] (-14,-1.98) -- (-14,8.02);
\draw [->] (-15,-0.98) -- (-4,-0.98);
\draw [->] (-5,-1.98) -- (-5,8.02);
\draw (-10.13,3.58) node[anchor=north west] {$L+S$};
\draw (-7.65,1.27) node[anchor=north west] {$S$};
\draw (-13.91,8.21) node[anchor=north west] {$\theta$};
\draw [color=qqqqff](-3.92,-0.2) node[anchor=north west] {$x_{1,l}$};
\draw (-3.94,-1.2) node[anchor=north west] {$x_{1,s }$};
\draw [shift={(3.61,-14.93)},color=qqqqff]  plot[domain=1.953:2.437,variable=\t]({1*23.1*cos(\t r)+0*23.1*sin(\t r)},{0*23.1*cos(\t r)+1*23.1*sin(\t r)});
\draw [shift={(-16.26,12.65)}] plot[domain=4.89:5.782,variable=\t]({1*12.83*cos(\t r)+0*12.83*sin(\t r)},{0*12.83*cos(\t r)+1*12.83*sin(\t r)});
\draw (-11.99,6.29) node[anchor=north west] {$L$};
\draw (8,-1.2) node[anchor=north west] {$x_{1,s }$};
\draw [color=qqqqff](8,-0.2) node[anchor=north west] {$x_{1,l}$};
\draw (-5.01,8.31) node[anchor=north west] {$\theta$};
\draw (-2,8.28) node[anchor=north west] {$\theta$};
\draw (7,8.28) node[anchor=north west] {$\theta$};
\draw (0.16,3.2) node[anchor=north west] {$L+S$};
\draw (0.38,0.93) node[anchor=north west] {$S$};
\draw [color=qqqqff](-11.41,3.67) node[anchor=north west] {$L$};
\draw [color=qqqqff](-11.02,3.9) node[anchor=north west] {$i$};
\draw [color=qqqqff](-10.73,4.06) node[anchor=north west] {$q$};
\draw [color=qqqqff](-10.39,4.3) node[anchor=north west] {$u$};
\draw [color=qqqqff](-10.03,4.6) node[anchor=north west] {$i$};
\draw [color=qqqqff](-9.81,4.79) node[anchor=north west] {$d$};
\draw [color=qqqqff](-9.47,4.88) node[anchor=north west] {$u$};
\draw [color=qqqqff](-9.18,5.13) node[anchor=north west] {$s$};
\draw (-10.64,1.44) node[anchor=north west] {$S$};
\draw (-10.22,1.55) node[anchor=north west] {$o$};
\draw (-9.88,1.82) node[anchor=north west] {$l$};
\draw (-9.59,2) node[anchor=north west] {$i$};
\draw (-9.3,2.3) node[anchor=north west] {$d$};
\draw (-8.91,2.4) node[anchor=north west] {$u$};
\draw (-8.52,2.7) node[anchor=north west] {$s$};
\draw (-15,0.96) node[anchor=north west] {$\theta_{f1}$};
\draw (-5,7.3) node[anchor=north west] {$\theta_{fus,2}$};
\draw (7.36,1.81) node[anchor=north west] {$\theta_{fus,1}$};
\draw (-3.6,6.42) node[anchor=north west] {$\theta_{fus,2}$};
\draw (-13.89,-1) node[anchor=north west] {$0$};
\begin{scriptsize}
\draw [color=black] (4.75,-0.01)-- ++(-2.5pt,-2.5pt) -- ++(5.0pt,5.0pt) ++(-5.0pt,0) -- ++(5.0pt,-5.0pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
\vspace{-0.3cm}
\noindent Premier cas : sans extremum. Ex : Cu/Ni.
\\
\noindent Second cas : avec extremum. M : point indifférent. Ex : Cu/Au.
\subsubsection{Analyse Thermique}
\vspace{-0.3cm}
\begin{center}
\definecolor{qqqqff}{rgb}{0,0,1}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
\clip(-15.55,-2.03) rectangle (6.67,8.33);
\draw [->] (-2,-2) -- (-2,8);
\draw [->] (-3,-1) -- (6.2,-1);
\draw [->] (-14,-1.98) -- (-14,8.02);
\draw [->] (-15,-0.98) -- (-4,-0.98);
\draw [->] (-5,-1.98) -- (-5,8.02);
\draw (-11.3,2.75) node[anchor=north west] {$L+S$};
\draw (-7.66,1.03) node[anchor=north west] {$S$};
\draw (-13.92,8.21) node[anchor=north west] {$\theta$};
\draw [color=qqqqff](-3.93,0.2) node[anchor=north west] {$x_{1,l}$};
\draw (-3.95,-0.74) node[anchor=north west] {$x_{1,s }$};
\draw [shift={(3.61,-14.93)},color=qqqqff]  plot[domain=1.953:2.437,variable=\t]({1*23.1*cos(\t r)+0*23.1*sin(\t r)},{0*23.1*cos(\t r)+1*23.1*sin(\t r)});
\draw [shift={(-16.26,12.65)}] plot[domain=4.89:5.782,variable=\t]({1*12.83*cos(\t r)+0*12.83*sin(\t r)},{0*12.83*cos(\t r)+1*12.83*sin(\t r)});
\draw (-11.45,7.53) node[anchor=north west] {$L$};
\draw (-4.99,8.31) node[anchor=north west] {$\theta$};
\draw (-2.01,8.28) node[anchor=north west] {$\theta$};
\draw (-15.5,0.96) node[anchor=north west] {$\theta_{fus,1}$};
\draw (-4.8,7.6) node[anchor=north west] {$\theta_{fus,2}$};
\draw (-13.9,-1) node[anchor=north west] {$0$};
\draw [dotted] (-7.41,7.36)-- (-7.39,5.38);
\draw [dash pattern=on 2pt off 2pt] (-7.39,5.38)-- (0,5.39);
\draw [dotted] (-7.39,5.38)-- (-7.39,3.38);
\draw [dash pattern=on 2pt off 2pt] (-7.39,3.38)-- (3.18,3.38);
\draw [dotted] (-7.39,3.38)-- (-10.48,3.38);
\draw (-7.39,3.38)-- (-7.4,2.24);
\draw (0,5.39)-- (3.18,3.38);
\draw [shift={(6.16,7.9)}] plot[domain=3.21:3.53,variable=\t]({1*6.65*cos(\t r)+0*6.65*sin(\t r)},{0*6.65*cos(\t r)+1*6.65*sin(\t r)});
\draw [shift={(13.29,5.65)}] plot[domain=3.36:3.69,variable=\t]({1*10.36*cos(\t r)+0*10.36*sin(\t r)},{0*10.36*cos(\t r)+1*10.36*sin(\t r)});
\draw (-10.96,4.64) node[anchor=north west] {$C$};
\draw (-8,8.28) node[anchor=north west] {$I$};
\draw (6.19,-0.72) node[anchor=north west] {$t$};
\draw (0.15,6.51) node[anchor=north west] {$A$};
\draw (3.28,4.45) node[anchor=north west] {$D$};
\begin{scriptsize}
\draw [color=black] (-7.41,7.36)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (0,5.39)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (3.18,3.38)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (-10.48,3.38)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
En A : apparition du premier microcristal. En D : disparition de la dernière goutte de liquide. En C : on a la composition de la dernière goutte.

\subsubsection{Cristallisation Fractionnée}
En effectuant plusieurs fois la double opération \textbf{cristallisation-fusion}, on obtient un solide de plus en plus riche (le moins fusible).

\subsection{Miscibilité nulle à l'état solide}
\subsubsection{Analyse Thermique}
\begin{center}
\definecolor{qqqqff}{rgb}{0,0,1}
\definecolor{ffqqqq}{rgb}{1,0,0}
\definecolor{qqzzqq}{rgb}{0,0.6,0}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
\clip(-15.3,-2.1) rectangle (9.32,8.72);
\draw [->] (-2,-2) -- (-2,8);
\draw [->] (-3,-1) -- (8,-1);
\draw [->] (7,-2) -- (7,8);
\draw [color=qqzzqq](8.07,-0.2) node[anchor=north west] {$x_{1,l}$};
\draw [->] (-14,-2) -- (-14,8);
\draw [->] (-15,-1) -- (-5,-1);
\draw (-5,-0.7) node[anchor=north west] {$t$};
\draw (-14.2,8.8) node[anchor=north west] {$\theta$};
\draw (-2.15,8.8) node[anchor=north west] {$\theta$};
\draw (-2.93,1.15) node[anchor=north west] {$\theta_E$};
\draw (7.37,3.92) node[anchor=north west] {$\theta_{fus1}$};
\draw (-3.55,6.54) node[anchor=north west] {$\theta_{fus2}$};
\draw (2.55,5.94) node[anchor=north west] {$L_{1+2}$};
\draw (1.5,0.55) node[anchor=north west] {$1(s)/2(s)$};
\draw (6.8,8.8) node[anchor=north west] {$\theta$};
\draw (-13.2,3.27) node[anchor=north west] {$X_1=0$};
\draw (-13.02,5.32)-- (-11.77,5.32);
\draw [line width=1.4pt,color=ffqqqq] (-9,0.99)-- (-7.45,0.99);
\draw [dotted] (-14,0.99)-- (7,0.99);
\draw (-15,1.79) node[anchor=north west] {$\theta_E$};
\draw [dash pattern=on 2pt off 2pt] (-10.17,4.29)-- (0.25,4.29);
\draw [line width=0.9pt,dash pattern=on 2pt off 2pt,color=qqqqff] (-2.01,0.99)-- (6.99,0.99);
\draw [shift={(-4.43,-3)},color=qqzzqq]  plot[domain=0.48:1.29,variable=\t]({1*8.67*cos(\t r)+0*8.67*sin(\t r)},{0*8.67*cos(\t r)+1*8.67*sin(\t r)});
\draw [shift={(7.55,-1.93)},color=qqzzqq]  plot[domain=1.68:2.54,variable=\t]({1*5.19*cos(\t r)+0*5.19*sin(\t r)},{0*5.19*cos(\t r)+1*5.19*sin(\t r)});
\draw (-1.4,3.39) node[anchor=north west] {$L+2(s)$};
\draw (4.5,2.22) node[anchor=north west] {$L+1(s)$};
\draw (0,-1.03) node[anchor=north west] {$0,2$};
\draw (-7.4,2.04) node[anchor=north west] {$D$};
\draw [dotted] (-11.77,5.32)-- (-2,5.32);
\draw (2.9,2.37) node[anchor=north west] {$E$};
\draw (-15.55,6.14) node[anchor=north west] {$\theta_{fus2}$};
\draw (-10.17,4.29)-- (-10.42,7.32);
\draw (-10.17,4.29)-- (-9,0.99);
\draw (-7.45,0.99)-- (-6.85,-0.18);
\draw (-13.02,5.32)-- (-13.5,7.32);
\draw (-11.77,5.32)-- (-10.52,1.32);
\draw (-9.05,2.87) node[anchor=north west] {$X_1=0,2$};
\draw [color=qqqqff](8.05,-1.2) node[anchor=north west] {$x_{1,s}$};
\draw (6.4,-1) node[anchor=north west] {$1$};
\draw (-1.85,-1) node[anchor=north west] {$0$};
\draw [line width=0.9pt,color=qqqqff] (-2,5.32)-- (-2,0.99);
\draw [line width=0.9pt,color=qqqqff] (7,3.23)-- (7,0.99);
\draw (-10.05,5.27) node[anchor=north west] {$A_{2(s)}$};
\draw (-13.57,5.32) node[anchor=north west] {$A_{2(s)}$};
\draw (-9.7,1.1) node[anchor=north west] {$A_{1(s)}$};
\draw (-12,6.29) node[anchor=north west] {$D$};
\begin{scriptsize}
\draw [color=black] (-13.02,5.32)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (-11.77,5.32)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (-10.17,4.29)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (-9,0.99)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (-7.45,0.99)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=qqzzqq] (0.25,4.29)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=qqzzqq] (3.26,0.99)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (0.22,-1)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
\noindent En \textcolor{red}{{\Large ---}}, cristallisation des 2 solides. 
\\
\noindent En D, disparition du liquide.
\\
\noindent En A(.), apparition de (.). 
\\
\noindent E, point \textbf{Eutectique}, à cette composition, la température de fusion est minimale. 
\\
\noindent En \textcolor{blue}{{\Large -\,-\,-}}, variation de $\theta$ avec la composition aux équilibres. 
\\ \\
Ex : Bi/Cd, Benzène/Naphtalène.
\subsubsection{Équation du liquidus}
$$ 2(s) \rightleftarrows 2(l) \hspace{2cm} K^0=\displaystyle\frac{a_2}{1}$$
$$ \displaystyle\frac{\mathrm d \ln{a_2}}{\mathrm dT}=\displaystyle\frac{\Delta_{fus}H^0_2}{RT^2}$$
$$ \text{Approximation \ d'\bsc{Ellingham} \ :} \int_1^{a_2}\mathrm d \ln{(a_2)}=\displaystyle\frac{\Delta_{fus}H^0_2}{R} \int_{T_{fus,2}}^T \displaystyle\frac{\mathrm dT}{T^2}$$
$$ \ln{(a_2)}=\displaystyle\frac{\Delta_{fus}H^0_2}{R} \left( \displaystyle\frac{1}{T_{fus,2}}-\displaystyle\frac{1}{T} \right)$$
\noindent Pour chaque valeur de T, le calcul fournit $a_2$. Le graphe fournit $x_2$, on en déduit $\gamma_2$ : $$ a_2=\gamma_2 x_2$$

\subsubsection{Cryométrie}
\noindent Première expérience : on repère $T_fus,S$, S le solvant.
\\
\noindent Seconde expérience : dans S(l) on ajoute un peu, très peu de soluté A. On laisse refroidir, on repère la T d'apparition du premier microcristal de S.
$$ T<T_{fus,S} \hspace{1cm} \Delta T=T_{fus}-T \ \longrightarrow \ \text{Abaissement \ cryoscopique}$$
Dans le liquide : S majoritaire, $a_S \simeq x_S=1-x_A$
$$ -x_A=\displaystyle\frac{\Delta_{fus}H^0_S}{R} \left( \displaystyle\frac{T-T_{fus}}{T_{fus}T} \right)\hspace{1cm} x_A \approx \displaystyle\frac{\Delta_{fus}H^0_S}{R} \displaystyle\frac{\Delta T}{T_{fus}^2}=\displaystyle\frac{m_A}{n_S}\displaystyle\frac{1}{M_A} $$
On détermine ainsi $M_A$.

\subsection{Existence de Composés Définis}
Lors d'une réaction entre 1 et 2, il peut se former un corps \textbf{PUR}, noté D, appelé composé \textbf{défini}.
$$ p1(s)+q2(s)\rightleftarrows \underbrace{1_p2_q}_{D}$$
\\
\textbf{Exemple} : ($cf$ CCP Chimie 1 2012)
\\
\noindent Binaire Al/Mg, composé défini : $\text{Al}_x\text{Mg}_y$, où $x$ et $y$ entiers, les plus petits possibles.
$$ D_1=\text{Al}_{x_1}\text{Mg} _{y_1} \hspace{1cm} \frac{x_1}{y_1}=\frac{0,4}{0,6}=\frac{2}{3} \ \longleftarrow \ \text{Al}_2\text{Mg}_3$$
$$ D_2=\text{Al}_{x_2}\text{Mg} _{y_2} \hspace{1cm} \frac{x_2}{y_2}=\frac{0,6}{0,4}=\frac{3}{2} \ \longleftarrow \ \text{Al}_3\text{Mg}_2$$
\textbf{Exemple} : Acide formique - Formamide. Où x est la fraction molaire en formamide.
\begin{center}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.2cm,y=1.2cm]
\clip(0.8,-3.24) rectangle (11.2,5.38);
\draw [->] (2,-3) -- (2,5);
\draw [->] (1,-2) -- (11,-2);
\draw [->] (10,-3) -- (10,5);
\draw (2,-1)-- (10,-1);
\draw (5.7,-1)-- (5.7,3);
\draw (2,1)-- (5.7,1);
\draw (5.7,0.42)-- (10,0.42);
\draw [shift={(-1.37,0.25)}] plot[domain=0.15:0.86,variable=\t]({1*5.16*cos(\t r)+0*5.16*sin(\t r)},{0*5.16*cos(\t r)+1*5.16*sin(\t r)});
\draw [shift={(6.26,0.49)}] plot[domain=1.79:2.94,variable=\t]({1*2.57*cos(\t r)+0*2.57*sin(\t r)},{0*2.57*cos(\t r)+1*2.57*sin(\t r)});
\draw [shift={(3.73,-0.68)}] plot[domain=0.27:1.08,variable=\t]({1*4.17*cos(\t r)+0*4.17*sin(\t r)},{0*4.17*cos(\t r)+1*4.17*sin(\t r)});
\draw [shift={(11.57,-0.21)}] plot[domain=1.99:2.98,variable=\t]({1*3.86*cos(\t r)+0*3.86*sin(\t r)},{0*3.86*cos(\t r)+1*3.86*sin(\t r)});
\draw (6.06,4.18) node[anchor=north west] {$L$};
\draw (2.4,2.92) node[anchor=north west] {$L$};
\draw (5.06,2.72) node[anchor=north west] {$L$};
\draw (6.1,2.42) node[anchor=north west] {$L$};
\draw (9.3,2.54) node[anchor=north west] {$L$};
\draw (2.68,2.4) node[anchor=north west] {$+$};
\draw (4.84,2.24) node[anchor=north west] {$+$};
\draw (6.35,1.86) node[anchor=north west] {$+$};
\draw (8.98,1.98) node[anchor=north west] {$+$};
\draw (2,1.86) node[anchor=north west] {$\text{Acide (s)}$};
\draw (8.2,1.3) node[anchor=north west] {$\text{Amide (s)}$};
\draw (2.8,0.26) node[anchor=north west] {$\text{Acide (s)}+D$};
\draw (6.8,-0.08) node[anchor=north west] {$\text{Amide (s)}+D$};
\draw (4.58,1.8) node[anchor=north west] {$D$};
\draw (6.65,1.38) node[anchor=north west] {$D$};
\draw (10.62,-2) node[anchor=north west] {$x$};
\draw (2.2,-2) node[anchor=north west] {$0$};
\draw (9.5,-2) node[anchor=north west] {$1$};
\draw (1.6,5) node[anchor=north west] {$\theta$};
\draw (10.1,5) node[anchor=north west] {$\theta$};
\draw (3.45,1) node[anchor=north west] {$E_1$};
\draw (7.5,0.45) node[anchor=north west] {$E_2$};
\begin{scriptsize}
\draw [color=black] (3.74,1)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (7.76,0.42)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}
\begin{center}
\definecolor{ffqqqq}{rgb}{1,0,0}
\definecolor{qqqqcc}{rgb}{0,0,0.8}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.92cm,y=0.92cm]
\clip(1.2,-10.65) rectangle (20.06,10.23);
\draw [->] (2,-10) -- (2,10);
\draw [->] (18,-10) -- (18,10);
\draw [->] (2,-10) -- (20,-10);
\draw (1.45,8.2) node[anchor=north west] {$0$};
\draw (0.78,-3.8) node[anchor=north west] {$-120$};
\draw (0.92,2.2) node[anchor=north west] {$-60$};
\draw (2,-1.03)-- (7.1,-1.03);
\draw (7.1,-1.03)-- (7.1,-10);
\draw (7.1,-1.03)-- (7.1,0.21);
\draw [shift={(1.3,-5.29)}] plot[domain=0.84:1.45,variable=\t]({1*5.71*cos(\t r)+0*5.71*sin(\t r)},{0*5.71*cos(\t r)+1*5.71*sin(\t r)});
\draw [shift={(7.12,-2.06)}] plot[domain=1.58:2.67,variable=\t]({1*2.27*cos(\t r)+0*2.27*sin(\t r)},{0*2.27*cos(\t r)+1*2.27*sin(\t r)});
\draw (7.1,-0.29)-- (10.53,-0.29);
\draw (10.53,-0.29)-- (10.52,0.93);
\draw [shift={(7.05,-4.52)}] plot[domain=1.11:1.56,variable=\t]({1*4.73*cos(\t r)+0*4.73*sin(\t r)},{0*4.73*cos(\t r)+1*4.73*sin(\t r)});
\draw [shift={(11.82,-1.86)}] plot[domain=2.01:2.61,variable=\t]({1*3.08*cos(\t r)+0*3.08*sin(\t r)},{0*3.08*cos(\t r)+1*3.08*sin(\t r)});
\draw [dotted] (2,8)-- (18,8);
\draw [shift={(6.96,-4.79)}] plot[domain=0:1.015,variable=\t]({1*6.74*cos(\t r)+0*6.74*sin(\t r)},{0*6.74*cos(\t r)+1*6.74*sin(\t r)});
\draw [shift={(34.83,-4.75)}] plot[domain=2.493:3.14,variable=\t]({1*21.13*cos(\t r)+0*21.13*sin(\t r)},{0*21.13*cos(\t r)+1*21.13*sin(\t r)});
\draw (6.6,5.33) node[anchor=north west] {$L$};
\draw (6.06,0.13) node[anchor=north west] {$L/S$};
\draw (11.17,-1.01) node[anchor=north west] {$L/S$};
\draw (2.27,0.2) node[anchor=north west] {$L$};
\draw (16.48,2.86) node[anchor=north west] {$L$};
\draw [color=qqqqcc](8.33,-4.92) node[anchor=north west] {$H_2(s)$};
\draw [color=ffqqqq](13.68,-7) node[anchor=north west] {$H_1(s)$};
\draw (3.67,-3.56) node[anchor=north west] {$NH_3(s)$};
\draw (13.43,-8) node[anchor=north west] {$H_2O(s)$};
\draw (15.45,-0.31) node[anchor=north west] {$H_2O(s)$};
\draw (10.01,4.67) node[anchor=north west] {$1 \varphi$};
\draw (4.04,-2.25) node[anchor=north west] {$2 \varphi$};
\draw (14.05,-6) node[anchor=north west] {$2 \varphi$};
\draw (8.53,-1.75) node[anchor=north west] {$2 \varphi$};
\draw (2.85,-0.3) node[anchor=north west] {$NH_{3(s)}$};
\draw (7.09,0.25) node[anchor=north west] {$L/S$};
\draw (9.55,0.68) node[anchor=north west] {$L/S$};
\draw (4.91,-1.09) node[anchor=north west] {$E_3$};
\draw (8.9,-0.31) node[anchor=north west] {$E_2$};
\draw (13.7,-4.2) node[anchor=north west] {$E_1$};
\draw (18.5,-9.5) node[anchor=north west] {$x_{eau}$};
\draw (2.4,10.06) node[anchor=north west] {$T(° C)$};
\draw [color=ffqqqq](11.46,-2.86) node[anchor=north west] {$H_1(s)$};
\draw [color=ffqqqq](9.3,0.3) node[anchor=north west] {$H_1(s)$};
\draw [color=ffqqqq](8.45,-3.48) node[anchor=north west] {$H_1(s)$};
\draw [color=qqqqcc](7.88,0.138) node[anchor=north west] {{\scriptsize $H_{2_{(s)}}$}};
\draw [color=qqqqcc](5.49,-0.45) node[anchor=north west] {$H_2(s)$};
\draw [color=qqqqcc](3.72,-5.17) node[anchor=north west] {$H_2(s)$};
\draw (10.53,-0.29)-- (10.53,-10);
\draw (13.7,-4.8)-- (10.53,-4.8);
\draw (13.7,-4.8)-- (18,-4.8);
\begin{scriptsize}
\draw [color=black] (2,8)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (2,-4)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (2,2)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (5.1,-1.03)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (9.17,-0.29)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (13.7,-4.8)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\end{scriptsize}
\end{tikzpicture}
Système binaire liquide-solide : $H_2O$/$NH_3$. $P=1 \ \text{atm}$. 
\end{center}

\subsection{Miscibilité Partielle}
\begin{center}
\definecolor{ffqqqq}{rgb}{1,0,0}
\definecolor{qqqqcc}{rgb}{0,0,0.8}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.9cm,y=0.9cm]
\clip(0.75,-10.89) rectangle (19.44,10.34);
\draw [->] (2,-10) -- (2,10);
\draw [->] (18,-10) -- (18,10);
\draw [->] (2,-10) -- (19,-10);
\draw (0.9,-7.75) node[anchor=north west] {$400$};
\draw (18.62,-9.5) node[anchor=north west] {$w$};
\draw (2.42,10.06) node[anchor=north west] {$T(° C)$};
\draw (0.74,-3.75) node[anchor=north west] {$600$};
\draw (0.78,0.25) node[anchor=north west] {$800$};
\draw (0.53,4.25) node[anchor=north west] {$1000$};
\draw (0.58,8.25) node[anchor=north west] {$1200$};
\draw (4,-0.77)-- (14.97,-0.77);
\draw [shift={(29.01,-12.56)}] plot[domain=2.063:2.61,variable=\t]({1*23.32*cos(\t r)+0*23.32*sin(\t r)},{0*23.32*cos(\t r)+1*23.32*sin(\t r)});
\draw [shift={(33.63,-2.32)}] plot[domain=2.558:3.06,variable=\t]({1*18.73*cos(\t r)+0*18.73*sin(\t r)},{0*18.73*cos(\t r)+1*18.73*sin(\t r)});
\draw [shift={(44.86,-10.73)}] plot[domain=2.902:3.124,variable=\t]({1*42.06*cos(\t r)+0*42.06*sin(\t r)},{0*42.06*cos(\t r)+1*42.06*sin(\t r)});
\draw [shift={(-4.89,-10.03)}] plot[domain=0.0015:0.437,variable=\t]({1*21.91*cos(\t r)+0*21.91*sin(\t r)},{0*21.91*cos(\t r)+1*21.91*sin(\t r)});
\draw (2.13,-10) node[anchor=north west] {$0$};
\draw (17.56,-10) node[anchor=north west] {$1$};
\draw [shift={(-4.67,-2.63)}] plot[domain=0.21:0.72,variable=\t]({1*8.87*cos(\t r)+0*8.87*sin(\t r)},{0*8.87*cos(\t r)+1*8.87*sin(\t r)});
\draw [shift={(-2.54,-12.58)}] plot[domain=0.8:1.29,variable=\t]({1*16.44*cos(\t r)+0*16.44*sin(\t r)},{0*16.44*cos(\t r)+1*16.44*sin(\t r)});
\draw (8.6,-0.7) node[anchor=north west] {$E$};
\draw (5.82,-3.65) node[anchor=north west] {$\text{2 phases solides non miscibles}$};
\draw (6.47,-5.28) node[anchor=north west] {$\text{(Cu+Ag) et (Cu+Ag)}$};
\draw (8.11,6.38) node[anchor=north west] {$L$};
\draw [color=qqqqcc](15.88,0.73) node[anchor=north west] {$\text{Ag}$};
\draw (15.92,-0.09) node[anchor=north west] {$\text{dans}$};
\draw [color=ffqqqq](16.04,-0.95) node[anchor=north west] {$\text{Cu}$};
\draw (2.22,-1.36) node[anchor=north west] {$\text{dans}$};
\draw [color=ffqqqq] (2.34,-0.54) node[anchor=north west] {$\text{Cu}$};
\draw [color=qqqqcc](2.38,-2.38) node[anchor=north west] {$\text{Ag}$};
\draw (3.24,2.2) node[anchor=north west] {$L$};
\draw (14.61,5.48) node[anchor=north west] {$L$};
\draw (13.71,4.37) node[anchor=north west] {$+$};
\draw (4.14,1.5) node[anchor=north west] {$+$};
\draw (2.58,0.49) node[anchor=north west] {$S$};
\draw (16.25,1.67) node[anchor=north west] {$S$};
\draw [color=qqqqcc](11.63,2.9) node[anchor=north west] {$\text{Ag}$};
\draw (11.99,2.04) node[anchor=north west] {$\text{dans}$};
\draw [color=ffqqqq] (12.93,1.26) node[anchor=north west] {$\text{Cu}$};
\draw [color=ffqqqq] (5.12,1) node[anchor=north west] {$\text{Cu}$};
\draw (4.79,0.44) node[anchor=north west] {$\text{dans}$};
\draw [color=qqqqcc](6,0.44) node[anchor=north west] {$\text{Ag}$};
\draw (9.25,-2.05) node[anchor=north west] {$S$};
\begin{scriptsize}
\draw [color=black] (2,-8)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (2,-4)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (2,0)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (2,4)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (2,8)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (8.89,-0.77)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\end{scriptsize}
\end{tikzpicture}
Système binaire liquide-solide. Composants formant des solutions solides. Ici exemple de Cu/Ag, à $P=1 \ \text{atm}$. $w$ : fraction massique en Cuivre.
\end{center}
\vspace{0.5cm}
\noindent Solution (par analogie aux solutions liquides) de \textcolor{red}{Cu} dans \textcolor{blue}{Ag} : solide où \textcolor{blue}{Ag} est majoritaire, \textcolor{red}{Cu} s'est inséré dans le réseau.

\section{Binaires Liquide/Liquide (Systèmes Fermés)}
\textbf{Exemple} : méthanol/n-heptane. La miscibilité des 2 composants est incomplète.
\begin{center}
\definecolor{ffqqqq}{rgb}{1,0,0}
\definecolor{qqqqff}{rgb}{0,0,1}
\definecolor{cqcqcq}{rgb}{0.75,0.75,0.75}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-0.24,-0.6) rectangle (10.6,9.18);
\draw [->] (0,0) -- (0,9);
\draw [->] (0,0) -- (10.54,0);
\draw [->] (10,0) -- (10,9);
\draw [shift={(10.14,0)},color=qqqqff]  plot[domain=2.525:3.141,variable=\t]({1*9.14*cos(\t r)+0*9.14*sin(\t r)},{0*9.14*cos(\t r)+1*9.14*sin(\t r)});
\draw [shift={(5.68,2.96)},color=qqqqff]  plot[domain=1.77:2.48,variable=\t]({1*3.79*cos(\t r)+0*3.79*sin(\t r)},{0*3.79*cos(\t r)+1*3.79*sin(\t r)});
\draw [shift={(5.41,4.28)},color=qqqqff]  plot[domain=0.626:1.77,variable=\t]({1*2.45*cos(\t r)+0*2.45*sin(\t r)},{0*2.45*cos(\t r)+1*2.45*sin(\t r)});
\draw [shift={(-1.51,0.13)},color=qqqqff]  plot[domain=-0.012:0.56,variable=\t]({1*10.51*cos(\t r)+0*10.51*sin(\t r)},{0*10.51*cos(\t r)+1*10.51*sin(\t r)});
\draw [dash pattern=on 2pt off 2pt,color=ffqqqq] (1.94,4.04)-- (8.25,4.04);
\draw [dotted] (4.52,4.04)-- (4.52,0);
\draw [dotted] (1.94,4.04)-- (1.94,0);
\draw [dotted] (8.25,4.04)-- (8.25,0);
\draw (0.06,9.18) node[anchor=north west] {$T(° C)$};
\draw (10.24,0.5) node[anchor=north west] {$x$};
\draw (5.36,7.4) node[anchor=north west] {$C$};
\draw (4.3,0.02) node[anchor=north west] {$x_G$};
\draw (1.7,0) node[anchor=north west] {$x_1$};
\draw (8,0.02) node[anchor=north west] {$x_2$};
\draw (-0.2,0.02) node[anchor=north west] {$0$};
\draw (9.8,0.02) node[anchor=north west] {$1$};
\draw (4.44,4.78) node[anchor=north west] {$I$};
\draw (2,8) node[anchor=north west] {$\text{1 phase liquide}\longrightarrow \text{2 constituants}$};
\draw (2.8,3) node[anchor=north west] {$\text{2 phases liq non miscibles}$};
\begin{scriptsize}
\draw [color=black] (5.46,6.73)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (1.94,4.04)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (8.25,4.04)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (4.52,4.04)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
\draw [color=black] (4.52,0)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (1.94,0)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\draw [color=black] (8.25,0)-- ++(-1.5pt,0 pt) -- ++(3.0pt,0 pt) ++(-1.5pt,-1.5pt) -- ++(0 pt,3.0pt);
\end{scriptsize}
\end{tikzpicture}

$x$ : fraction molaire en méthanol. $P=1\,\text{atm}$

\textcolor{blue}{Courbe d'Équilibre}
\end{center}

\noindent Si l'on part de I, avec pour le méthanol, une fraction molaire globale $x_G$, on aura deux phases liquides, une de fraction $x_1$ en méthanol, l'autre de fraction $x_2$ en méthanol.