## Essential University Physics: Volume 1 (3rd Edition)

As $mc(T_h-T_f)=mc(T_f-T_c)$ $\implies T_f=\frac{T_h+T_c}{2}$ Now $\Delta S_{hot}=mc\ln(\frac{T_f}{T_h})=mc\ln(\frac{\frac{T_h+T_c}{2}}{T_h})$ and Now $\Delta S_{cold}=mc\ln(\frac{T_f}{T_c})=mc\ln(\frac{\frac{T_h+T_c}{2}}{T_c})$ $\Delta S_{system}=\Delta S_{hot}+\Delta S_{cold}$ $\implies \Delta S_{system}=mc\ln(\frac{\frac{T_h+T_c}{2}}{T_h})+mc\ln(\frac{\frac{T_h+T_c}{2}}{T_c})$ $\implies \Delta S_{system}=mc\ln(\frac{1+(T_h-T_c)^2}{4T_hT_c})$ Thus, the entropy change of the system is positive because x is greater than 1.