Entropy generated by cooling a finite body in a reservoir ($\Delta S_{\mathrm{tot}}=C[\ln (T_0/T_i)+T_i/T_0-1]$)

For a body of constant heat capacity $C$ cooling from $T_i$ to a thermal reservoir at $T_0>0$, the body's entropy change is $C\ln (T_0/T_i)$, while the reservoir gains $C(T_i-T_0)/T_0$. Their sum is nonnegative because $r-1-\ln r\ge 0$ for $r=T_i/T_0>0$. The logarithmic body contribution is computed along a hypothetical reversible path; the actual finite-temperature-difference heat transfer is irreversible.