
import numpy as np 
import sympy as sp
from sympy.interactive import printing 
printing.init_printing(use_latex = True)

Definition of symbolic variables


g_fo_o,g_fa_o,y_fo,y_fa,T,R,W = sp.symbols('g_fo^o,g_fa^o,y_fo,y_fa,T,R,W')

Definition of gibbs energy of the olivine solid solution


g = y_fo*g_fo_o + (1-y_fo)*g_fa_o + R*T*(2*y_fo*sp.log(y_fo) + 2*(1-y_fo)*sp.log(1-y_fo)) + W*y_fo*(1-y_fo)


display(g)

\begin{matrix} (1) & R T (2 y_{f o} \log (y_{f o}) + (2 - 2 y_{f o}) \log (1 - y_{f o})) + W y_{f o} (1 - y_{f o}) + g_{f a}^{o} (1 - y_{f o}) + g_{f o}^{o} y_{f o} \end{matrix}

Definition of the gibss energy of forsterite in the olivine solid solution


g_fo_ss = g + (1-y_fo)*sp.diff(g,y_fo)
display(sp.simplify(g_fo_ss))

\begin{matrix} (2) & 2 R T \log (y_{f o}) + W y_{f o}^{2} - 2 W y_{f o} + W + g_{f o}^{o} \end{matrix}

Definition of activity

$g_{fo}$ $g_{fo}^o$

\begin{matrix} (3) & a_{f o} = e x p (\frac{g_{f o} - g_{f o}^{o}}{R T}) \end{matrix}


a_fo = sp.exp((g_fo_ss-g_fo_o)/R/T)
sp.simplify(a_fo)

\begin{matrix} (4) & y_{f o}^{2} e^{\frac{W (y_{f o}^{2} - 2 y_{f o} + 1)}{R T}} \end{matrix}


print(sp.simplify(a_fo))


y_fo**2*exp(W*(y_fo**2 - 2*y_fo + 1)/(R*T))

$RT\ln a_{fo}$


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RTlna_fo = R*T*sp.log(sp.exp((g_fo_ss-g_fo_o)/R/T))


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sp.simplify(RTlna_fo)

\begin{matrix} (5) & R T \log (y_{f o}^{2} e^{\frac{W y_{f o}^{2} - 2 W y_{f o} + W + g_{f o} - g_{f o}^{o}}{R T}}) \end{matrix}

Just note that


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sp.simplify(g_fo_ss-g_fo_o)

\begin{matrix} (6) & 2 R T \log (y_{f o}) + W y_{f o}^{2} - 2 W y_{f o} + W \end{matrix}


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sp.collect(W*y_fo**2-2*W*y_fo+W,W)

\begin{matrix} (7) & W (y_{f o}^{2} - 2 y_{f o} + 1) \end{matrix}

and that


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sp.factor(W*y_fo**2-2*W*y_fo+W,y_fo)

\begin{matrix} (8) & W {(y_{f o} - 1)}^{2} \end{matrix}

so ,

\begin{matrix} (9) & R T \ln (a_{f o}) = 2 R T \ln (y_{f o}) + W {(y_{f o} - 1)}^{2} \end{matrix}