# Does this system violate the second law of thermodynamics?

Consider a system (Blundell and Blundell 2nd edition, page 135) where a cycle consists of connecting each point to a reservoir at temperature $$T_i$$ and a heat amount $$\delta Q_i$$ enters that point. A whole cycle results in a work output $$\Delta W$$, which, since over a cycle $$\Delta U = 0$$, by the First Law, is given by: $$\Delta W = \sum_{cycle} \delta Q_i$$ However, Kelvin's statement of the Second law states that no process can have the full conversion of heat into work as a unique result. How is this not happening here? There is no colder reservoir so it looks to me like all the individual heat ammounts are fully converted into work.

The index $$i$$ indicates that multiple reservoirs at temperature $$T_i$$ are being used, each delivering infinitesimal heat $$\text{đ}Q_i$$. We conclude that some values of $$\text{đ}Q_i$$ must be negative to avoid requiring entropy to disappear, which—as you note—is forbidden by the Second Law.