Entropy generation in finite-temperature heat transfer ($d{S}_{\mathrm{tot}}=dQ(1/T-1/T_R)$)

When a system at instantaneous temperature $T$ absorbs heat $dQ$ from a reservoir at $T_R$, with no other entropy production in the system, its entropy increases by $dQ/T$ while the reservoir changes by $-dQ/T_R$. The total is positive for spontaneous heat transfer across a finite temperature difference. A quasistatic mechanical path can therefore be thermally irreversible.

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.