Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A1 / b / Solution

For the symmetric regular solution model, the dimensionless mixing free energy is $f/(k_{B}T)=\varphi \log \varphi +(1-\varphi )\log (1-\varphi )+\chi \varphi (1-\varphi )$. At the thermodynamic critical point, its second and third derivatives in composition vanish. The third derivative gives ${\varphi }_c=1/2$; the second then gives $4-2{\chi }_c=0$. ThereforeThis uses the equal-size, composition-symmetric mixture model.