$($a$)$ Starting from the entropy increase principle and assuming that all processes

in the system are reversible, find the coefficient eta$\_($R$)$ as a function of T$\_(1)$ and T$\_(2).$

$($b$)$ What relation is satisfied by the coefficient of performance eta$\_($R$)$ of a similar

device, but in the presence of irreversibilities? The device works in a cycle and

uses the same two thermostats. Justify your answer.