Entropy generation by mixing equal thermal masses ($\Delta S=2M{c}_P\ln \frac{T_1+T_2}{2\sqrt{T_1{T}_2}}$)

For equal masses $M$ of the same liquid with constant specific heat $c_P$, adiabatic constant-pressure mixing gives final temperature $T_f=(T_1+T_2)/2$ by enthalpy conservation. The entropy change is $M{c}_P\ln (T_f/T_1)+M{c}_P\ln (T_f/T_2)=2M{c}_P\ln [(T_1+T_2)/(2\sqrt{T_1{T}_2})]$. The arithmetic-geometric mean inequality proves nonnegativity, with equality only for equal initial temperatures. There is no separate compositional mixing entropy for identical fluids.