Editing Eutectic system (section)

==Eutectic calculation==
The composition and temperature of a eutectic can be calculated from enthalpy and entropy of fusion of each components.<ref>{{cite journal |last1=Brunet |first1=Luc E. |last2=Caillard |first2=Jean |last3=André |first3=Pascal |title=Thermodynamic Calculation of ''n''-component Eutectic Mixtures |journal=International Journal of Modern Physics C |date=June 2004 |volume=15 |issue=5 |pages=675–687 |doi=10.1142/S0129183104006121 |publisher=World Scientific|bibcode=2004IJMPC..15..675B }}</ref>

The Gibbs free energy ''G'' depends on its own differential:
: <math>
G = H - TS
\Rightarrow 
\begin{cases}
 H = G + TS \\
 \left(\frac{\partial G}{\partial T}\right)_P = -S
\end{cases}
\Rightarrow
H = G - T \left(\frac{\partial G}{\partial T}\right)_P.
</math>

Thus, the ''G''/''T'' derivative at constant pressure is calculated by the following equation:
: <math>
    \left(\frac{\partial G / T}{\partial T}\right)_P
    =
    \frac{1}{T} \left(\frac{\partial G}{\partial T}\right)_P - \frac{1}{T^2}G
    =
    -\frac{1}{T^2} \left(G - T\left(\frac{\partial G}{\partial T}\right)_P\right)
    = -\frac{H}{T^2}.
</math>

The chemical potential <math>\mu_i</math> is calculated if we assume that the activity is equal to the concentration:
: <math>
\mu_i = \mu_i^\circ + RT\ln \frac{a_i}{a} \approx \mu_i^\circ + RT\ln x_i.
</math>

At the equilibrium, <math>\mu_i = 0</math>, thus <math>\mu_i^\circ</math> is obtained as
: <math>
\mu _i = \mu _i^\circ + RT\ln x_i = 0 \Rightarrow \mu_i^\circ = -RT\ln x_i.
</math>

Using{{clarify|date=January 2019}} and integrating gives
: <math>
 \left(\frac{\partial \mu_i / T}{\partial T}\right)_P =
 \frac{\partial}{\partial T}\left(R\ln x_i\right) \Rightarrow
 R\ln x_i =
 -\frac{H_i^\circ}{T} + K.
</math>

The integration constant ''K'' may be determined for a pure component with a melting temperature <math>T^\circ</math> and an enthalpy of fusion <math>H^\circ</math>:
: <math>
x_i = 1 \Rightarrow T = T_i^\circ \Rightarrow K = \frac{H_i^\circ}{T_i^\circ}.
</math>

We obtain a relation that determines the molar fraction as a function of the temperature for each component:
: <math>
R\ln x_i = -\frac{H_i^\circ}{T} + \frac{H_i^\circ}{T_i^\circ}.
</math>

The mixture of ''n'' components is described by the system
: <math>
\begin{cases}
 \ln x_i + \frac{H_i^\circ}{RT} - \frac{H_i^\circ}{RT_i^\circ } = 0, \\
 \sum\limits_{i = 1}^n x_i = 1.
\end{cases}
</math>
: <math>
\begin{cases}
 \forall i < n \Rightarrow \ln x_i + \frac{H_i^\circ}{RT} - \frac{H_i^\circ}{RT_i^\circ} = 0, \\
 \ln \left(1 - \sum\limits_{i = 1}^{n - 1} x_i\right) + \frac{H_n^\circ}{RT} - \frac{H_n^\circ}{RT_n^\circ} = 0, 
\end{cases}
</math>

which can be solved by
: <math>
\begin{array}{c}
\left[ {{\begin{array}{*{20}c}
 {\Delta x_1 } \\
 {\Delta x_2 } \\
 {\Delta x_3 } \\
 \vdots \\
 {\Delta x_{n - 1} } \\
 {\Delta T} \\
\end{array} }} \right] = \left[ {{\begin{array}{*{20}c}
 {1 / x_1 } & 0 & 0 & 0 & 0 & { - \frac{H_1^\circ }{RT^{2}}} \\
 0 & {1 / x_2 } & 0 & 0 & 0 & { - \frac{H_2^\circ }{RT^{2}}} \\
 0 & 0 & {1 / x_3 } & 0 & 0 & { - \frac{H_3^\circ }{RT^{2}}} \\
 \vdots & \ddots & \ddots & \ddots & \ddots & { \vdots} \\
 0 & 0 & 0 & 0 & {1 / x_{n - 1} } & { - \frac{H_{n - 1}^\circ }{RT^{2}}}
\\
 {\frac{ - 1}{1 - \sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{i = 1}^{n - 1} {x_i } }} & { -
\frac{H_n^\circ }{RT^{2}}} \\
\end{array} }} \right]^{ - 1}

.\left[ {{\begin{array}{*{20}c}
 {\ln x_1 + \frac{H_1 ^\circ }{RT} - \frac{H_1^\circ }{RT_1^\circ }}
\\
 {\ln x_2 + \frac{H_2 ^\circ }{RT} - \frac{H_2^\circ }{RT_2^\circ }}
\\
 {\ln x_3 + \frac{H_3 ^\circ }{RT} - \frac{H_3^\circ }{RT_3^\circ }}
\\
 \vdots \\
 {\ln x_{n - 1} + \frac{H_{n - 1} ^\circ }{RT} - \frac{H_{n - 1}^\circ
}{RT_{n - 1}^\circ }} \\
 {\ln \left({1 - \sum\limits_{i = 1}^{n - 1} {x_i } } \right) + \frac{H_n
^\circ }{RT} - \frac{H_n^\circ }{RT_n^\circ }} \\
\end{array} }} \right]
 \end{array}
</math>