Consider a closed system undergoing an irreversible process, where heat Q is transferred to the system from a thermal reservoir at constant temperature $T_{\text{r}\text{e}\text{s}\text{e}\text{r}\text{v}\text{o}\text{i}\text{r}},$ and the system itself undergoes a change in state from state $1$ to state $2.$

Derive an expression for the change in entropy of the system $S_{\text{s}\text{y}\text{s}\text{t}\text{e}\text{m}}$ when the process occurs at a constant temperature. Assume the heat transferred is $Q_{in}.$

Using the second law of thermodynamics, explain why the total entropy of the system and the surroundings combined must increase during this irreversible heat transfer process. How would the situation differ if the process were reversible?

In a different scenario, consider a heat engine operating between two thermal reservoirs at temperatures $T_{\text{h}\text{i}\text{g}\text{h}}$ and $T_{\text{l}\text{o}\text{w}}.$ If the engine absorbs a heat $Q_{\text{h}\text{i}\text{g}\text{h}}$ from the high$-$temperature reservoir, derive the expression for the change in entropy of the system $($heat engine$)$ and the two reservoirs. What is the condition for maximum efficiency in terms of entropy change for a reversible cycle?